A deep neural network for operator learning enhanced by attention and gating mechanisms for long-time forecasting of tumor growth

Abstract

Forecasting tumor progression and assessing the uncertainty of predictions play a crucial role in clinical settings, especially for determining disease outlook and making informed decisions about treatment approaches. In this work, we propose TGM-ONets, a deep neural operator learning (PI-DeepONet) based computational framework, which combines bioimaging and tumor growth modeling (TGM) for enhanced prediction of tumor growth. Deep neural operators have recently emerged as a powerful tool for learning the solution maps between the function spaces, and they have demonstrated their generalization capability in making predictions based on unseen input instances once trained. Incorporating the physics laws into the loss function of the deep neural operator can significantly reduce the amount of the training data. The novelties of the design of TGM-ONets include the employment of a convolutional block attention module (CBAM) and a gating mechanism (i.e., mixture of experts (MoE)) to extract the features of the input images. Our results show that the TGM-ONets not only can capture the detailed morphological characteristics of the mild and aggressive tumors within and outside the training domain but also can be used to predict the long-term dynamics of both mild and aggressive tumor growth for up to 6 months with a maximum error of less than 6.7 $\times 10^{-2}$ for unseen input instances with two or three snapshots added. We also systematically study the effects of the number of training snapshots and noisy data on the performance of TGM-ONets as well as quantify the uncertainty of the model predictions. We demonstrate the efficiency and accuracy by comparing the performance of TGM-ONets with three state-of-the-art (SOTA) baseline models. In summary, we propose a new deep learning model capable of integrating the TGM and sequential observations of tumor morphology to improve the current approaches for predicting tumor growth and thus provide an advanced computational tool for patient-specific tumor prognosis.

Appendices

Appendices

A Convergence of spectral/hp element (Nektar) results for TGMs

We give a brief introduction to the spectral/hp element method which we use to solve the PDEs for tumor growth and generate the synthetic data. More details about the spectral/hp element method can be found in [110]. We first define the weak form of the PDE and impose the boundary conditions. Then we discretize the computational domain into subdomains. Below we use the one-dimensional Poisson equation in the interval $0< x \le 1$ for illustration. $\bigtriangleup u + f$ = 0, where $u(x = 0)$ = $g_D$ = 1 and $\frac{\partial u(x = 1)}{\partial x}$ = $g_N$ = 1.

• 1. We get the weak form by multiplying the problem by a discrete test space and integrating the second-order derivative by parts: $\int _{0}^{1}\frac{\partial v^{\delta }}{\partial x}\frac{\partial u^{\delta }}{\partial x} = \int v^{\delta }f dx + v^{\delta }(1)g_{N}.$

• 2. We lift a known solution from the problem by decomposing into a known solution satisfying the Dirichlet boundary conditions and a homogeneous solution such that $u^{\delta } = u^{D}+u^{H}$, and the weak solution becomes $\int _{0}^{1}\frac{\partial v^{\delta }}{\partial x}\frac{\partial u^{H}}{\partial x} = \int v^{\delta }f dx + v^{\delta }(1)g_{N}-\int _{0}^{1}\int _{0}^{1}\frac{\partial v^{\delta }}{\partial x}\frac{\partial u^{D}}{\partial x}.$

We use piecewise linear functions as basis functions and decompose the domain into two subdomains. We can use finer mesh which can give h-convergence and higher-order polynomials as basis functions which can give p-type convergence. For the linear two-subdomain case the approximate expansion has the form $u^{\delta } = \sum _{i = 0}^{2} \hat{u_{i}}\Phi _i(x)$, where $\Phi _i(x)$ are the piecewise linear functions. Then we represent f in terms of basis functions $f(x) = \sum _{i = 0}^2\hat{f_i}\Phi _i(x)$. Finally, we can solve the linear system of equations to get the numerical solution for u, which is also a finite element approximation for this example.

We solve the Eqs. 1–4 in Sect. 2.1 for tumor growth using a spectral/hp element Nektar solver with $\Delta t$ = $1.0\,\times \,10^{-2}$, $1.0\,\times \,10^{-3}$, and $1.0\,\times \,10^{-4}$. The characteristic length and time are 1 mm and 1 day. We found that at $\Delta t \le 1.0\,\times \,10^{-3}$, the differences in the results between $\Delta t$s are marginal. For aggressive tumors, the maximum difference (i.e., the maximum pointwise absolute difference between $\phi$ from two simulation runs) between $\Delta t = 1.0\,\times \,10^{-2}$ and $\Delta t = 1.0\,\times \,10^{-5}$ is $2.08\,\times \,10^{-2}$, $\Delta t = 1.0\,\times \,10^{-3}$ and $\Delta t = 1.0\,\times \,10^{-5}$ is $2.99\,\times \,10^{-3}$, and $\Delta t = 1.0\,\times \,10^{-4}$ and $\Delta t = 1.0\,\times \,10^{-5}$ is $1.32\,\times \,10^{-3}$. For all the numerical simulations conducted by Nektar, we use polynomial order = 3, time step size $\Delta t$ = $1.0\,\times \,10^{-3}$, mesh size $6.67\,\times \,10^{-3}$ in both x- and y- directions which resulted in 22,500 quadrilateral elements, and run the solver with 256 CPU nodes in parallel. Table 43 shows the parameters used in the simulations. It takes about 1.1 h to run one mild tumor case up to 80 days and 3.9 h to run one aggressive tumor case up to 200 days.

Table 43 Parameters used for mechanistic simulations using phase-field model

B Forecast the tumor growth using the initial density of nutrients as the input for the branch net

Fig. 32

Prediction for tumor cells and nutrient dynamics for mild tumor cases mapping from the initial density of nutrients. a Prediction errors for training and testing datasets. The blue lines represent the mean of prediction errors in training datasets. The red lines represent the mean of prediction errors in testing datasets. The shaded region represents the region encompassed by the maximum and minimum of the prediction errors. b Predictions of the tumor morphologies $\phi$ at different time. left: R = 0.07 mm; right: R = 0.21 mm (R: the length of the minor axis of the initial ellipsoidal tumor)

Fig. 33

Prediction for tumor cells and nutrient dynamics for aggressive tumor cases mapping from the initial density of nutrients. a Prediction errors for training and testing datasets. The blue lines represent the mean of prediction errors in training datasets. The red lines represent the mean of prediction errors in testing datasets. The shaded region represents the region encompassed by the maximum and minimum of the prediction errors. b Predictions of the tumor morphologies $\phi$ at different time. left: R = 0.07 mm; right: R = 0.23 mm (R: the length of the minor axis of the initial ellipsoidal tumor)

We also test the performances of TGM-ONets to forecast tumor growth using the initial density of nutrients as the input for the branch net. The hyper-parameters (i.e., the initial learning rate, the decay step, the $\omega _{PDE}$ and the $\omega _{data}$) are selected to be the same as in Sect. 3.1.1. We parameterized the initial density of nutrients for both mild and aggressive tumors as:

$$\begin{aligned} \sigma (0, x, y) = 1 - 0.8\phi (0, x, y), \end{aligned}$$

(24)

which represents an ellipsoidal nutrients field corresponding to an ellipsoidal tumor in the computational domain for which the y-axis has double length compared to the x-axis. We use the same training datasets as in Sect. 3.1.1 to train TGM-ONets.

For mild tumor cases, prediction errors for all training and testing cases given by TGM-ONets are represented in Fig. 32(a-b), which shows that the average of prediction errors for both $\phi$ and $\sigma$ are under $1.0\,\times \,10^{-3}$ in training datasets and $2.0\,\times \,10^{-2}$ in testing datasets. the maximum prediction error are around $2.0\,\times \,10^{-3}$ for $\phi$ and $5.0\,\times \,10^{-4}$ for $\sigma$ in training datasets while $6.0\,\times \,10^{-2}$ for $\phi$ and $5.0\,\times \,10^{-2}$ for $\sigma$ in testing datasets. Predictions for two specific cases of R = 0.07 mm (in-distribution) and 0.21 mm (out-of-distribution) are illustrated in Fig. 32c, d.

For aggressive tumor cases, prediction errors for all training and testing cases given by TGM-ONets are represented in Fig. 33a, b, which shows that the average of prediction errors for both $\phi$ and $\sigma$ are under $5.0\,\times \,10^{-4}$ in training datasets and $2.0\,\times \,10^{-2}$ in testing datasets. the maximum prediction error are under $2.0\,\times \,10^{-3}$ for both $\phi$ and $\sigma$ in training datasets while $4.0\,\times \,10^{-2}$ for both $\phi$ and $\sigma$ in testing datasets. Predictions for two specific cases of R = 0.07 mm (in-distribution) and 0.23 mm (out-of-distribution) are illustrated in Fig. 33c, d.

C Forecast the tumor growth using the initial density of tumor cells with varying shapes as the input for the branch net

We evaluate the performance of TGM-ONets to forecast tumor growth using the initial density of tumor cells with varying shapes as the input for the branch net. For mild tumors, we vary the ratios of the y-semiaxes to the x-semiaxes ($\delta$) (C.1), positions within the domain (C.2), the length of the minor axis of the initial ellipsoidal tumor centered at (0.5,0.5) (C.3) and centered not at (0.5,0.5) (C.4), circular shapes (C.5) and oblique ellipsoidal tumors (C.6). For aggressive tumors, we vary the ratios of the y-semiaxes to the x-semiaxes ($\delta$) (C.7) and positions (C.8). The hyper-parameters (i.e., initial learning rate, the decay step, the $\omega _{PDE}$ and the $\omega _{data}$) are selected to be the same as Sect. 3.1.1.

