Numerical analysis of physics-informed neural networks for solving PDEs

Abstract

This thesis presents an investigation into the theory, analysis, and enhancement of Physics-Informed Neural Networks (PINNs) for solving partial differential equations (PDEs). While PINNs have gained popularity for approximating PDE solutions using machine learning, their mathematical foundations, error control mechanisms, and integration with classical numerical methods remain underdeveloped. This work addresses these gaps by developing novel energy-norm and residual-based error estimates, thereby providing the first precise convergence characterizations of PINNs in Sobolev spaces. These bounds not only establish the well-posedness of PINN formulations but also enable their use in a posteriori error estimation and adaptive refinement schemes. Building on this foundation, the thesis introduces a hybrid numerical framework that utilizes PINN residuals to identify high-error regions in Finite Difference Method (FDM) solutions to nonlinear PDEs, as a way to guide adaptive mesh refinement. The resulting FDM solvers are shown to achieve significantly improved accuracy with fewer degrees of freedom for Burgers’ and the 2D incompressible Navier-Stokes equations. This constitutes a novel and practical use of PINNs not merely as solvers, but as refinement advisors for classical methods. In addition, the thesis addresses the computational bottlenecks inherent in training large-scale PINNs by introducing a hierarchical matrix (H-matrix) compression strategy for the residual Jacobian and neural tangent kernel. This structured approximation significantly reduces memory usage and computational cost while preserving the spectral and convergence properties of the original network. Theoretical guarantees are provided for spectral stability, training error propagation, and convergence recovery under adaptive compression. These methods allow for energy-efficient PINN deployment on constrained hardware, enabling real-time PDE inference without compromising accuracy. Beyond algorithmic improvements, the thesis develops a rigorous mathematical framework for analyzing PINNs as numerical solvers. Specifically, it establishes formal definitions and proofs for energy-norm convergence, variational stability, and operator consistency in the context of both linear and nonlinear PDEs. Using tools from functional analysis and variational theory, the work derives upper and lower bounds on solution error based on the empirical residual, and shows that under mild regularity assumptions, PINNs exhibit convergence properties analogous to Galerkin methods. A sequence of results; including coercivity-driven error bounds, Ritz-type projection identities, and Gamma-convergence of the discrete residual loss; demonstrate that residual minimization in PINNs leads to convergence in Sobolev and uniform norms. These results position PINNs as a basis-free, adaptive numerical scheme that inherits the core theoretical guarantees of classical methods, while remaining flexible and mesh-free. This synthesis of theory and practice offers a principled framework for reliable PINN deployment in scientific and engineering problems governed by PDEs.