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

TL;DR

The paper introduces Holistic Physics Mixer (HPM), a unified framework combining spectral and physical information to improve PDE solving by enhancing flexibility, accuracy, and generalization over existing methods.

Contribution

HPM unifies spectral and attention-based PDE solvers into a single framework, enabling more powerful interactions and overcoming limitations of each approach.

Findings

HPM outperforms state-of-the-art methods in accuracy.

HPM demonstrates superior computational efficiency.

HPM maintains strong generalization with limited data.

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 simple 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.