1.1 C.1 Forecast the mild tumor growth using the initial density of tumor cells with varying ratio ($\delta$) of y-semiaxis to the x-semiaxis

For mild tumor cases, we use TGM-ONets to learn the mapping from the initial density of tumor cells with varying $\delta$ to the solutions of tumor cells and nutrients on the entire computation domain. The growth rate and the length of the minor axis of the initial ellipsoidal tumor R remain the same as 1.5 and 0.05. We sample 1000 values of $\delta$ from a uniform distribution U(1.0, 2.6). Assuming we have 8 cases of data recording the density of tumor cells and nutrients for every 0.5 days up to 70.5 days with different values of $\delta$ sampled from U(1.0, 2.6), we follow the same training procedure as in Sect. 3.1.1. We evaluate the performance of TGM-ONets on testing datasets with different values of $\delta$ sampled from U(1.0, 2.9). The accuracy of predictions associated with both the training and testing datasets are represented in Fig. 34a, from which we can see the maximum prediction error for $\phi$ and $\sigma$ are bounded by $6.0\,\times \,10^{-4}$ and $3.0\,\times \,10^{-4}$ in training datasets while $8.0\,\times \,10^{-4}$ and $2.0\,\times \,10^{-3}$ in testing datasets. The average prediction errors for $\phi$ and $\sigma$ are under $3.0\,\times \,10^{-4}$ in training datasets while $5.0\,\times \,10^{-4}$ in testing datasets. Predictions for two specific cases of $\delta$ = 1.4 (in-distribution) and $\delta$ = 2.9 (out-of-distribution) are illustrated in Fig. 34b.

Fig. 34

Prediction for tumor cells and nutrient dynamics for mild tumor cases mapping from the initial density of tumor cells with varying $\delta$. a Prediction errors for training and testing datasets. The blue lines represent the mean of prediction errors in training datasets. The red lines represent the mean of prediction errors in testing datasets. The shaded region represents the region encompassed by the maximum and minimum of the prediction errors. b Predictions of the tumor morphologies $\phi$ at different time. left: $\delta$ = 1.4; right: $\delta$ = 2.9 ($\delta$: the ratios of the y-semiaxes to the x-semiaxes)

1.2 C.2 Forecast the mild tumor growth using the initial density of tumor cells with varying positions within the domain

In this subsection, we use TGM-ONets to learn the mapping from the initial density of tumor cells with varying positions within the domain to the solutions of tumor cells and nutrients on the entire computation domain for mild tumor cases. The growth rate and the length of the minor axis of the initial ellipsoidal tumor R remain the same as 1.5 mm and 0.05 mm. Let $(x^{*}, y^{*})$ denote the position of tumor cells and nutrients, we sample 1000 values of $x^{*}$ and $y^{*}$ from a uniform distribution U(0.4, 0.6). Assuming we have 8 cases of data recording the density of tumor cells and nutrients for every 0.5 days up to 70.5 days with different values of $x^{*}$ and $y^{*}$ sampled from U(0.4, 0.6), we follow the same training procedure as in Sect. 3.1.1. We evaluate the performance of TGM-ONets on testing datasets with different values of $x^{*}$ and $y^{*}$ sampled from U(0.4, 0.6). The accuracy of predictions associated with both the training and testing datasets are represented in Fig. 35a, from which we can see the maximum prediction error for $\phi$ and $\sigma$ are bounded by $1.0\,\times \,10^{-3}$ in training datasets while $1.5\times10^{-1}$ in testing datasets. The average prediction errors for $\phi$ and $\sigma$ are under $1.0\,\times \,10^{-2}$ in training datasets while $8.0\,\times \,10^{-2}$ in testing datasets. Predictions for two specific cases of $(x^{*}, y^{*})$ = (0.58, 0.42) (in-distribution) and $(x^{*}, y^{*})$ = (0.42, 0.58) (in-distribution) are illustrated in Fig. 35b.

Fig. 35

Prediction for tumor cells and nutrient dynamics for mild tumor cases mapping from the initial density of tumor cells with varying positions within the computation domain. a Prediction errors for training and testing datasets. The blue lines represent the mean of prediction errors in training datasets. The red lines represent the mean of prediction errors in testing datasets. The shaded region represents the region encompassed by the maximum and minimum of the prediction errors. b Predictions of the tumor morphologies $\phi$ at different time. left: $(x^{*}, y^{*})$ = (0.58, 0.42); right: $(x^{*}, y^{*})$ = (0.42, 0.58) ($(x^{*}, y^{*})$: the center position of tumor cells and nutrients within computation domain)

1.3 C.3 Forecast the mild tumor growth using the initial density of tumor cells with varying length of the minor axis of the initial ellipsoidal tumor centered at (0.5,0.5)

In this subsection, we use TGM-ONets to learn the mapping from the initial density of tumor cells with varying lengths of the minor axis of the initial ellipsoidal tumor R and a larger ratio of the y-semiaxes to the x-semiaxes ($\delta$ = 3) to the solutions of tumor cells and nutrients on the entire computation domain. The growth rate remains the same as 1.5. We sample 1000 values of R from a uniform distribution U(0.06, 0.20). Assuming we have 9 cases of data with different values of R sampled from U(0.06, 0.20) and an additional case of data with R = 0.22 mm recording the density of tumor cells and nutrients for every 0.5 days up to 70.5 days, we follow the same training procedure as in Sect. 3.1.1. We evaluate the performance of TGM-ONets on testing datasets with different values of R sampled from U(0.06, 0.23). The accuracy of predictions associated with both the training and testing datasets are represented in Fig. 36a, from which we can see the maximum prediction error for $\phi$ and $\sigma$ are bounded by $2.0\,\times \,10^{-3}$ and $6.0\,\times \,10^{-4}$ in training datasets while $1.3\,\times \,10^{-2}$ and $1.0\,\times \,10^{-2}$ in testing datasets. The average prediction errors for $\phi$ and $\sigma$ are under $1.0\,\times \,10^{-3}$ in training datasets while $5.0\,\times \,10^{-3}$ in testing datasets. Predictions for two specific cases of R = 0.07 mm (in-distribution) and R = 0.23 mm (out-of-distribution) are illustrated in Fig. 36b.

Fig. 36

Prediction for tumor cells and nutrient dynamics for mild tumor cases mapping from the initial density of tumor cells with varying lengths of the minor axis of the initial ellipsoidal tumor R and a larger $\delta$ = 3. a Prediction errors for training and testing datasets. The blue lines represent the mean of prediction errors in training datasets. The red lines represent the mean of prediction errors in testing datasets. The shaded region represents the region encompassed by the maximum and minimum of the prediction errors. b Predictions of the tumor morphologies $\phi$ at different time. left: R = 0.07 mm; right: R = 0.23 mm. (R: the length of the minor axis of the initial ellipsoidal tumor)

1.4 C.4 Forecast the mild tumor growth using the initial density of tumor cells with varying length of the minor axis of the initial ellipsoidal tumor centered not at (0.5,0.5)

In this subsection, we use TGM-ONets to learn the mapping from the non-center initial density of tumor cells with varying lengths of the minor axis of the initial ellipsoidal tumor R to the solutions of tumor cells and nutrients on the entire computation domain. The growth rate remains the same as 1.5 1/day. We sample 1000 values of R from a uniform distribution U(0.06, 0.20). Assuming we have 9 cases of data with different values of R sampled from U(0.06, 0.20) and additional data with R = 0.22 mm recording the density of tumor cells and nutrients for every 0.5 days up to 70.5 days, we follow the same training procedure as in Sect. 3.1.1. We evaluate the performance of TGM-ONets on testing datasets with different values of R sampled from U(0.06, 0.23). The accuracy of predictions associated with both training and testing datasets are represented in Fig. 37a, from which we can see the maximum prediction error for $\phi$ and $\sigma$ are bounded by $3.0\,\times \,10^{-3}$ in training datasets while $5.0\,\times \,10^{-2}$ in testing datasets. The average prediction errors for $\phi$ and $\sigma$ are under $1.0\,\times \,10^{-3}$ in training datasets while $2.0\,\times \,10^{-2}$ in testing datasets. Predictions for two specific cases of R = 0.07 mm (in-distribution) and R = 0.23 mm (out-of-distribution) are illustrated in Fig. 37b.

Fig. 37

Prediction for tumor cells and nutrient dynamics for mild tumor cases mapping from the non-center initial density of tumor cells with varying scaling factors of R. a Prediction errors for training and testing datasets. The blue lines represent the mean of prediction errors in training datasets. The red lines represent the mean of prediction errors in testing datasets. The shaded region represents the region encompassed by the maximum and minimum of the prediction errors. b Predictions of the tumor morphologies $\phi$ at different time. left: R = 0.07 mm; right: R = 0.23 mm. (R: the length of the minor axis of the initial ellipsoidal tumor)

1.5 C.5 Forecast the mild tumor growth using the initial density of tumor cells with varying radii of the initial circular tumor

In this subsection, we use TGM-ONets to learn the mapping from the initial density of tumor cells with varying radii of the initial circular tumor R to the solutions of tumor cells and nutrients on the entire computation domain for mild tumor cases. The growth rate remains the same as 1.5. We sample 1000 values of R from a uniform distribution U(0.06, 0.20). Assuming we have 9 cases of data with different values of R sampled from U(0.06, 0.20) and an additional case of data with R = 0.22 recording the density of tumor cells and nutrients for every 0.5 days up to 70.5 days, we follow the same training procedure as in Sect. 3.1.1. We evaluate the performance of TGM-ONets on testing datasets with different values of R sampled from U(0.06, 0.23). The accuracy of predictions associated with both the training and testing datasets are represented in Fig. 38a, from which we can see the maximum prediction error for $\phi$ and $\sigma$ are bounded by $8.0\,\times \,10^{-4}$ and $5.0\,\times \,10^{-4}$ in training datasets while $1.0\,\times \,10^{-2}$ in testing datasets. The average prediction errors for $\phi$ and $\sigma$ are under $4.0\,\times \,10^{-4}$ in training datasets while $5.0\,\times \,10^{-3}$ in testing datasets. Predictions for two specific cases of R = 0.07(in-distribution) and R = 0.23(out-of-distribution) are illustrated in Fig. 38b.

