Generalization Error Bounds for Picard-Type Operator Learning in Nonlinear Parabolic PDEs
2026-05-11 • Machine Learning
Machine Learning
AI summaryⓘ
The authors study how to teach computers to solve complex math problems called nonlinear parabolic PDEs, which describe things like heat flow over time. They focus on a method called Picard iteration, viewing it as a system that steps through states, and build a general theory around it. Their results show that using more steps in this method reduces errors without causing the model to become too complicated to learn. They also look at making predictions over long times by applying the model repeatedly. Finally, they test their ideas on a specific example involving heat equations using a special neural network based on Fourier analysis.
Operator learningPartial differential equationsNonlinear parabolic PDEsDuhamel–Picard iterationState-transition modelGeneralization error boundsEstimation errorPicard truncation errorFourier neural operatorLong-time prediction
Authors
Koichi Taniguchi, Sho Sonoda
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.