Holistic Physics Solver: Learning PDEs in a Unified Spectral-Physical Space

Xihang Yue, Yi Yang, Linchao Zhu

TL;DR: We introduce a novel neural operator that combines attention's flexibility with spectral methods' continuity constraints through learning PDEs in a unified spectral-physical space, achieving superior performance in PDE solving.

Abstract: Recent advances in operator learning have produced two distinct approaches for solving partial differential equations (PDEs): attention-based methods offering point-level adaptability but lacking spectral constraints, and spectral-based methods providing domain-level continuity priors but limited in local flexibility. This dichotomy has hindered the development of PDE solvers with both strong flexibility and generalization capability. This work introduces Holistic Physics Mixer (HPM), a novel framework that bridges this gap by integrating spectral and physical information in a unified space. HPM unifies both approaches as special cases while enabling more powerful spectral-physical interactions beyond either method alone. This enables HPM to inherit both the strong generalization of spectral methods and the flexibility of attention mechanisms while avoiding their respective limitations. Through extensive experiments across diverse PDE problems, we demonstrate that HPM consistently outperforms state-of-the-art methods in both accuracy and computational efficiency, while maintaining strong generalization capabilities with limited training data and excellent zero-shot performance on unseen resolutions.

Lay Summary:

Scientists and engineers often need to solve complex equations, known as PDEs, to predict things like fluid flow or structural stress. Current AI methods struggle here: some capture the big picture but miss fine details, while others excel at specifics but aren't always reliable with new scenarios or limited data.

We developed the Holistic Physics Mixer (HPM), an AI framework that overcomes this challenge. It learns by processing information in a special "holistic spectral space," where these global and local perspectives are intelligently blended.

HPM significantly surpasses existing methods in accuracy and efficiency. It consistently makes accurate predictions even with sparse training data and adapts well to new conditions, such as varying levels of discretization.

Link To Code: https://github.com/yuexihang/HPM

Primary Area: Applications->Chemistry, Physics, and Earth Sciences

Keywords: PDE, Neural Operator, Attention Mechanism

Submission Number: 3357