Fig. 38

Prediction for tumor cells and nutrient dynamics for mild tumor cases mapping from the circular initial density of tumor cells with varying scaling factors of R. a Prediction errors for training and testing datasets. The blue lines represent the mean of prediction errors in training datasets. The red lines represent the mean of prediction errors in testing datasets. The shaded region represents the region encompassed by the maximum and minimum of the prediction errors. b Predictions of the tumor morphologies $\phi$ at different time. left: R = 0.07 mm; right: R = 0.23 mm. (R: the radius of the initial circular tumor)

1.6 C.6 Forecast the mild tumor growth using the initial density of tumor cells with varying oblique ($\theta$) and the ratios of the y-semiaxes to the x-semiaxes ($\delta$ = 2 or 4)

In this subsection, we use TGM-ONets to learn the mapping from the initial density of tumor cells with varying oblique($\theta$) and the ratios of the y-semiaxes to the x-semiaxes($\delta$ = 2 or 4) to the solutions of tumor cells and nutrients on the entire computation domain. The growth rate remains the same as 1.5. We sample 1000 values of $\theta$ from a uniform distribution $U(0, 2\pi )$. Assuming we have 8 cases of data with different values of $\theta$ sampled from $U(0, 2\pi )$, we follow the same training procedure as in Sect. 3.1.1. We evaluate the performance of TGM-ONets on testing datasets with different values of $\theta$ sampled from $U(0, 2\pi )$. The accuracy of predictions associated with both training and testing datasets are represented in Fig. 39a, from which we can see the maximum prediction error for $\phi$ and $\sigma$ are around $3.0\,\times \,10^{-3}$ and $1.5\,\times \,10^{-3}$ in training datasets while $2.0\,\times \,10^{-1}$ and $8.0\,\times \,10^{-2}$ in testing datasets. The average prediction errors for $\phi$ and $\sigma$ are under $2.0\,\times \,10^{-3}$ in training datasets while $1.0\,\times \,10^{-1}$ in testing datasets. Predictions for two specific cases of ($\theta$ = 0.3$\pi$, $\delta$ = 4) (in-distribution) and ($\theta$ = 0.1$\pi$, $\delta$ = 2) (in-distribution) are illustrated in Fig. 39b.

Fig. 39

Prediction for tumor cells and nutrient dynamics for mild tumor cases mapping from the initial density of tumor cells with varying oblique $\theta$ and the ratios of the y-semiaxes to the x-semiaxes ($\delta$ = 2 or 4). a Prediction errors for training and testing datasets. The blue lines represent the mean of prediction errors in training datasets. The red lines represent the mean of prediction errors in testing datasets. The shaded region represents the region encompassed by the maximum and minimum of the prediction errors. b Predictions of the tumor morphologies $\phi$ at different time. left: $\theta$ = 0.3$\pi$, $\delta$ = 4; right: $\theta$ = 0.1$\pi$, $\delta$ = 2. ($\theta$: oblique of the initial density of tumor cells, $\delta$: the ratios of the y-semiaxes to the x-semiaxes)

1.7 C.7 Forecast the aggressive tumor growth using the initial density of tumor cells with varying the ratio of the x-semiaxis to y-semiaxis ($\delta$)

For aggressive tumor cases, here we use TGM-ONets to learn the mapping from the ellipsoidal initial density of tumor cells with varying $\delta$ to the solutions of tumor cells and nutrients on the entire computation domain. The growth rate, the length of the minor axis of the initial ellipsoidal tumor R and $\gamma _{c}$ remain the same as 1.0, 0.05 and 17.5. We sample 1000 values of $\delta$ from a uniform distribution U(1.0, 3.0). Assuming we have 8 cases of data recording the density of tumor cells and nutrients for every 0.5 days up to 200.5 days with different values of $\delta$ sampled from U(1.0, 3.0), we follow the same training procedure as in Sect. 3.1.1. We evaluate the performance of TGM-ONets on testing datasets with different values of $\delta$ sampled from U(1.0, 3.0). The accuracy of predictions associated with both the training and testing datasets are represented in Fig. 40a, from which we can see the maximum prediction error for $\phi$ and $\sigma$ are bounded by $4.0\,\times \,10^{-3}$ and $8.0\,\times \,10^{-3}$ in both training and testing datasets. The average prediction errors for $\phi$ and $\sigma$ are under $5\,\times \,10^{-3}$ in both training and testing datasets. Predictions for two specific cases of $\delta$ = 1.4 (in-distribution) and 2.7 (in-distribution) are illustrated in Fig. 40b.

Fig. 40

Prediction for tumor cells and nutrient dynamics for aggressive tumor cases mapping from the ellipsoidal initial density of tumor cells with varying $\delta$. a Prediction errors for training and testing datasets. The blue lines represent the mean of prediction errors in training datasets. The red lines represent the mean of prediction errors in testing datasets. The shaded region represents the region encompassed by the maximum and minimum of the prediction errors. b Predictions of the tumor morphologies $\phi$ at different time. left: $\delta$ = 1.4; right: $\delta$ = 2.7. ($\delta$: the ratios of the y-semiaxes to the x-semiaxes)

1.8 C.8 Forecast the aggressive tumor growth using the initial density of tumor cells with varying positions within the domain

In this subsection, we use TGM-ONets to learn the mapping from the initial density of tumor cells with varying positions within the domain to the solutions of tumor cells and nutrients on the entire computation domain for aggressive tumor cases. The growth rate, the length of the minor axis of the initial ellipsoidal tumor R and $\gamma _{c}$ remain the same as 1.0 and 0.05 and 17.5. Let $(x^{*}, y^{*})$ denote the position of tumor cells and nutrients, we sample 1000 values of $x^{*}$ and $y^{*}$ from a uniform distribution U(0.4, 0.6). Assuming we have 10 cases of data recording the density of tumor cells and nutrients for every 0.5 days up to 200.5 days with different values of $x^{*}$ and $y^{*}$ sampled from U(0.4, 0.6), we follow the same training procedure as in Sect. 3.1.1. We evaluate the performance of TGM-ONets on testing datasets with different values of $x^{*}$ and $y^{*}$ sampled from U(0.4, 0.6). The accuracy of predictions associated with both the training and testing datasets are represented in Fig. 41a, from which we can see the maximum prediction error for $\phi$ and $\sigma$ are bounded by $3.0\,\times \,10^{-3}$ in training datasets while $3.0\,\times \,10^{-2}$ in testing datasets. The average prediction errors for $\phi$ and $\sigma$ are under $2.0\,\times \,10^{-3}$ in training datasets while $2.0\,\times \,10^{-2}$ in testing datasets. Predictions for two specific cases of $(x^{*}, y^{*})$ = (0.54, 0.54) (in-distribution) and $(x^{*}, y^{*})$ = (0.54, 0.46) (in-distribution) are illustrated in Fig. 41b.

Fig. 41

Prediction for tumor cells and nutrient dynamics for aggressive tumor cases mapping from the ellipsoidal initial density of tumor cells with varying positions within computation domain. a Prediction errors for training and testing datasets. The blue lines represent the mean of prediction errors in training datasets. The red lines represent the mean of prediction errors in testing datasets. The shaded region represents the region encompassed by the maximum and minimum of the prediction errors. b Predictions of the tumor morphologies $\phi$ at different time. left: ($x^{*}, y^{*}$) = (0.54, 0.54); right: ($x^{*}, y^{*}$) = (0.54, 0.46). (($x^{*}, y^{*}$): the center position of tumor cells and nutrients within computation domain)

D Significance tests for the difference of prediction errors obtained by TGM-ONets using different input functions for the branch net

In this section, we conduct Kruskal–Wallis and Dunns test to check if the differences in prediction errors obtained by TGM-ONets using varying input functions are statistically significant. We take the average prediction errors over time to compute the statistics for Kruskal–Wallis and Dunns test.

For mild tumors, we vary the input functions from the initial density of tumor cells to the growth rate. The results of Kruskal–Wallis test are summarized on the $1^{st}$ two rows in Table 44, from which we can see the p-values are only lower than $1.0\,\times \,10^{-2}$ in training datasets, indicating that the $\epsilon _{\phi }$ and $\epsilon _{\sigma }$ in training datasets obtained by TGM-ONets using initial density of tumor cells and growth rate as inputs for the branch net are significantly different. However, the generalization ability of TGM-ONets for unseen datasets of mild tumors is consistent and irrespective of input functions. Results of the median of $\phi$ and $\sigma$ in training datasets are summarized in Table 45. These results suggest that TGM-ONets perform better in fitting training datasets for mild tumors mapping from the initial density of tumor cells.

For aggressive tumor cases, we vary the input functions from the initial density of tumor cells to the nutrient uptake. The results of Kruskal–Wallis test are summarized on the last two rows in Table 44 and the results of the median of $\epsilon _{\phi }$ and $\epsilon _{\sigma }$ in training datasets are summarized in Table 46. These results also suggest that TGM-ONets performances are consistent and irrespective of input functions for unseen datasets while significantly better fitting training datasets for aggressive tumors mapping from the initial density of tumor cells. We infer that the CBAM blocks utilized in CNN-based branch nets are the reasons accounting for the better performance in fitting training datasets for mild and aggressive tumors mapping from the initial density of tumor cells.

