Imposing Boundary Conditions on Neural Operators via Learned Function Extensions

TL;DR

This paper introduces a framework for neural operators to handle complex boundary conditions by learning boundary-to-domain extensions, significantly improving accuracy across diverse PDE problems.

Contribution

The authors propose a novel method to condition neural operators on complex boundary conditions using learned function extensions, enhancing their flexibility and accuracy.

Findings

Achieved state-of-the-art accuracy on 18 challenging PDE datasets.

Outperformed baseline methods by large margins without hyperparameter tuning.

Demonstrated robustness across diverse geometries and boundary types.

Abstract

Neural operators have emerged as powerful surrogates for the solution of partial differential equations (PDEs), yet their ability to handle general, highly variable boundary conditions (BCs) remains limited. Existing approaches often fail when the solution operator exhibits strong sensitivity to boundary forcings. We propose a general framework for conditioning neural operators on complex non-homogeneous BCs through function extensions. Our key idea is to map boundary data to latent pseudo-extensions defined over the entire spatial domain, enabling any standard operator learning architecture to consume boundary information. The resulting operator, coupled with an arbitrary domain-to-domain neural operator, can learn rich dependencies on complex BCs and input domain functions at the same time. To benchmark this setting, we construct 18 challenging datasets spanning Poisson, linear elasticity, and hyperelasticity problems, with highly variable, mixed-type, component-wise, and multi-segment BCs on diverse geometries. Our approach achieves state-of-the-art accuracy, outperforming baselines by large margins, while requiring no hyperparameter tuning across datasets. Overall, our results demonstrate that learning boundary-to-domain extensions is an effective and practical strategy for imposing complex BCs in existing neural operator frameworks, enabling accurate and robust scientific machine learning models for a broader range of PDE-governed problems.