A Geometry-Aware Operator Learning Framework for Interface Problems on Varying Domains

TL;DR

This paper introduces a geometry-aware neural operator framework for efficiently solving PDE interface problems on varying domains, combining TFPM with neural networks to improve accuracy and reduce computational costs.

Contribution

It extends operator learning to general linear interface problems on varying domains by integrating TFPM, with theoretical guarantees and state-of-the-art numerical performance.

Findings

Achieves state-of-the-art accuracy in interface PDE problems

Reduces memory consumption and alleviates curse of dimensionality

Provides theoretical error estimates and domain perturbation continuity

Abstract

Solving Partial Differential Equation (PDE) interface problems on varying domains is a critical task in design and optimization, yet it remains computationally prohibitive for traditional solvers. Although operator learning has shown promise on fixed geometries, its potential for geometry-dependent interface problems has been largely unexplored. To bridge this gap, we propose an extension-based neural operator framework applicable to general linear interface problems. A key innovation of our method is the integration of the Tailored Finite Point Method (TFPM) with our base network, which reduces memory consumption and effectively alleviates the curse of dimensionality. On the theoretical front, we establish the continuity of the Helmholtz operator with respect to domain perturbations and provide rigorous error estimates for the proposed encodings. Comprehensive numerical experiments demonstrate that our framework achieves state-of-the-art accuracy and robustness. Consequently, this work provides a powerful, data-efficient tool for varying-domain simulations, offering new possibilities for real-time shape optimization.