HYCO: A Formalism for Hybrid-Cooperative PDE Modelling

TL;DR

HYCO is a novel hybrid modeling framework that combines physics-based and data-driven models through mutual regularization, improving robustness and accuracy in PDE problems.

Contribution

It introduces a cooperative, parallelizable scheme for hybrid PDE modeling that treats models as co-trained agents, with a game-theoretic interpretation.

Findings

HYCO recovers accurate solutions under noisy, sparse data.

It demonstrates robustness on static and dynamic benchmark problems.

The framework can be interpreted as finding a Nash equilibrium in a game.

Abstract

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 .