Holistic Physics Solver: Learning PDEs in a Unified Spectral-Physical Space
This addresses a key bottleneck in PDE solver development for scientific computing, offering a unified approach that is not incremental but integrates existing paradigms.
The paper tackles the problem of solving partial differential equations (PDEs) by bridging the gap between attention-based and spectral-based methods, resulting in a framework that outperforms state-of-the-art methods in accuracy and computational efficiency with strong generalization on limited data and unseen resolutions.
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.