Physics-embedded Fourier Neural Network for Partial Differential Equations
This addresses the lack of physical law enforcement and interpretability in frequency domain-based methods for PDEs, with applications like flood simulations, though it appears incremental as it builds on existing Fourier neural operators.
The paper tackles the problem of solving complex spatiotemporal dynamical systems governed by PDEs by introducing Physics-embedded Fourier Neural Networks (PeFNN), which enforce momentum conservation and improve interpretability, achieving state-of-the-art results in solving PDEs and generalizing well across resolutions.
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.