Bridging Continuous and Discrete Physics: A Hybrid PINN Framework with Differentiable Solvers

Guillermo Ricardo Moreno Carrillo

Keywords: Differentiable physics, Phisically-Informed Machinea learning, Partial Differential Equations

Abstract: Physics-Informed Neural Networks (PINNs) provide a flexible approach for solving partial differential equations (PDEs) without labeled data, yet their performance often degrades under varying boundary conditions or coarse discretizations. We propose a hybrid framework that integrates PINNs with differentiable solvers to learn mappings between imperfect and refined physical representations. Instead of enforcing hard boundary or initial constraints, our model leverages PDE residual optimization to assist a differentiable-physics component that refines coarse or low-accuracy numerical solutions. The network acts as a correction operator, improving solver outputs through gradient feedback derived from the discrete numerical process. We evaluate the method on single-phase Darcy flow, diffusion, and Poisson equations, training on fine simulations and testing under coarse spatial--temporal discretizations or higher solver tolerances. Results show improved generalization across discretization regimes, suggesting that coupling continuous residual learning with differentiable solvers offers a promising direction for robust, data-free PDE modeling.

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Submission Number: 46