标题： Sidecar：一种保结构的神经网络求解偏微分方程框架 显示英文标题

标题： Sidecar: A Structure-Preserving Framework for Solving Partial Differential Equations with Neural Networks

摘要： 用神经网络（NNs）求解偏微分方程（PDEs）在各个科学和工程领域展现了巨大的潜力。然而，大多数现有的神经网络求解器主要关注于满足给定的PDEs，而没有明确考虑诸如质量守恒或能量耗散等内在物理特性。这一局限性可能导致不稳定或非物理的解，特别是在长期模拟中。为了解决这个问题，我们提出了Sidecar，这是一种新颖的框架，通过引入保结构知识来增强现有神经网络求解器的准确性和物理一致性。受时间相关谱重整化（TDSR）方法的启发，我们的Sidecar框架引入了一个小型副驾驶网络，该网络被训练用来指导现有的神经网络求解器以保留物理结构。该框架设计得非常灵活，能够将来自不同PDEs的保结构原理融入到各种神经网络求解器中。我们在基准PDE上的实验结果表明，现有神经网络求解器在准确性以及与保结构性质的一致性方面得到了提升。 显示英文摘要

摘要： Solving partial differential equations (PDEs) with neural networks (NNs) has shown great potential in various scientific and engineering fields. However, most existing NN solvers mainly focus on satisfying the given PDEs, without explicitly considering intrinsic physical properties such as mass conservation or energy dissipation. This limitation can result in unstable or nonphysical solutions, particularly in long-term simulations. To address this issue, we propose Sidecar, a novel framework that enhances the accuracy and physical consistency of existing NN solvers by incorporating structure-preserving knowledge. Inspired by the Time-Dependent Spectral Renormalization (TDSR) approach, our Sidecar framework introduces a small copilot network, which is trained to guide the existing NN solver in preserving physical structure. This framework is designed to be highly flexible, enabling the incorporation of structure-preserving principles from diverse PDEs into a wide range of NN solvers. Our experimental results on benchmark PDEs demonstrate the improvement of the existing neural network solvers in terms of accuracy and consistency with structure-preserving properties.