Table 44 Results of Kruskal–Wallis test for comparisons between TGM-ONets with varying input functions

Table 45 Median of prediction errors in training datasets for mild tumors with varying input functions

Table 46 Median of prediction errors in training datasets for aggressive tumors with varying input functions

E Long-time predictions using TGM-ONets

In this subsection, we demonstrate more results about long-time prediction using TGM-ONets. Three scenarios are the same as in Sect. 3.2: (1) assuming we have sparse observations at the last time step in the testing domain, (2) assuming we have sparse observations at the early stage in the testing domain, and (3) assuming we have no additional observations in the testing domain.

Tables 47 and 48 show the mean relative $L^2$ error for $\phi$ and $\sigma$ in long-time predictions for mild tumors mapping from the initial density of tumor cells in scenario 1, respectively. Tables 49 - 50 show the mean relative $L^2$ error for $\phi$ and $\sigma$ in long-time predictions for aggressive tumors mapping from the initial density of tumor cells in scenario 1, respectively. We see that with the time and the initial density of tumor cells increasing, the mean relative $L^2$ errors increase a little bit. However, even at T = 270 with R > 0.2 mm, the maximum mean relative $L^2$ errors for $\phi$ and $\sigma$ are still under around $7\times \,10^{-2}$.

Tables 51 and 52 show the mean relative $L^2$ error for $\phi$ and $\sigma$ in long-time predictions for mild tumors mapping from the initial density of tumor cells in scenario 2, respectively. Table 53 and 54 shows the mean relative $L^2$ error for $\phi$ and $\sigma$ in long-time predictions for aggressive tumors mapping from the initial density of tumor cells in scenario 2, respectively.

Tables 55 and 56 show the mean relative $L^2$ error for $\phi$ and $\sigma$ in long-time predictions for mild tumors mapping from the initial density of tumor cells in scenario 3, respectively. Tables 57 and 58 show the mean relative $L^2$ error for $\phi$ and $\sigma$ in long-time predictions for aggressive tumors mapping from the initial density of tumor cells in scenario 3, respectively.

Tables 59 and 60 show the mean relative $L^2$ error for $\phi$ and $\sigma$ in long-time prediction for mild tumors mapping from the concomitant changes in the initial density of tumor cells and growth rate in scenario 1, respectively. Tables 61 and 62 show the mean relative $L^2$ error for $\phi$ and $\sigma$ in long-time prediction for aggressive tumors mapping from the concomitant changes in the initial density of tumor cells and nutrient uptake in scenario 1, respectively.

Tables 63 and 64 show the mean relative $L^2$ error for $\phi$ and $\sigma$ in long-time prediction for mild tumors mapping from the concomitant changes in the initial density of tumor cells and growth rate in scenario 2, respectively. Tables 65 and 66 show the mean relative $L^2$ error for $\phi$ and $\sigma$ in long-time prediction for aggressive tumors mapping from the concomitant changes in the initial density of tumor cells and nutrient uptake in scenario 2, respectively.

Tables 67 and 68 show the mean relative $L^2$ error for $\phi$ and $\sigma$ in long-time prediction for mild tumors mapping from the concomitant changes in the initial density of tumor cells and growth rate in scenario 3, respectively. Tables 69 and 70 show the mean relative $L^2$ error for $\phi$ and $\sigma$ in long-time prediction for aggressive tumors mapping from the concomitant changes in the initial density of tumor cells and nutrient uptake in scenario 3, respectively.

Table 47 Long-time prediction for mild tumor $\phi$ scenario 1: Mean relative $L^2$ error for $\phi$ of the mild tumor growth mapping from the initial density of tumor cells using additional snapshots at T = 200 and T = 270

Table 48 Long-time prediction for mild tumor $\sigma$ in scenario 1: Mean relative $L^2$ error for $\sigma$ of the mild tumor growth mapping from the initial density of tumor cells using additional snapshots at T = 200 and T = 270

Table 49 Long-time prediction for aggressive tumor $\phi$ in scenario 1: Mean relative $L^2$ error for $\phi$ of the aggressive tumor growth mapping from the initial density of tumor cells using additional snapshots at T = 200, 300 and 400

Table 50 Long-time prediction for aggressive tumor $\sigma$ in scenario 1: Mean relative $L^2$ error for $\sigma$ of the aggressive tumor growth mapping from the initial density of tumor cells using additional snapshots at T = 200, 300 and 400

Table 51 Long-time prediction for mild tumor $\phi$ in scenario 2: Mean relative $L^2$ error for $\phi$ of the mild tumor growth mapping from the initial density of tumor cells using an additional snapshot at T = 80

Table 52 Long-time prediction for mild tumor $\sigma$ in scenario 2: Mean relative $L^2$ error for $\sigma$ of the mild tumor growth mapping from the initial density of tumor cells using an additional snapshot at T = 80

Table 53 Long-time prediction for aggressive tumor $\phi$ in scenario 2: Mean relative $L^2$ error for $\phi$ of the aggressive tumor growth mapping from the initial density of tumor cells using additional snapshots at T = 230 and T = 280

Table 54 Long-time prediction for aggressive tumor $\sigma$ in scenario 2: Mean relative $L^2$ error for $\sigma$ of the aggressive tumor growth mapping from the initial density of tumor cells using additional snapshots at T = 230 and T = 280

Table 55 Long-time prediction for mild tumor $\phi$ in scenario 3: Mean relative $L^2$ error for $\phi$ of the mild tumor growth mapping from the initial density of tumor cells using no additional snapshots in testing domain

Table 56 Long-time prediction for mild tumor $\sigma$ in scenario 3: Mean relative $L^2$ error for $\sigma$ of the mild tumor growth mapping from the initial density of tumor cells using no additional snapshots in testing domain

Table 57 Long-time prediction for aggressive tumor $\phi$ in scenario 3: Mean relative $L^2$ error for $\phi$ of the aggressive tumor growth mapping from the initial density of tumor cells using no additional snapshots in testing domain

Table 58 Long-time prediction for aggressive tumor $\sigma$ in scenario 3: Mean relative $L^2$ error for $\sigma$ of the aggressive tumor growth mapping from the initial density of tumor cells using no additional snapshots in testing domain

Table 59 Long-time prediction for mild tumor $\phi$ accommodating multiple input functions in scenario 1: Mean relative $L^2$ error for $\phi$ of the mild tumor growth mapping from the initial density of tumor cells and growth rate using additional snapshots at T = 200 and T = 270

Table 60 Long-time prediction for mild tumor $\sigma$ accommodating multiple input functions in scenario 1: Mean relative $L^2$ error for $\sigma$ of the mild tumor growth mapping from the initial density of tumor cells and growth rate using additional snapshots at T = 200 and T = 270

Table 61 Long-time prediction for aggressive tumor $\phi$ accommodating multiple input functions in scenario 1: Mean relative $L^2$ error for $\phi$ of the aggressive tumor growth mapping from the initial density of tumor cells and nutrient uptake using additional snapshots at T = 200, 300 and 400

Table 62 Long-time prediction for aggressive tumor $\sigma$ accommodating multiple input functions in scenario 1: Mean relative $L^2$ error for $\sigma$ of the aggressive tumor growth mapping from the initial density of tumor cells and nutrient uptake using 3 additional snapshots at T = 200, 300 and 400

Table 63 Long-time prediction for mild tumor $\phi$ accommodating multiple input functions in scenario 2: Mean relative $L^2$ error for $\phi$ of the mild tumor growth mapping from the initial density of tumor cells and growth rate using an additional snapshot at T = 80

Table 64 Long-time prediction for mild tumor $\sigma$ accommodating multiple input functions in scenario 2: Mean relative $L^2$ error for $\sigma$ of the mild tumor growth mapping from the initial density of tumor cells and growth rate using an additional snapshot at T = 80

Table 65 Long-time prediction for aggressive tumor $\phi$ accommodating multiple input functions in scenario 2: Mean relative $L^2$ error for $\phi$ of the aggressive tumor growth mapping from the initial density of tumor cells and nutrient uptake using 2 additional snapshots T = 230 and T = 280

Table 66 Long-time prediction for aggressive tumor $\sigma$ accommodating multiple input functions in scenario 2: Mean relative $L^2$ error for $\sigma$ of the aggressive tumor growth mapping from the initial density of tumor cells and nutrient uptake using 2 additional snapshots T = 230 and T = 280

Table 67 Long-time prediction for mild tumor $\phi$ accommodating multiple input functions in scenario 3: Mean relative $L^2$ error for $\phi$ of the mild tumor growth mapping from the initial density of tumor cells using no additional snapshots in testing domain

Table 68 Long-time prediction for mild tumor $\sigma$ accommodating multiple input functions in scenario 3: Mean relative $L^2$ error for $\sigma$ of the mild tumor growth mapping from the initial density of tumor cells using no additional snapshots in testing domain

Table 69 Long-time prediction for aggressive tumor $\phi$ accommodating multiple input functions in scenario 3: Mean relative $L^2$ error for $\phi$ of the aggressive tumor growth mapping from the initial density of tumor cells and nutrient uptake $\gamma _{c}$ using no additional snapshots in testing domain

Table 70 Long-time prediction for aggressive tumor $\sigma$ accommodating multiple input functions in scenario 3: Mean relative $L^2$ error for $\sigma$ of the aggressive tumor growth mapping from the initial density of tumor cells using no additional snapshots in testing domain

F Ranges of prediction errors in training datasets for examining the robustness of TGM-ONets

In this subsection, we provide the ranges of prediction errors in training datasets for examining the robustness of TGM-ONets.

