用小网络辅助主模型，让神经网络解偏微分方程更物理可信。
A Structure-Preserving Framework for Solving Parabolic Partial Differential Equations with Neural Networks
- 引入小型协管网络，引导主网络保持质量、能量等物理守恒性。
- 在典型算例上显著提升解的精度与长期稳定性。
- 适用于多种偏微分方程，可嵌入现有神经网络求解器中。
用神经网络求解偏微分方程（PDE）在科学与工程领域展现出巨大潜力。然而，现有神经网络求解器主要关注强或弱形式下满足PDE方程，未显式考虑质量守恒、动量守恒或能量耗散等内在物理特性，导致长期模拟中可能出现非物理解或不稳定现象。为此，我们提出「Sidecar」框架，通过受时间依赖谱归一化启发的小型协管网络，增强主流神经网络求解器对结构保持性质的尊重。该框架高度灵活，可将不同PDE所需的物理守恒量集成至广泛神经网络求解器中。在若干基准问题上的实验表明，所提框架显著提升了现有神经网络求解器的准确性和结构保持能力。
原文摘要 · Abstract (English)
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 PDE formulas in the strong or weak sense, without explicitly considering some intrinsic physical properties, such as mass and momentum conservation, or energy dissipation. This limitation may result in nonphysical or unstable numerical solutions, particularly in long-term simulations. To address this issue, we propose ``Sidecar'', a novel framework that enhances the physical consistency of existing NN solvers for solving parabolic PDEs. Inspired by the time-dependent spectral renormalization approach, our Sidecar framework introduces a small network as a copilot, guiding the primary function-learning NN solver to respect the structure-preserving properties. Our framework is highly flexible, allowing the preservation of various physical quantities for different PDEs to be incorporated into a wide range of NN solvers. Experimental results on some benchmark problems demonstrate significant improvements brought by the proposed framework to both accuracy and structure preservation of existing NN solvers.
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