Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

HYCO: A Formalism for Hybrid-Cooperative PDE Modelling

Lorenzo Liverani, Enrique Zuazua

TL;DR

Hybrid-Cooperative Learning is presented, a hybrid modeling framework that integrates physics-based and data-driven models through mutual regularization and is naturally parallelizable and demonstrates robustness to sparse and noisy data.

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 .

HYCO: A Formalism for Hybrid-Cooperative PDE Modelling

TL;DR

Hybrid-Cooperative Learning is presented, a hybrid modeling framework that integrates physics-based and data-driven models through mutual regularization and is naturally parallelizable and demonstrates robustness to sparse and noisy data.

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 .
Paper Structure (12 sections, 20 equations, 3 figures, 3 tables, 1 algorithm)

This paper contains 12 sections, 20 equations, 3 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Advantages of HYCO
  3. Methodology
  4. Problem Formulation
  5. The HYCO Algorithm
  6. Ghost Points and Efficient Computation
  7. Stopping Criterion
  8. Related Work
  9. Numerical Experiments
  10. Gray-Scott Reaction-Diffusion System
  11. Helmholtz Equation: Static Inverse Problem
  12. Conclusions and Future Work

Figures (3)

  • Figure 4.1: Gray-Scott experiment: evolution of the $u$-component. Rows show HYCO Physical, HYCO Synthetic, pure NN, PINN, and ground truth.
  • Figure 4.2: Recovered coefficients for Helmholtz equation with observations in $\mathsf{Q}_2$: $\kappa$ (top row), $\eta$ (middle row). Columns show FEM, PINN, HYCO Physical, and ground truth. Red dots indicate sensor positions.
  • Figure 4.3: Error evolution for Helmholtz experiment with observations in $\mathsf{Q}_2$. Top: data mismatch. Middle: solution error. Bottom: parameter error.