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Physics-embedded Fourier Neural Network for Partial Differential Equations

Qingsong Xu, Nils Thuerey, Yilei Shi, Jonathan Bamber, Chaojun Ouyang, Xiao Xiang Zhu

TL;DR

The paper tackles the challenge of learning spatiotemporal PDEs with data-driven methods that respect physical laws and remain interpretable. It introduces Physics-embedded Fourier Neural Network (PeFNN), which enforces momentum conservation via momentum-conserving Fourier (MC-Fourier) layers and achieves interpretable nonlinear expressivity through an element-wise product, augmented by controllable error and zero-shot super-resolution. Key contributions include the MC-Fourier design with translation and rotation invariances grounded in Noether's theorem, a compact, plug-and-play architecture, and state-of-the-art performance on the incompressible Navier–Stokes and shallow water equations, plus a real-world flood forecasting benchmark with strong cross-regional transfer. The work demonstrates robust long-horizon predictions and enhanced generalization across resolutions, offering practical impact for large-scale, physics-consistent PDE solving and rapid flood forecasting, while outlining future directions to extend conservation laws beyond momentum and to non-fluid PDEs.

Abstract

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.

Physics-embedded Fourier Neural Network for Partial Differential Equations

TL;DR

The paper tackles the challenge of learning spatiotemporal PDEs with data-driven methods that respect physical laws and remain interpretable. It introduces Physics-embedded Fourier Neural Network (PeFNN), which enforces momentum conservation via momentum-conserving Fourier (MC-Fourier) layers and achieves interpretable nonlinear expressivity through an element-wise product, augmented by controllable error and zero-shot super-resolution. Key contributions include the MC-Fourier design with translation and rotation invariances grounded in Noether's theorem, a compact, plug-and-play architecture, and state-of-the-art performance on the incompressible Navier–Stokes and shallow water equations, plus a real-world flood forecasting benchmark with strong cross-regional transfer. The work demonstrates robust long-horizon predictions and enhanced generalization across resolutions, offering practical impact for large-scale, physics-consistent PDE solving and rapid flood forecasting, while outlining future directions to extend conservation laws beyond momentum and to non-fluid PDEs.

Abstract

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.
Paper Structure (34 sections, 23 equations, 18 figures, 5 tables)

This paper contains 34 sections, 23 equations, 18 figures, 5 tables.

Table of Contents

  1. Introduction
  2. Physics-aware spatiotemporal PDE learning
  3. Physics-embedded Fourier neural network
  4. Symmetries and momentum conservation laws
  5. Momentum-conserving Fourier layer
  6. Architecture and implementation
  7. Experiments
  8. Datasets
  9. Experimental settings
  10. Results and analysis
  11. Incompressible Navier-Stokes equations
  12. Shallow water equations
  13. Real-world flood simulation
  14. Conclusion
  15. Theoretical proof
  16. ...and 19 more sections

Figures (18)

  • Figure 1: The trainable variables are negated and rotated by the center point. This results in a rotation-invariant kernel. The single rotation kernel (a) exhibits superior feature expressibility, while the multiple rotation kernel (b) entails fewer parameters.
  • Figure 2: Schematic architecture of PeFNN. (a) The network with PeFNN for the Recurrent or Markov training technique; (b) Overview of the Momentum-conserving Fourier layer; (c) properties of PeFNN.
  • Figure 3: Ablation study of MC-Fourier layers on the NS equation with $\nu=1 \times 10^{-4}, T=30$.
  • Figure 4: Results and transferable results on flood simulation, all trained with the Markov strategy.
  • Figure 5: Pakistan Flood 2022: (a) Location of the study area and elevation information; (b) land use and land cover. 1: Cultivated land, 2: Forest, 3: Grass land, 4: Shrubland, 5: Wetland, 6: Water body, 7: Artificial surfaces, 8: Bare land; (c) Flood depth is determined using FwDET hawker202230, FABDEM, and SAR-based flood mapping from GloFAS Global Flood Monitoring roth2022sentinel on August 18, 2022.
  • ...and 13 more figures