HYCO: A Formalism for Hybrid-Cooperative PDE Modelling
For researchers in scientific machine learning, HYCO offers a robust and parallelizable approach to integrate physics and data, improving performance in ill-posed inverse problems.
HYCO introduces a hybrid-cooperative framework for PDE modeling that co-trains physics-based and data-driven models via mutual regularization, achieving accurate solution and parameter recovery under sparse and noisy data conditions.
We present Hybrid-Cooperative Learning (HYCO), a hybrid modeling framework that integrates physics-based and data-driven models through mutual regularization. Unlike traditional approaches that impose physical constraints directly on synthetic models, HYCO treats both components as co-trained agents nudged toward agreement. This cooperative scheme is naturally parallelizable and demonstrates robustness to sparse and noisy data. Numerical experiments on static and time-dependent benchmark problems show that HYCO can recover accurate solutions and model parameters under ill-posed conditions. The framework admits a game-theoretic interpretation as a Nash equilibrium problem, enabling alternating optimization. This paper is based on the extended preprint: arXiv:2509.14123 .