For the results of examining the effects of the number of training snapshots, Fig. 42 shows the ranges of prediction errors in training datasets for mild tumor cases mapping from the initial density of tumor cells, Fig. 43 shows the ranges of prediction errors in training datasets for aggressive tumor cases mapping from the initial density of tumor cells, and Fig. 44 shows the ranges of prediction errors in training datasets for aggressive tumor cases mapping from the initial density of tumor cells using sparser data.

For the results of examining the effects of the noisy measurements, Fig. 45 shows the ranges of prediction errors in training datasets for mild tumor cases mapping from the initial density of tumor cells, and Fig. 46 shows the ranges of prediction errors in training datasets for aggressive tumor cases mapping from the initial density of tumor cells.

Fig. 42

Prediction errors in training datasets for examining the effects of the number of training points for mild tumor cases mapping from the initial density of tumor cells. The blue lines represent the mean of prediction errors in training datasets. The shaded region represents the region encompassed by the maximum and minimum of the prediction errors

Fig. 43

Prediction errors in training datasets for examining the effects of the number of training points for aggressive tumor cases mapping from the initial density of tumor cells. The blue lines represent the mean of prediction errors in training datasets. The shaded region represents the region encompassed by the maximum and minimum of the prediction errors

Fig. 44

Prediction errors in training datasets for examining the effects of the number of training points for aggressive tumor cases mapping from the initial density of tumor cells using sparser data. The blue lines represent the mean of prediction errors in training datasets. The shaded region represents the region encompassed by the maximum and minimum of the prediction errors

Fig. 45

Prediction errors in training datasets for examining the effects of the noisy measurements for mild tumor cases mapping from the initial density of tumor cells. The blue lines represent the mean of prediction errors in training datasets. The shaded region represents the region encompassed by the maximum and minimum of the prediction errors

Fig. 46

Prediction errors in training datasets for examining the effects of the noisy measurements for aggressive tumor cases mapping from the initial density of tumor cells. The blue lines represent the mean of prediction errors in training datasets. The shaded region represents the region encompassed by the maximum and minimum of the prediction errors

Fig. 47

Ablation study: Prediction errors for inferring state variables for aggressive tumor cases mapping from the initial density of tumor cells with varying network structures. a The average of prediction errors for training datasets. b The average of prediction errors for testing datasets

Fig. 48

Ablation study: Prediction errors for inferring state variables for mild tumor cases mapping from the growth rate with varying network structures. a The average of prediction errors for training datasets. b The average of prediction errors for testing datasets

Fig. 49

Grid study: Prediction errors for inferring state variables for mild tumor cases mapping from the nutrient uptake $\gamma _{c}$ with varying number of total hidden layers. a The average of prediction errors for training datasets. b The average of prediction errors for testing datasets

Fig. 50

Grid study: Prediction errors for inferring state variables for mild tumor cases mapping from the initial density of tumor cells with varying number of total hidden layers. a The average of prediction errors for training datasets. b The average of prediction errors for testing datasets

Fig. 51

Grid study: Prediction errors for inferring state variables for mild tumor cases mapping from the growth rate $\rho$ with varying number of total hidden layers. a The average of prediction errors for training datasets. b The average of prediction errors for testing datasets

Fig. 52

Grid study: Prediction errors for inferring state variables for aggressive tumor cases mapping from the initial density of tumor cells with varying number of total hidden layers. a The average of prediction errors for training datasets. b The average of prediction errors for testing datasets

G Ablation & grid studies of TGM-ONets

For all cases considered in Sect. 4 and in this subsection, the distributions for the parameters of the mechanistic model for training and testing are the same as in Sect. 3.

In this subsection, we first consider the contributions of CBAM and MoE blocks in aggressive tumor cases mapping from the initial density of tumor cells as well as in mild tumor cases mapping from the growth rates $\rho$. We use the same settings mentioned in the first ablation study in Sect. 4. Model performances are summarized in Figs. 47 and 48. We also conduct single-tailed Wilcoxon tests to check if the prediction errors given by Vanilla PI-DeepONet are significantly greater than those given by our proposed methods. The null hypothesis and the alternative hypothesis are the same as in Sect. 4. We take the average prediction errors over time for each training and test sample to compute the statistics for the single-tailed Wilcoxon tests. The results are summarized in Tables 71 and 72, from which we can see the p-values are lower than $1.0\,\times \,10^{-2}$ for $\epsilon _{\phi }$ and $\epsilon _{\sigma }$ in testing datasets for both mild tumors and aggressive tumor cases. These results further showcase the improvements in the generalization ability of TGM-ONets for unseen datasets by utilizing our proposed methods.

Additionally, we further investigate the effects of the number of total hidden layers used in TGM-ONets for mild tumors mapping from the nutrient uptake $\gamma _{c}$, mild tumors mapping from the initial density of tumor cells, mild tumors mapping from the growth rate $\rho$ and aggressive tumors mapping from the initial density of tumor cells. Model performances are summarized in Figs. 49, 50, 51 and  52. We also conduct Kruskal–Wallis test and Dunns test to check if the difference of the prediction errors given by TGM-ONets with varying numbers of total hidden layers are statistically significant. We take the average prediction errors over time for each training and test sample to compute the statistics for Kruskal–Wallis and Dunns test. The results of Kruskal–Wallis test are summarized in Tables 73 - 76, from which we can see the effects of the number of total hidden layers are different from case to case. For mild tumors mapping from the nutrient uptake $\gamma _{c}$, the p-values are lower than $1.0\,\times \,10^{-2}$ in training and testing datasets. For mild tumors mapping from the initial density of tumor cells, the p-values are lower than $1.0\,\times \,10^{-2}$ in training datasets. For mild tumors mapping from the growth rate $\rho$, the p-values are greater than $1.0\,\times \,10^{-2}$ in training and testing datasets. For aggressive tumors mapping from the initial density of tumor cells, the p-values are only lower than $1.0\,\times \,10^{-2}$ in training datasets. These results indicate that increasing the number of total hidden layers does enhance the performance, but the degree of enhancement achieved through this approach can be relatively minor sometimes and may not offset the increased computational cost associated with an increased number of model parameters. Further results of Dunns test for each case are summarized in Tables 77 - 83.

Quantitative results of enhancement derived from utilizing CBAM and MoE blocks are also provided in this section. Tables 84 - 89 show the mean relative $L^{2}$ errors for $\phi$ and $\sigma$ for mild tumors mapping from the initial density of tumor cells with R = 0.05 mm, 0.18 mm and 0.23 mm using varying architectures of deep operator networks. Tables 90 - 95 show the mean relative $L^{2}$ errors for $\phi$ and $\sigma$ for aggressive tumors mapping from the nutrient uptake by tumor cells with $\gamma _{c}$ = 16.1 g/L/day, 16.9 g/L/day and 18.9 g/L/day using varying architectures of deep operator networks. Tables 96 - 101 show the mean relative $L^{2}$ errors for $\phi$ and $\sigma$ for aggressive tumors mapping from the initial density of tumor cells with $R = 0.09$ mm, 0.16 mm and 0.21 mm using varying architectures of deep operator networks. Tables 102, 103, 104, 105, 106 and 107 show the mean relative $L^{2}$ errors for $\phi$ and $\sigma$ for mild tumors mapping from the growth rate with $\rho$ = 1.2 1/day, 2.1 1/day and 2.7 1/day using varying architectures of deep operator networks. In all the tables, bolded are the best results, and underlined are the second-best ones.

We also provide the ranges for prediction errors for each case considered in Sect. 4 and in this subsection. From Figs. 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68 and 69, we can see that ranges for the training error are roughly no larger than the ranges for the testing errors for all the cases considered in ablation and grid studies. In Figs. 53, 54 and 65, we can see that TGM-ONets with CBAM and MoE blocks have smaller ranges of prediction errors compared with vanilla PI-DeepONets. In Fig. 55 and 56, we can see that $w_{data} = 100$ has smaller ranges for the prediction errors than other values for $w_{data}$. In Figs. 57, 66, 67, 68 and 69, we can see that 8 or 9 hidden layers have smaller ranges for the prediction errors than others. In Fig. 58, we can see that 2 MoE has smaller ranges for the testing errors but 3 MoE has smaller ranges for the training errors. In Fig. 59, we can see that $1\,\times \,10^{-3}$, $6\,\times \,10^{-4}$, and $2\,\times \,10^{-4}$ learning rate have smaller ranges for the prediction errors. In Fig. 60, 1000 decay steps have smaller ranges for the prediction errors. Figure 61 shows that continuous activation functions have smaller ranges for the prediction errors. Figure 62 shows different numbers of training datasets (i.e., 7, 8, 9, 10) have roughly the same ranges for the prediction errors. Figures 63 and 64 show that with or without boundary loss, ranges for the prediction errors do not change much.

