融合谱信息与物理特性，提升偏微分方程求解的通用性与灵活性。
Holistic Physics Solver: Learning PDEs in a Unified Spectral-Physical Space
- 在统一空间中联合谱域与物理域信息，实现双重优势互补。
- 在多种偏微分方程上表现优于现有方法，零样本泛化能力强。
- 适合需要高精度、少数据训练的科学计算场景。
近年来，算子学习在求解偏微分方程（PDEs）方面发展出两类方法：基于注意力的方法具备点级自适应能力但缺乏谱约束；基于谱的方法提供全局连续性先验但局部灵活性不足。这一分歧阻碍了兼具强灵活性与泛化能力的PDE求解器发展。本文提出全息物理混合器（HPM），通过在统一谱-物理空间中整合谱信息与物理信息，将两类方法视为特例，并实现超越二者的能力。该框架继承谱方法的强泛化能力与注意力机制的灵活适应性，同时规避其局限性。在多种典型PDE问题上的实验表明，HPM在精度与计算效率上均优于当前最优方法，且在少量训练数据下仍具强泛化能力，在未见分辨率上表现优异，实现卓越零样本性能。
原文摘要 · Abstract (English)
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.
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