Full error analysis for deep learning PDE solvers

Establish a complete error analysis for deep learning-based approximation schemes for partial differential equations (including physics-informed neural networks, deep Galerkin methods, deep BSDE methods, and the deep Kolmogorov method) by rigorously bounding the overall approximation error between the exact PDE solution and the neural network realization, accounting simultaneously for approximation, sampling/generalization, and optimization errors under reasonable assumptions.

Background

The paper develops an error analysis for the deep Kolmogorov method applied to heat equations, providing bounds that decompose the total error into contributions from architecture size, sample size in the loss function, and optimization error. While classical numerical methods for PDEs, such as finite differences and finite elements, have well-established error analyses, analogous comprehensive results for deep learning methods remain limited.

The authors emphasize that, beyond partial results for specific algorithms and settings, a general and complete theory that rigorously quantifies the overall error for deep learning solvers of PDEs is still lacking. Their work advances the state of the art for the deep Kolmogorov method, but a unified full error analysis for broad classes of deep learning PDE solvers is identified as an open research problem.