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Neural Caches for Monte Carlo Partial Differential Equation Solvers

Published: 11 December 2023 Publication History

Abstract

This paper presents a method that uses neural networks as a caching mechanism to reduce the variance of Monte Carlo Partial Differential Equation solvers, such as the Walk-on-Spheres algorithm [Sawhney and Crane 2020]. While these Monte Carlo PDE solvers have the merits of being unbiased and discretization-free, their high variance often hinders real-time applications. On the other hand, neural networks can approximate the PDE solution, and evaluating these networks at inference time can be very fast. However, neural-network-based solutions may suffer from convergence difficulties and high bias. Our hybrid system aims to combine these two potentially complementary solutions by training a neural field to approximate the PDE solution using supervision from a WoS solver. This neural field is then used as a cache in the WoS solver to reduce variance during inference. We demonstrate that our neural field training procedure is better than the commonly used self-supervised objectives in the literature. We also show that our hybrid solver exhibits lower variance than WoS with the same computational budget: it is significantly better for small compute budgets and provides smaller improvements for larger budgets, reaching the same performance as WoS in the limit.

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Cited By

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  • (2024)Solving poisson equations using neural walk-on-spheresProceedings of the 41st International Conference on Machine Learning10.5555/3692070.3693583(37277-37292)Online publication date: 21-Jul-2024
  • (2024)Walkin’ Robin: Walk on Stars with Robin Boundary ConditionsACM Transactions on Graphics10.1145/365815343:4(1-18)Online publication date: 19-Jul-2024
  • (2024)Neural Control Variates with Automatic IntegrationACM SIGGRAPH 2024 Conference Papers10.1145/3641519.3657395(1-9)Online publication date: 13-Jul-2024

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cover image ACM Conferences
SA '23: SIGGRAPH Asia 2023 Conference Papers
December 2023
1113 pages
ISBN:9798400703157
DOI:10.1145/3610548
Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].

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Published: 11 December 2023

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Author Tags

  1. Geometry Processing
  2. Monte Carlo
  3. Neural Fields
  4. PDE Solver

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SA '23: SIGGRAPH Asia 2023
December 12 - 15, 2023
NSW, Sydney, Australia

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View all
  • (2024)Solving poisson equations using neural walk-on-spheresProceedings of the 41st International Conference on Machine Learning10.5555/3692070.3693583(37277-37292)Online publication date: 21-Jul-2024
  • (2024)Walkin’ Robin: Walk on Stars with Robin Boundary ConditionsACM Transactions on Graphics10.1145/365815343:4(1-18)Online publication date: 19-Jul-2024
  • (2024)Neural Control Variates with Automatic IntegrationACM SIGGRAPH 2024 Conference Papers10.1145/3641519.3657395(1-9)Online publication date: 13-Jul-2024

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