2025/09/17 by Lorenzo Liverani, Enrique Zuazua, Liverani, Lorenzo +2 · 1 citation
#math.OC #cs.NA #math.AP #math.NA
paper · pdf · doi:10.48550/arxiv.2509.14123
We introduce Hybrid-Cooperative Learning (HYCO), a framework for data-driven PDE modeling in which a physics-based solver and a flexible synthetic model are trained as two independent but cooperating components. Rather than imposing the governing equation as a residual on a single network, HYCO alternates between updating the physical parameters and the synthetic model, coupling them through agreement of their predictions.Each component therefore fits the information available to it while acting as a regularizer for the other. This modular formulation accommodates sparse, heterogeneous, or disjoint datasets, avoids differentiating continuous PDE residuals, and combines standard solvers with general data-driven architectures. We further show that HYCO defines an exact potential game and establish equilibrium existence for a convex measure-relaxed model, providing a first structural interpretation of the alternating procedure. On inverse problems for reaction-diffusion systems, a heterogeneous Helmholtz equation, and a shock-forming traffic-flow model, HYCO reconstructs solutions and identifies parameters from sparse or localized observations, improving parameter recovery and extrapolation over uncoupled models, PINN-type methods, and classical solver-based inversion.