A priori error analysis of consistent PINNs for parabolic PDEs

TL;DR

This paper introduces a new a priori analysis for collocation methods in parabolic PDEs, establishing optimal recovery rates based on pointwise data and a novel loss function that reflects the PDE structure.

Contribution

It develops a new theoretical framework for analyzing consistent PINNs for parabolic PDEs, including a novel loss function and error bounds under Besov regularity.

Findings

Error bounds demonstrate near optimal recovery.

Numerical experiments confirm the effectiveness of the proposed methods.

The new loss function effectively controls approximation errors.

Abstract

We present a new a priori analysis of a class of collocation methods for parabolic PDEs that rely only on pointwise data of force term, boundary data, and initial data. Under Besov regularity assumptions, we characterize the optimal recovery rate of the solution u based on sample complexity. We establish error bounds by constructing a new consistent loss function that effectively controls the approximation error. This loss incorporates contributions from the interior, boundary, and initial data in a discretized form and is designed to reflect the true PDE structure. Our theoretical results demonstrate that minimizing this loss function yields near optimal recovery under suitable conditions of regularity and sampling. Novel practical variants of the loss function are discussed, and numerical experiments confirm the effectiveness.