Deep Learning of PDE Correction and Mesh Adaption Without Automatic Differentiation

Abstract

Deep learning has shown promise in solving partial differential equations (PDEs) in computational fluid dynamics, particularly for enhancing solutions from coarse-mesh simulations. However, integrating deep learning with traditional PDE solvers requires these solvers to support automatic differentiation, a feature often unavailable in existing black-box solvers. This study explores a novel training framework for hybrid models combining a black-box PDE solver and a graph neural network. By replacing the gradient of the mesh nodes positions with its estimation, we optimize both mesh parameters and network weights without requiring solver differentiation. Although the method underperforms exact differentiation in some cases, it surpasses models trained on fixed meshes. With a warm-start strategy, we achieve faster convergence and improved generalization. Our approach demonstrates that effective hybrid modeling is possible even with non-differentiable solvers, expanding accessibility to standard PDE correction workflows.