ComPhy: Composing Physical Models with end-to-end Alignment

Real-world phenomena typically involve multiple, interwoven dynamics that can be elegantly captured by systems of Partial Differential Equations (PDEs). However, accurately solving such systems remains a challenge. In this paper, we introduce ComPhy (CP), a novel modular framework designed to leverage the inherent physical structure of the problem to solve systems of PDEs. CP assigns each PDE to a dedicated learning module, each capable of incorporating state-of-the-art methodologies such as Physics-Informed Neural Networks or Neural Conservation Laws. Crucially, CP introduces an end-to-end alignment mechanism, explicitly designed around the physical interplay of shared variables, enabling knowledge transfer between modules, and promoting solutions that are the result of the collective effort of all modules. CP is the first approach specifically designed to tackle systems of PDEs, and our results show that it outperforms state-of-the-art approaches where a single model is trained on all PDEs at once.