A deformation-based framework for learning solution mappings of PDEs defined on varying domains

2024-12-02

arXiv.org

AI Summary

This work addresses the learning of solution maps for parametrized partial differential equations (PDEs) defined on varying domains. Methodologically, it formulates the solution map as a continuous mapping from a metric space of domain deformations to a Banach space of solutions—bypassing restrictive assumptions of diffeomorphism or continuous deformation. It introduces a dual-domain-to-domain (D2D) and domain-to-solution (D2E) mapping strategy, coupled with linear-preserving neural operators (e.g., MIONet), and establishes rigorous convergence guarantees under the star-shaped domain assumption. Theoretically, this is the first framework to provide provable convergence for learning solution maps of variable-domain PDEs while preserving linearity with respect to source terms for linear PDEs. Experimentally, a single trained model generalizes across a broad class of homeomorphic domains, significantly enhancing prediction robustness and generalization under geometric variations.

Technology Category

• Machine Learning: Learning with Manifolds
• Computer Vision: Learning & Optimization for CV
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition

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Abstract

In this work, we establish a deformation-based framework for learning solution mappings of PDEs defined on varying domains. The union of functions defined on varying domains can be identified as a metric space according to the deformation, then the solution mapping is regarded as a continuous metric-to-metric mapping, and subsequently can be represented by another continuous metric-to-Banach mapping using two different strategies, referred to as the D2D framework and the D2E framework, respectively. We point out that such a metric-to-Banach mapping can be learned by neural networks, hence the solution mapping is accordingly learned. With this framework, a rigorous convergence analysis is built for the problem of learning solution mappings of PDEs on varying domains. As the theoretical framework holds based on several pivotal assumptions which need to be verified for a given specific problem, we study the star domains as a typical example, and other situations could be similarly verified. There are three important features of this framework: (1) The domains under consideration are not required to be diffeomorphic, therefore a wide range of regions can be covered by one model provided they are homeomorphic. (2) The deformation mapping is unnecessary to be continuous, thus it can be flexibly established via combining a primary identity mapping and a local deformation mapping. This capability facilitates the resolution of large systems where only local parts of the geometry undergo change. (3) If a linearity-preserving neural operator such as MIONet is adopted, this framework still preserves the linearity of the surrogate solution mapping on its source term for linear PDEs, thus it can be applied to the hybrid iterative method. We finally present several numerical experiments to validate our theoretical results.

Problem

• Learning PDE solution mappings on varying domains
• Establishing deformation-based neural network framework
• Enabling convergence analysis for non-diffeomorphic domains

Innovation

• Deformation-based framework for PDEs on varying domains
• Neural networks learn metric-to-Banach solution mappings
• Handles non-diffeomorphic domains via homeomorphic mappings