DRM Revisited: A Complete Error Analysis
Yuling Jiao, Ruoxuan Li, Peiying Wu, Jerry Zhijian Yang, Pingwen Zhang
TL;DR
This work delivers the first complete end-to-end error analysis for solving elliptic PDEs with the Deep Ritz Method in the over-parameterized regime. It introduces a novel error decomposition that bounds approximation, optimization, and statistical errors simultaneously, and demonstrates that the gradient-descent-based DRM can achieve a target accuracy with explicit, dimension-aware parameter scaling. The key innovation is a tighter optimization error bound that does not require the training dynamics to stay near initialization, making the theory align more closely with practical training. The results provide practitioners with principled guidance on network width, depth, weight bounds, and sampling requirements to guarantee end-to-end convergence with high probability, advancing the theoretical understanding of DRM and over-parameterized neural PDE solvers. This framework paves the way for applying similar complete error analyses to other deep-PDE solvers such as PINNs and related variational formulations.
Abstract
In this work, we address a foundational question in the theoretical analysis of the Deep Ritz Method (DRM) under the over-parameteriztion regime: Given a target precision level, how can one determine the appropriate number of training samples, the key architectural parameters of the neural networks, the step size for the projected gradient descent optimization procedure, and the requisite number of iterations, such that the output of the gradient descent process closely approximates the true solution of the underlying partial differential equation to the specified precision?