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

2025-04-14

AI Summary

Existing neural PDE solvers often neglect intrinsic physical structures—such as mass conservation and energy dissipation—leading to numerical instability or physically implausible solutions in long-term simulations. To address this, we propose a “co-piloting” collaborative control framework: a lightweight, plug-and-play auxiliary network is introduced to inject structure-preserving priors into the main solver without modifying its architecture. Leveraging time-dependent spectral renormalization (TDSR), we design a co-training mechanism that jointly optimizes physics-informed loss terms and an adaptive guidance strategy. The method exhibits strong generalizability across diverse PDE types and solver backbones. On multiple benchmark tasks, it significantly improves long-term stability, physical consistency, and structural fidelity—achieving superior accuracy over state-of-the-art baselines.

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Abstract

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.

Problem

• Enhancing accuracy of neural network PDE solvers
• Preserving physical properties in PDE solutions
• Improving long-term simulation stability with Sidecar

Innovation

• Structure-preserving framework for PDEs
• Copilot network guides NN solvers
• Enhances accuracy and physical consistency