We consider solving complex spatiotemporal dynamical systems governed by partial differential equations (PDEs) using frequency domain-based discrete learning approaches, such as Fourier neural operators. Despite their widespread use for approximating nonlinear PDEs, the majority of these methods neglect fundamental physical laws and lack interpretability. We address these shortcomings by introducing Physics-embedded Fourier Neural Networks (PeFNN) with flexible and explainable error control. PeFNN is designed to enforce momentum conservation and yields interpretable nonlinear expressions by utilizing unique multi-scale momentum-conserving Fourier (MC-Fourier) layers and an element-wise product operation. The MC-Fourier layer is by design translation- and rotation-invariant in the frequency domain, serving as a plug-and-play module that adheres to the laws of momentum conservation. PeFNN establishes a new state-of-the-art in solving widely employed spatiotemporal PDEs and generalizes well across input resolutions. Further, we demonstrate its outstanding performance for challenging real-world applications such as large-scale flood simulations.

翻译：我们考虑使用基于频域的离散学习方法，例如傅里叶神经算子，来求解由偏微分方程（PDEs）支配的复杂时空动力系统。尽管这些方法被广泛用于逼近非线性偏微分方程，但大多数忽略了基本的物理定律且缺乏可解释性。我们通过引入具有灵活且可解释误差控制的物理嵌入傅里叶神经网络（PeFNN）来解决这些不足。PeFNN旨在强制动量守恒，并通过利用独特的多尺度动量守恒傅里叶（MC-Fourier）层和逐元素乘积操作，产生可解释的非线性表达式。MC-Fourier层在设计上在频域内具有平移和旋转不变性，作为一个即插即用模块，遵循动量守恒定律。PeFNN在求解广泛应用的时空偏微分方程方面建立了新的最先进水平，并且在不同的输入分辨率上具有良好的泛化能力。此外，我们展示了其在具有挑战性的现实世界应用（如大规模洪水模拟）中的出色性能。