PhysPDE: Rethinking PDE Discovery and a Physical Hypothesis Selection Benchmark

Existing works on recovering PDE expressions from experimental observations often involve symbolic regression. This method generally lacks the explicit incorporation of physical insights, which weaken the interpretations and effectiveness, especially in the presence of large noises. Recognizing that the primary interest of Machine Learning for Science (ML4Sci) often lies in understanding the underlying physical mechanisms or even discovering new physical laws rather than simply obtaining mathematical expressions, this paper introduces a novel ML4Sci task paradigm. It focuses on interpreting experimental data within the framework of prior physical hypotheses and theories, thereby guiding and constraining the discovery of PDE expressions. Technically, the approach is formulated as a nonlinear mixed-integer programming (MIP) problem, addressed through an efficient search scheme developed for this purpose. The experimental results on our newly designed Fluid Mechanics and Laser Fusion datasets demonstrate the interpretability and feasibility of our method. Source code and benchmarks are publicly available.