HYCO: Hybrid-Cooperative Learning for Data-Driven PDE Modeling

We present Hybrid-Cooperative Learning (HYCO), a hybrid modeling framework that iteratively integrates physics-based and data-driven models through a mutual regularization mech- anism. Unlike traditional approaches that impose physical constraints directly on synthetic models, HYCO treats the physical and synthetic components as co-trained agents: the physical and syn- thetic models are nudged toward agreement, while the synthetic model is enhanced to better fit the available data. This cooperative learning scheme is naturally parallelizable and improves robust- ness to noise as well as to sparse or heterogeneous data. Extensive numerical experiments on both static and time-dependent problems demonstrate that HYCO outperforms classical physics-based and data-driven methods, recovering accurate solutions and model parameters even under ill-posed conditions. The method also admits a natural game-theoretic interpretation, enabling alternating optimization and paving the way for future theoretical developments.