NEXT：物理信息神经谱指数时间差分架构

中文摘要 AI 辅助

物理信息神经网络（PINNs）构建时间相关偏微分方程（PDE）解的神经表示，自然融合物理知识与观测数据，使其非常适合正问题和逆问题。然而，PINNs 已知存在谱偏差和缺乏因果性的问题。神经谱架构（NeuSA）作为近期提出的 PINNs 替代方案，缓解了这两个问题，但其数值积分在处理许多相关物理问题中出现的刚性微分方程时变得不稳定。本研究提出神经谱指数时间差分架构（NEXT），将 NeuSA 中 PDE 解的谱表示与高阶指数积分器相结合。在该方法中，由 PDE 引起的向量场的线性刚性部分通过矩阵指数精确积分，而可能非线性的余项由神经网络建模。NEXT 的有效性通过一组刚性 PDE 的基准实验得到验证，在这些实验中，NEXT 保持稳定且准确，而 NeuSA 则数值发散。研究还表明，NEXT 可应用于逆问题，即模型需从稀疏数据中学习未知参数或边界条件。本工作使用的所有代码公开于：此 https URL。

英文摘要

Physics-Informed Neural Networks (PINNs) build neural representations of time-dependent PDE solutions, naturally incorporating physics knowledge and observational data, which makes them well suited to both forward and inverse PDE problems. PINNs, however, are known to suffer from spectral bias and lack of causality. Neuro-Spectral Architectures (NeuSA), a recently proposed alternative to PINNs, mitigate both issues, but their numerical integration becomes unstable for stiff differential equations arising in many relevant physical problems. This study proposes Neuro-Spectral Exponential Time Differencing Architectures (NEXT), which combines the spectral representation of the PDE solution in NeuSA with high-order exponential integrators. Within this approach, the linear stiff part of the vector field induced by the PDE is integrated exactly through matrix exponentials, while the possibly nonlinear remainder is modeled by a neural network. The effectiveness of NEXT is verified through benchmark experiments on a set of stiff PDEs, in which NEXT is stable and accurate while NeuSA diverges numerically. It is also shown that NEXT can be applied to inverse problems, where the model has to learn unknown parameters or boundary conditions from sparse data. All code used in this work is publicly available at: https://github.com/marcioh2m/next.git .

发表机构

• Instituto de Matemática Pura e Aplicada (IMPA)（巴西纯粹与应用数学研究所）
• Universidade do Estado do Rio de Janeiro (UERJ)（里约热内卢州立大学）
• Universidade Federal Fluminense (UFF)（弗鲁米嫩塞联邦大学）

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