Generalization Error Bounds for Picard-Type Operator Learning in Nonlinear Parabolic PDEs

TL;DR

This paper develops a theoretical framework for Picard-type operator learning in nonlinear parabolic PDEs, providing generalization error bounds and insights into model depth and long-term prediction.

Contribution

It introduces a novel analysis of Picard iteration-based operator learning, deriving bounds that separate implementation and estimation errors, and demonstrates the approach on nonlinear heat equations.

Findings

Increasing Picard depth reduces truncation error.

Long-time prediction can be achieved by rolling out local models.

Fourier neural operator effectively implements the theory.

Abstract

Operator learning for partial differential equations (PDEs) aims to learn solution operators on infinite-dimensional function spaces from finite-resolution data. In this setting, it is important for the learned model to be discretization-invariant, or resolution-robust, and to reflect PDE-specific structure. It is therefore natural to ask how such structure should be encoded in the model architecture, hypothesis class, or learning procedure. In this paper, we study operator learning for solution operators of nonlinear parabolic PDEs based on Duhamel--Picard iteration. We formulate Picard iteration as an abstract state-transition model and present a theoretical framework for Picard-type operator learning. We derive implementation-agnostic generalization error bounds that separate the implementation error from the estimation error associated with the abstract state-transition model induced by Picard iteration. A key consequence is that increasing the Picard depth reduces the Picard truncation error without causing an unbounded growth of the entropy-based estimation error. We also extend the analysis to long-time prediction by rolling out the same learned local model over successive time blocks. Finally, we illustrate the theory for nonlinear heat equations on the torus using a Picard-type Fourier neural operator as a concrete implementation.