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PeSANet: Physics-encoded Spectral Attention Network for Simulating PDE-Governed Complex Systems

Han Wan, Rui Zhang, Qi Wang, Yang Liu, Hao Sun

TL;DR

PeSANet addresses forecasting PDE-governed complex systems under incomplete physics and scarce data by combining a hard-constrained physics-encoded local operator learner with a spectral attention-based global module. The physics-encoded block uses a $\Pi$-block to approximate unknown nonlinear operators and a PyConv layer to incorporate known FD stencils, while the spectral-enhanced block processes global dependencies in the frequency domain via an FFT-based encoder/decoder and a novel spectral attention mechanism that models inter-spectrum relationships. Experiments on 2D Burgers', FitzHugh-Nagumo, Gray-Scott, and Navier–Stokes equations show state-of-the-art long-term forecasting and strong generalization with only 2–5 training trajectories, including transfer learning across Reynolds numbers. The approach offers a data-efficient PDE surrogate framework with practical applicability to real-world systems with partial physics.

Abstract

Accurately modeling and forecasting complex systems governed by partial differential equations (PDEs) is crucial in various scientific and engineering domains. However, traditional numerical methods struggle in real-world scenarios due to incomplete or unknown physical laws. Meanwhile, machine learning approaches often fail to generalize effectively when faced with scarce observational data and the challenge of capturing local and global features. To this end, we propose the Physics-encoded Spectral Attention Network (PeSANet), which integrates local and global information to forecast complex systems with limited data and incomplete physical priors. The model consists of two key components: a physics-encoded block that uses hard constraints to approximate local differential operators from limited data, and a spectral-enhanced block that captures long-range global dependencies in the frequency domain. Specifically, we introduce a novel spectral attention mechanism to model inter-spectrum relationships and learn long-range spatial features. Experimental results demonstrate that PeSANet outperforms existing methods across all metrics, particularly in long-term forecasting accuracy, providing a promising solution for simulating complex systems with limited data and incomplete physics.

PeSANet: Physics-encoded Spectral Attention Network for Simulating PDE-Governed Complex Systems

TL;DR

PeSANet addresses forecasting PDE-governed complex systems under incomplete physics and scarce data by combining a hard-constrained physics-encoded local operator learner with a spectral attention-based global module. The physics-encoded block uses a -block to approximate unknown nonlinear operators and a PyConv layer to incorporate known FD stencils, while the spectral-enhanced block processes global dependencies in the frequency domain via an FFT-based encoder/decoder and a novel spectral attention mechanism that models inter-spectrum relationships. Experiments on 2D Burgers', FitzHugh-Nagumo, Gray-Scott, and Navier–Stokes equations show state-of-the-art long-term forecasting and strong generalization with only 2–5 training trajectories, including transfer learning across Reynolds numbers. The approach offers a data-efficient PDE surrogate framework with practical applicability to real-world systems with partial physics.

Abstract

Accurately modeling and forecasting complex systems governed by partial differential equations (PDEs) is crucial in various scientific and engineering domains. However, traditional numerical methods struggle in real-world scenarios due to incomplete or unknown physical laws. Meanwhile, machine learning approaches often fail to generalize effectively when faced with scarce observational data and the challenge of capturing local and global features. To this end, we propose the Physics-encoded Spectral Attention Network (PeSANet), which integrates local and global information to forecast complex systems with limited data and incomplete physical priors. The model consists of two key components: a physics-encoded block that uses hard constraints to approximate local differential operators from limited data, and a spectral-enhanced block that captures long-range global dependencies in the frequency domain. Specifically, we introduce a novel spectral attention mechanism to model inter-spectrum relationships and learn long-range spatial features. Experimental results demonstrate that PeSANet outperforms existing methods across all metrics, particularly in long-term forecasting accuracy, providing a promising solution for simulating complex systems with limited data and incomplete physics.
Paper Structure (18 sections, 11 equations, 3 figures, 4 tables)

This paper contains 18 sections, 11 equations, 3 figures, 4 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Data-driven learning methods
  4. Physics-aware learning methods
  5. Hybrid learning methods
  6. Methodology
  7. Problem Formulation
  8. PeSANet Architecture
  9. Physics-encoded Block
  10. Spectral-enhanced Block
  11. Frequency Domain Operator
  12. Spectral Attention Mechanism
  13. Experiments
  14. Experimental Setup
  15. Main Results
  16. ...and 3 more sections

Figures (3)

  • Figure 1: The architecture of PeSANet. (a) Core blocks: physics-encoded and spectral-enhanced blocks to learn local and global features, respectively. (b) Physics-based convolution for known PDE terms (such as $\nabla ^2 \mathbf{u}$) and $\Pi$-block for unknown terms. (c) Spectral-enhanced block: including the FFT, encoder, frequency domain operator, IFFT, and decoder. (d) Frequency domain operator. (e) Spectral attention mechanism.
  • Figure 2: An overview comparison between PeSANet and other baselines: error propagation curves (left), error boxplots (middle), and final prediction plots (right). Figures (a-d) respectively show the qualitative results for the 2D Burgers, 2D FN, 2D GS, and 2D NSE cases.
  • Figure 3: Generalization test. The error distribution and propagation of PeSANet for generalization over different Reynolds numbers.