Replacing sensors with indirect sensing techniques contributes to retaining the flexibility of soft robots. By combining physical models with recurrent neural networks (which we term a physics-informed recurrent neural network [PIRNN] approach), we implemented a hybrid prediction scheme on two typical soft pneumatic actuators: a McKibben pneumatic artificial muscle and a pneumatic-based soft finger made of silicone. The results showed that this hybrid scheme robustly enhanced the prediction accuracy to a great extent, even when combined with an inaccurate physical model. We also present the broad applicability of the PIRNN approach, showing its effectiveness for diverse types of RNNs and soft robotics platforms. Our work fills the gaps in the literature by applying a physics-informed machine-learning approach to practical engineering problems in soft robotics.