Table 71 Results of Wilcoxon test for comparisons between vanilla PI-DeepONet and TGM-ONets with CBAM and MoE block for aggressive tumors mapping from the initial density of tumor cells

Table 72 Results of Wilcoxon test for comparisons between vanilla PI-DeepONet and TGM-ONets with MoE block for mild tumors mapping from the growth rate $\rho$

Table 73 Results of Kruskal–Wallis test for examing the effectiveness of number of hidden layers for mild tumors mapping from the nutrient uptake $\gamma _{c}$

Table 74 Results of Kruskal–Wallis test for examing the effectiveness of number of hidden layers for mild tumors mapping from the initial density of tumor cells

Table 75 Results of Kruskal–Wallis test for examing the effectiveness of number of hidden layers for mild tumors mapping from the growth rate $\rho$

Table 76 Results of Kruskal–Wallis test for examing the effectiveness of number of hidden layers for aggressive tumors mapping from the initial density of tumor cells

Table 77 Results of Dunns test for $\epsilon _{\phi }$ in training datasets for mild tumors mapping from the nutrient uptake $\gamma _{c}$ with varying number of total hidden layers. Each element represents the result of the p-value of the corresponding post hoc pairwise test for multiple comparisons

Table 78 Results of Dunns test for $\epsilon _{\sigma }$ in training datasets for mild tumors mapping from the nutrient uptake $\gamma _{c}$ with varying number of total hidden layers. Each element represents the result of the p-value of the corresponding post hoc pairwise test for multiple comparisons

Table 79 Results of Dunns test for $\epsilon _{\phi }$ in testing datasets for mild tumors mapping from the nutrient uptake $\gamma _{c}$ with varying number of total hidden layers. Each element represents the result of the p-value of the corresponding post hoc pairwise test for multiple comparisons

Table 80 Results of Dunns test for $\epsilon _{\sigma }$ in testing datasets for mild tumors mapping from the nutrient uptake $\gamma _{c}$ with varying number of total hidden layers. Each element represents the result of the p-value of the corresponding post hoc pairwise test for multiple comparisons

Table 81 Results of Dunns test for $\epsilon _{\phi }$ in training datasets for mild tumor cases mapping from the initial density of tumor cells with varying number of total hidden layers. Each element represents the result of the p-value of the corresponding post hoc pairwise test for multiple comparisons

Table 82 Results of Dunns test for $\epsilon _{\phi }$ in training datasets for aggressive tumor cases mapping from the initial density of tumor cells with varying number of total hidden layers. Each element represents the result of the p-value of the corresponding post hoc pairwise test for multiple comparisons

Table 83 Results of Dunns test for $\epsilon _{\sigma }$ in training datasets for aggressive tumor cases mapping from the initial density of tumor cells with varying number of total hidden layers. Each element represents the result of the p-value of the corresponding post hoc pairwise test for multiple comparisons

Table 84 Ablation study for mild tumor $\phi$: Mean relative $L^2$ error for $\phi$ of the mild tumor growth mapping from the initial density of tumor cells with R = 0.05 mm (test case)

Table 85 Ablation study for mild tumor $\sigma$: Mean relative $L^2$ error for $\sigma$ of the mild tumor growth mapping from the initial density of tumor cells with R = 0.05 mm (test case)

Table 86 Ablation study for mild tumor $\phi$: Mean relative $L^2$ error for $\phi$ of the mild tumor growth mapping from the initial density of tumor cells with R = 0.18 mm (test case)

Table 87 Ablation study for mild tumor $\sigma$: Mean relative $L^2$ error for $\sigma$ of the mild tumor growth mapping from the initial density of tumor cells with R = 0.18 mm(test case)

Table 88 Ablation study for mild tumor $\phi$: Mean relative $L^2$ error for $\phi$ of the mild tumor growth mapping from the initial density of tumor cells with R = 0.23 mm (test case)

Table 89 Ablation study for mild tumor $\sigma$: Mean relative $L^2$ error for $\sigma$ of the mild tumor growth mapping from the initial density of tumor cells with R = 0.23 mm (test case)

Table 90 Ablation study for aggressive tumor $\phi$: Mean relative $L^2$ error for $\phi$ of the aggressive tumor growth mapping from the nutrient uptake by tumor cells with $\gamma _{c}$ = 16.1 g/L/day (test case)

Table 91 Ablation study for aggressive tumor $\sigma$: Mean relative $L^2$ error for $\sigma$ of the aggressive tumor growth mapping from the nutrient uptake by tumor cells with $\gamma _{c}$ = 16.1 g/L/day (test case)

Table 92 Ablation study for aggressive tumor $\phi$: Mean relative $L^2$ error for $\phi$ of the aggressive tumor growth mapping from the nutrient uptake by tumor cells with $\gamma _{c}$ = 16.9 g/L/day (test case)

Table 93 Ablation study for aggressive tumor $\sigma$: Mean relative $L^2$ error for $\sigma$ of the aggressive tumor growth mapping from the nutrient uptake by tumor cells with $\gamma _{c}$ = 16.9 g/L/day (test case)

Table 94 Ablation study for aggressive tumor $\phi$: Mean relative $L^2$ error for $\phi$ of the aggressive tumor growth mapping from the nutrient uptake by tumor cells with $\gamma _{c}$ = 18.9 g/L/day (test case)

Table 95 Ablation study for aggressive tumor $\sigma$: Mean relative $L^2$ error for $\sigma$ of the aggressive tumor growth mapping from the nutrient uptake by tumor cells with $\gamma _{c}$ = 18.9 g/L/day (test case)

Table 96 Ablation study for aggressive tumor $\phi$: Mean relative $L^2$ error for $\phi$ of the aggressive tumor growth mapping from the initial density of tumor cells with R = 0.09 mm (test case)

Table 97 Ablation study for aggressive tumor $\sigma$: Mean relative $L^2$ error for $\sigma$ of the aggressive tumor growth mapping from the initial density of tumor cells with R = 0.09 mm (test case)

Table 98 Ablation study for aggressive tumor $\phi$: Mean relative $L^2$ error for $\phi$ of the aggressive tumor growth mapping from the initial density of tumor cells with R = 0.16 mm (test case)

Table 99 Ablation study for aggressive tumor $\sigma$: Mean relative $L^2$ error for $\sigma$ of the aggressive tumor growth mapping from the initial density of tumor cells with R = 0.16 mm (test case)

Table 100 Ablation study for aggressive tumor $\phi$: Mean relative $L^2$ error for $\phi$ of the aggressive tumor growth mapping from the initial density of tumor cells with R = 0.21 mm (test case)

Table 101 Ablation study for aggressive tumor $\sigma$: Mean relative $L^2$ error for $\sigma$ of the aggressive tumor growth mapping from the initial density of tumor cells with R = 0.21 mm (test case)

Table 102 Ablation study for mild tumor $\phi$: Mean relative $L^2$ error for $\phi$ of the mild tumor growth mapping from the growth rate with $\rho$ = 1.2 1/day (test case)

Table 103 Ablation study for mild tumor $\sigma$: Mean relative $L^2$ error for $\sigma$ of the mild tumor growth mapping from the growth rate with $\rho$ = 1.2 1/day (test case)

Table 104 Ablation study for mild tumor $\phi$: Mean relative $L^2$ error for $\phi$ of the mild tumor growth mapping from the growth rate with $\rho$ = 2.1 1/day (test case)

Table 105 Ablation study for mild tumor $\sigma$: Mean relative $L^2$ error for $\sigma$ of the mild tumor growth mapping from the growth rate with $\rho$ = 2.1 1/day (test case)

Table 106 Ablation study for mild tumor $\phi$: Mean relative $L^2$ error for $\phi$ of the mild tumor growth mapping from the growth rate with $\rho$ = 2.7 1/day (test case)

Table 107 Ablation study for mild tumor $\sigma$: Mean relative $L^2$ error for $\sigma$ of the mild tumor growth mapping from the growth rate with $\rho$ = 2.7 1/day (test case)

Fig. 53

Ablation study: Prediction errors for inferring state variables for mild tumor cases mapping from the initial density of tumor cells with varying network structures. a Prediction errors for training datasets. The blue lines represent the mean of prediction errors in training datasets. b Prediction errors for testing datasets. The red lines represent the mean of prediction errors in testing datasets. The shaded region represents the region encompassed by the maximum and minimum of the prediction errors

Fig. 54

Ablation study: Prediction errors for inferring state variables for aggressive tumor cases mapping from the nutrient uptake by tumor cells with varying network structures. a Prediction errors for training datasets. The blue lines represent the mean of prediction errors in training datasets. b Prediction errors for testing datasets. The red lines represent the mean of prediction errors in testing datasets. The shaded region represents the region encompassed by the maximum and minimum of the prediction errors

Fig. 55

Grid study: Prediction errors for inferring state variables for mild tumor cases mapping from the initial density of tumor cells with varying $\omega _{data}$. a Prediction errors for training datasets. The blue lines represent the mean of prediction errors in training datasets. b Prediction errors for testing datasets. The red lines represent the mean of prediction errors in testing datasets. The shaded region represents the region encompassed by the maximum and minimum of the prediction errors

Fig. 56

Grid study: Prediction errors for inferring state variables for aggressive tumor cases mapping from the initial density of tumor cells with varying $\omega _{data}$. a Prediction errors for training datasets. The blue lines represent the mean of prediction errors in training datasets. b Prediction errors for testing datasets. The red lines represent the mean of prediction errors in testing datasets. The shaded region represents the region encompassed by the maximum and minimum of the prediction errors

Fig. 57

Grid study: Prediction errors for inferring state variables for aggressive tumor cases mapping from the nutrient uptake by tumor cells with varying number of total hidden layers. a Prediction errors for training datasets. The blue lines represent the mean of prediction errors in training datasets. b Prediction errors for testing datasets. The red lines represent the mean of prediction errors in testing datasets. The shaded region represents the region encompassed by the maximum and minimum of the prediction errors

Fig. 58

Grid study: Prediction errors for inferring state variables for mild tumor cases mapping from the initial density of tumor cells with varying number of expert networks in MoE block. a Prediction errors for training datasets. The blue lines represent the mean of prediction errors in training datasets. b Prediction errors for testing datasets. The red lines represent the mean of prediction errors in testing datasets. The shaded region represents the region encompassed by the maximum and minimum of the prediction errors

Fig. 59

Grid study: Prediction errors for inferring state variables for mild tumor cases mapping from the growth rate with varying initial learning rate. a Prediction errors for training datasets. The blue lines represent the mean of prediction errors in training datasets. b Prediction errors for testing datasets. The red lines represent the mean of prediction errors in testing datasets. The shaded region represents the region encompassed by the maximum and minimum of the prediction errors

Fig. 60

Grid study: Prediction errors for inferring state variables for mild tumor cases mapping from the growth rate with varying decay steps for the optimizer. a Prediction errors for training datasets. The blue lines represent the mean of prediction errors in training datasets. b Prediction errors for testing datasets. The red lines represent the mean of prediction errors in testing datasets. The shaded region represents the region encompassed by the maximum and minimum of the prediction errors

Fig. 61

Grid study: Prediction errors for inferring state variables for mild tumor cases mapping from the growth rate with varying activation functions. a Prediction errors for training datasets. The blue lines represent the mean of prediction errors in training datasets. b Prediction errors for testing datasets. The red lines represent the mean of prediction errors in testing datasets. The shaded region represents the region encompassed by the maximum and minimum of the prediction errors

Fig. 62

Grid study: Prediction errors for inferring state variables for mild tumor cases mapping from the growth rate with varying number of training datasets. a Prediction errors for training datasets. The blue lines represent the mean of prediction errors in training datasets. b Prediction errors for testing datasets. The red lines represent the mean of prediction errors in testing datasets. The shaded region represents the region encompassed by the maximum and minimum of the prediction errors

Fig. 63

Grid study: Prediction errors for inferring state variables for mild tumor cases mapping from the growth rate with or without boundary loss. a Prediction errors for training datasets. The blue lines represent the mean of prediction errors in training datasets. b Prediction errors for testing datasets. The red lines represent the mean of prediction errors in testing datasets. The shaded region represents the region encompassed by the maximum and minimum of the prediction errors

Fig. 64

Grid study: Prediction errors for inferring state variables for aggressive tumor cases mapping from the nutrient uptake by tumor cells with or without boundary loss. a Prediction errors for training datasets. The blue lines represent the mean of prediction errors in training datasets. b Prediction errors for testing datasets. The red lines represent the mean of prediction errors in testing datasets. The shaded region represents the region encompassed by the maximum and minimum of the prediction errors

Fig. 65

Ablation study: Prediction errors for inferring state variables for aggressive tumor cases mapping from the initial density of tumor cells with varying network structures. a Prediction errors for training datasets. The blue lines represent the mean of prediction errors in training datasets. b Prediction errors for testing datasets. The red lines represent the mean of prediction errors in testing datasets. The shaded region represents the region encompassed by the maximum and minimum of the prediction errors

Fig. 66

Grid study: Prediction errors for inferring state variables for mild tumor cases mapping from the nutrient uptake $\gamma _{c}$ with varying number of total hidden layers. a Prediction errors for training datasets. The blue lines represent the mean of prediction errors in training datasets. b Prediction errors for testing datasets. The red lines represent the mean of prediction errors in testing datasets. The shaded region represents the region encompassed by the maximum and minimum of the prediction errors

Fig. 67

Grid study: Prediction errors for inferring state variables for mild tumor cases mapping from the initial density of tumor cells with varying number of total hidden layers. a Prediction errors for training datasets. The blue lines represent the mean of prediction errors in training datasets. b Prediction errors for testing datasets. The red lines represent the mean of prediction errors in testing datasets. The shaded region represents the region encompassed by the maximum and minimum of the prediction errors

Fig. 68

Grid study: Prediction errors for inferring state variables for mild tumor cases mapping from the growth rate $\rho$ with varying number of total hidden layers. a Prediction errors for training datasets. The blue lines represent the mean of prediction errors in training datasets. b Prediction errors for testing datasets. The red lines represent the mean of prediction errors in testing datasets. The shaded region represents the region encompassed by the maximum and minimum of the prediction errors

Fig. 69

Grid study: Prediction errors for inferring state variables for aggressive tumor cases mapping from the initial density of tumor cells with varying number of total hidden layers. a Prediction errors for training datasets. The blue lines represent the mean of prediction errors in training datasets. b Prediction errors for testing datasets. The red lines represent the mean of prediction errors in testing datasets. The shaded region represents the region encompassed by the maximum and minimum of the prediction errors

Fig. 70

Comparison with SOTA models. a Prediction errors for mild tumor case with R = 0.16 mm in scenario 1 (i.e., using a sparse measurement at T = 270). b Prediction errors for aggressive tumor case with R = 0.18 mm in scenario 1 (i.e., using sparse measurements at T = 200, 300 and 400). c Prediction errors for mild tumor case with R = 0.16 mm in scenario 2 (i.e., using no additional measurements in testing domain). d Prediction errors for aggressive tumor case with R = 0.18 mm in scenario 2 (i.e., using no additional measurements in testing domain). In scenario 1, the prediction range is T$\in$[0, 270] for mild tumors while T$\in$[0, 400] for aggressive tumors. In scenario 2, the prediction range is T$\in$[0, 130] for mild tumors while T$\in$[0, 230] for aggressive tumors

Fig. 71

Comparison with SOTA models. a Prediction errors for mild tumor case with R = 0.21 mm in scenario 1 (i.e., using 1 snapshot at T = 270). b Prediction errors for aggressive tumor case with R = 0.21 mm in scenario 1 (i.e., using 3 snapshots at T = 200, 300 and 400). c Prediction errors for mild tumor case with R = 0.21 mm in scenario 2 (i.e., using no additional snapshots in testing domain). d Prediction errors for aggressive tumor case with R = 0.21 mm in scenario 2 (i.e., using no additional snapshots in testing domain). In scenario 1, the prediction range is T$\in$[0, 270] for mild tumors while T$\in$[0, 400] for aggressive tumors. In scenario 2, the prediction range is T$\in$[0, 130] for mild tumors while T$\in$[0, 230] for aggressive tumors

H. Comparison with three state-of-the-art (SOTA) models

In this subsection, we further compare the TGM-ONets ability in predicting mild tumor growth with R = 0.16 mm, 0.21 mm and aggressive tumor growth with R = 0.18 mm, 0.21 mm with three SOTA models. We consider the same two scenarios as in Sect. 5.

For the first scenario, prediction errors for mild tumor cases with R = 0.16 mm and 0.21 mm are displayed in Fig. 70a and Fig. 71a, prediction errors for aggressive tumor cases with R = 0.18 mm and 0.21 mm are displayed in Fig. 70b and Fig. 71b.

For the second scenario, prediction errors for mild tumor cases with R = 0.16 mm and 0.21 mm are displayed in Fig. 70c and Fig. 71. c, prediction errors for aggressive tumor cases with R = 0.18 mm and 0.21 mm are displayed in Fig. 70d and Fig. 71d.

From the results presented above, we can see our fine-tuning method can provide more stable and accurate prediction results compared with the other three SOTA models, which demonstrate the effectiveness and efficiency of our proposed method.

Quantitative results are also provided for all cases considered in Sect. 5 and in this subsection. Tables 108 and 109 show the mean relative $L^2$ error for mild tumor case with R = 0.08 mm in scenario 1. Tables 110 and 111 show the mean relative $L^2$ error for mild tumor case with R = 0.16 mm in scenario 1. Tables 112 and 113 show the mean relative $L^2$ error for mild tumor case with R = 0.21 mm in scenario 1.

Tables 114 and 115 show the mean relative $L^2$ error for aggressive tumor case with R = 0.05 mm in scenario 1. Tables 116 and 117 show the mean relative $L^2$ error for aggressive tumor cases with R = 0.18 mm in scenario 1. Tables 118 and 119 show the mean relative $L^2$ error for aggressive tumor cases with R = 0.21 mm in scenario 1.

Tables 120 and 121 show the mean relative $L^2$ error for mild tumor case with R = 0.08 mm in scenario 2. Tables 122 and 123 show the mean relative $L^{2}$ errors for mild tumor case with R = 0.16 mm in scenario 2. Tables 124 and 125 show the mean relative $L^{2}$ errors for mild tumor case with R = 0.21 mm in scenario 2.

Tables 126 and 127 show the mean relative $L^2$ error for aggressive tumor case with R = 0.05 mm in scenario 2. Tables 128 and 129 show the mean relative $L^2$ error for aggressive tumor case with R = 0.18 mm in scenario 2. Tables 130 and 131 show the mean relative $L^2$ error for aggressive tumor case with R = 0.21 mm in scenario 2.

Table 108 Comparison with SOTA for mild tumor $\phi$ (R = 0.08 mm) in scenario 1: Mean relative $L^2$ error for $\phi$ of the mild tumor growth mapping from the initial density of tumor cells with different SOTA methods using an additional snapshot at T = 270. TGM-ONets are trained in T$\in$[0, 70.5] while tested in T$\in$[0, 270]. $\phi$ and $\sigma$ are seen during the training of TGM-ONets in T$\in$[0, 70.5]

Table 109 Comparison with SOTA for mild tumor $\sigma$ (R = 0.08 mm) in scenario 1: Mean relative $L^2$ error for $\sigma$ of the mild tumor growth mapping from the initial density of tumor cells with different SOTA methods using an additional snapshot at T = 270. TGM-ONets are trained in T$\in$[0, 70.5] while tested in T$\in$[0, 270]. $\phi$ and $\sigma$ are seen during the training of TGM-ONets in T$\in$[0, 70.5]

Table 110 Comparison with SOTA for mild tumor $\phi$ (R = 0.16 mm) in scenario 1: Mean relative $L^2$ error for $\phi$ of the mild tumor growth mapping from the initial density of tumor cells with different SOTA methods using an additional snapshot at T = 270. TGM-ONets are trained in T$\in$[0, 70.5] while tested in T$\in$[0, 270]. $\phi$ and $\sigma$ are unseen during the training of TGM-ONets in T$\in$[0, 70.5]

Table 111 Comparison with SOTA for mild tumor $\sigma$ (R = 0.16 mm) in scenario 1: Mean relative $L^2$ error for $\sigma$ of the mild tumor growth mapping from the initial density of tumor cells with different SOTA methods using an additional snapshot at T = 270. TGM-ONets are trained in T$\in$[0, 70.5] while tested in T$\in$[0, 270]. $\phi$ and $\sigma$ are unseen during the training of TGM-ONets in T$\in$[0, 70.5]

Table 112 Comparison with SOTA for mild tumor $\phi$ (R = 0.21 mm) in scenario 1: Mean relative $L^2$ error for $\phi$ of the mild tumor growth mapping from the initial density of tumor cells with different SOTA methods using an additional snapshot at T = 270. TGM-ONets are trained in T$\in$[0, 70.5] while tested in T$\in$[0, 270]. $\phi$ and $\sigma$ are unseen during the training of TGM-ONets in T$\in$[0, 70.5]

Table 113 Comparison with SOTA for mild tumor $\sigma$ (R = 0.21 mm) in scenario 1: Mean relative $L^2$ error for $\sigma$ of the mild tumor growth mapping from the initial density of tumor cells with different SOTA methods using an additional snapshot at T = 270. TGM-ONets are trained in T$\in$[0, 70.5] while tested in T$\in$[0, 270]. $\phi$ and $\sigma$ are unseen during the training of TGM-ONets in T$\in$[0, 70.5]

Table 114 Comparison with SOTA for aggressive tumor $\phi$ (R = 0.05 mm) in scenario 1: Mean relative $L^2$ error for $\phi$ of the aggressive tumor growth mapping from the initial density of tumor cells with different SOTA methods using additional snapshots at T = 200, 300 and 400. TGM-ONets are trained in T$\in$[0, 200] while tested in T$\in$[0, 400]. $\phi$ and $\sigma$ are seen during the training of TGM-ONets in T$\in$[0, 200]

Table 115 Comparison with SOTA for aggressive tumor $\sigma$ (R = 0.05 mm) in scenario 1: Mean relative $L^2$ error for $\sigma$ of the aggressive tumor growth mapping from the initial density of tumor cells with different SOTA methods using additional 3 snapshots at T = 200, 300 and 400. TGM-ONets are trained in T$\in$[0, 200] while tested in T$\in$[0, 400]. $\phi$ and $\sigma$ are seen during the training of TGM-ONets in T$\in$[0, 200]

Table 116 Comparison with SOTA for aggressive tumor $\phi$ (R = 0.18 mm) in scenario 1: Mean relative $L^2$ error for $\phi$ of the aggressive tumor growth mapping from the initial density of tumor cells with different SOTA methods using 3 additional snapshots at T = 200, 300, 400. TGM-ONets are trained in T$\in$[0, 200] while tested in T$\in$[0, 400]. $\phi$ and $\sigma$ are unseen during the training of TGM-ONets in T$\in$[0, 200]

Table 117 Comparison with SOTA for aggressive tumor $\sigma$ (R = 0.18 mm) in scenario 1: Mean relative $L^2$ error for $\sigma$ of the aggressive tumor growth mapping from the initial density of tumor cells with different SOTA methods using 3 additional snapshots at T = 200, 300, 400. TGM-ONets are trained in T$\in$[0, 200] while tested in T$\in$[0, 400]. $\phi$ and $\sigma$ are unseen during the training of TGM-ONets in T$\in$[0, 200]

Table 118 Comparison with SOTA for aggressive tumor $\phi$ (R = 0.21 mm) in scenario 1: Mean relative $L^2$ error for $\phi$ of the aggressive tumor growth mapping from the initial density of tumor cells with different SOTA methods using 3 additional snapshots at T = 200, 300, 400. TGM-ONets are trained in T$\in$[0, 200] while tested in T$\in$[0, 400]. $\phi$ and $\sigma$ are unseen during the training of TGM-ONets in T$\in$[0, 200]

Table 119 Comparison with SOTA for aggressive tumor $\sigma$ (R = 0.21 mm) in scenario 1: Mean relative $L^2$ error for $\sigma$ of the aggressive tumor growth mapping from the initial density of tumor cells with different SOTA methods using 3 additional snapshots at T = 200, 300, 400. TGM-ONets are trained in T$\in$[0, 200] while tested in T$\in$[0, 400]. $\phi$ and $\sigma$ are unseen during the training of TGM-ONets in T$\in$[0, 200]

Table 120 Comparison with SOTA for mild tumor $\phi$ (R = 0.08 mm) in scenario 2: Mean relative $L^2$ error for $\phi$ of the mild tumor growth mapping from the initial density of tumor cells with different SOTA methods using no additional snapshots in testing domain. TGM-ONets are trained in T$\in$[0, 70.5] while tested in T$\in$[0, 130]. $\phi$ and $\sigma$ are seen during the training of TGM-ONets in T$\in$[0, 70.5]

Table 121 Comparison with SOTA for mild tumor $\sigma$ (R = 0.08 mm) in scenario 2: Mean relative $L^2$ error for $\sigma$ of the mild tumor growth mapping from the initial density of tumor cells with different SOTA methods using no additional snapshots in testing domain. TGM-ONets are trained in T$\in$[0, 70.5] while tested in T$\in$[0, 130]. $\phi$ and $\sigma$ are seen during the training of TGM-ONets in T$\in$[0, 70.5]

Table 122 Comparison with SOTA for mild tumor $\phi$ (R = 0.16 mm) in scenario 2: Mean relative $L^2$ error for $\phi$ of the mild tumor growth mapping from the initial density of tumor cells with different SOTA methods using no additional snapshots in testing domain. TGM-ONets are trained in T$\in$[0, 70.5] while tested in T$\in$[0, 130]. $\phi$ and $\sigma$ are unseen during the training of TGM-ONets in T$\in$[0, 70.5]

Table 123 Comparison with SOTA for mild tumor $\sigma$ (R = 0.16 mm) in scenario 2: Mean relative $L^2$ error for $\sigma$ of the mild tumor growth mapping from the initial density of tumor cells with different SOTA methods using no additional snapshots in testing domain. TGM-ONets are trained in T$\in$[0, 70.5] while tested in T$\in$[0, 130]. $\phi$ and $\sigma$ are unseen during the training of TGM-ONets in T$\in$[0, 70.5]

Table 124 Comparison with SOTA for mild tumor $\phi$ (R = 0.21 mm) in scenario 2: Mean relative $L^2$ error for $\phi$ of the mild tumor growth mapping from the initial density of tumor cells with different SOTA methods using no additional snapshots in testing domain. TGM-ONets are trained in T$\in$[0, 70.5] while tested in T$\in$[0, 130]. $\phi$ and $\sigma$ are unseen during the training of TGM-ONets in T$\in$[0, 70.5]

Table 125 Comparison with SOTA for mild tumor $\sigma$ (R = 0.21 mm) in scenario 2: Mean relative $L^2$ error for $\sigma$ of the mild tumor growth mapping from the initial density of tumor cells with different SOTA methods using no additional snapshots in testing domain. TGM-ONets are trained in T$\in$[0, 70.5] while tested in T$\in$[0, 130]. $\phi$ and $\sigma$ are unseen during the training of TGM-ONets in T$\in$[0, 70.5]

Table 126 Comparison with SOTA for aggressive tumor $\phi$ (R = 0.05 mm) in scenario 2: Mean relative $L^2$ error for $\phi$ of the aggressive tumor growth mapping from the initial density of tumor cells with different SOTA methods using no additional snapshots in testing domain. TGM-ONets are trained in T$\in$[0, 200] while tested in T$\in$[0, 230]. $\phi$ and $\sigma$ are seen during the training of TGM-ONets in T$\in$[0, 200]

Table 127 Comparison with SOTA for aggressive tumor $\sigma$ (R = 0.05 mm) in scenario 2: Mean relative $L^2$ error for $\sigma$ of the aggressive tumor growth mapping from the initial density of tumor cells with different SOTA methods using no additional snapshots in testing domain. TGM-ONets are trained in T$\in$[0, 200] while tested in T$\in$[0, 230]. $\phi$ and $\sigma$ are seen during the training of TGM-ONets in T$\in$[0, 200]

Table 128 Comparison with SOTA for aggressive tumor $\phi$ (R = 0.18 mm) in scenario 2: Mean relative $L^2$ error for $\phi$ of the aggressive tumor growth mapping from the initial density of tumor cells with different SOTA methods using no additional snapshots in testing domain. TGM-ONets are trained in T$\in$[0, 200] while tested in T$\in$[0, 230]. $\phi$ and $\sigma$ are unseen during the training of TGM-ONets in T$\in$[0, 200]

Table 129 Comparison with SOTA for aggressive tumor $\sigma$ (R = 0.18 mm) in scenario 2: Mean relative $L^2$ error for $\sigma$ of the aggressive tumor growth mapping from the initial density of tumor cells with different SOTA methods using no additional snapshots in testing domain. TGM-ONets are trained in T$\in$[0, 200] while tested in T$\in$[0, 230]. $\phi$ and $\sigma$ are unseen during the training of TGM-ONets in T$\in$[0, 200]

Table 130 Comparison with SOTA for aggressive tumor $\phi$ (R = 0.21 mm) in scenario 2: Mean relative $L^2$ error for $\phi$ of the aggressive tumor growth mapping from the initial density of tumor cells with different SOTA methods using no additional snapshots in testing domain. TGM-ONets are trained in T$\in$[0, 200] while tested in T$\in$[0, 230]. $\phi$ and $\sigma$ are unseen during the training of TGM-ONets in T$\in$[0, 200]

Table 131 Comparison with SOTA for aggressive tumor $\sigma$ (R = 0.21 mm) in scenario 2: Mean relative $L^2$ error for $\sigma$ of the aggressive tumor growth mapping from the initial density of tumor cells with different SOTA methods using no additional snapshots in testing domain. TGM-ONets are trained in T