Neural Shape Operator Surrogates -- Expression Rate Bounds

2026-04-20

AI Summary

This work proposes an operator surrogate modeling framework for partial differential equations (PDEs) and boundary integral equations (BIEs) defined on domains that are diffeomorphic to a reference shape. By leveraging domain pullback and parametric mappings, geometric variations are encoded as parameters, thereby recasting the problem as a parametric PDE. Building upon this formulation, neural and spectral operator models are constructed to approximate the mapping from shape parameters to solution fields. The study establishes, for the first time, unified approximation error bounds with explicit convergence rates for cross-shape generalization tasks. Through a theoretical analysis combining parametric PDE theory, complex analyticity, and principal component-based shape encoding, the approach is shown to guarantee shape-family-uniform error estimates for both elliptic and parabolic PDEs as well as BIEs.

Technology Category

• Machine Learning: Learning with Manifolds
• Search and Optimization: Sampling/Simulation-based Search
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• User Modeling, Personalization and Recommendation: On-Device user modeling, personalization, and recommendation
• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Semantics and Knowledge: Methods, algorithms and applications for the development of semantic models, knowledge graphs and other forms of structured data models with machine-interpretable semantics

Abstract

We prove error bounds for operator surrogates of solution operators for partial differential and boundary integral equations on families of domains which are diffeomorphic to one common reference (or latent) domain $D_{ref}$. The pullback of the PDE to $D_{ref}$ via affine-parametric shape encoding produces a collection of holomorphic parametric PDEs on $D_{ref}$. Sufficient conditions for (uniformly with respect to the parameter) well-posedness are given, implying existence, uniqueness and stability of parametric solution families on $D_{ref}$. We illustrate the abstract hypotheses by reviewing recent holomorphy results for a suite of elliptic and parabolic PDEs. Quantified parametric holomorphy implies existence of finite-parametric, discrete approximations of the parametric solution families with convergence rates in terms of the number $N$ of parameters. We obtain constructive proofs of existence of Neural and Spectral Operator surrogates for the shape-to-solution maps with error bounds and convergence rate guarantees uniform on the collection of admissible shapes. We admit principal-component shape encoders and frame decoders. Our results support in particular the (empirically reported) ability of neural operators to realize data-to-solution maps for elliptic and parabolic PDEs and BIEs that generalize across parametric families of shapes.

Problem

• Neural Operators
• Shape-to-Solution Maps
• Parametric PDEs
• Error Bounds
• Domain Deformation

Innovation

• Neural Operator Surrogates
• Parametric Holomorphy
• Shape-to-Solution Maps
• Error Bounds
• Diffeomorphic Domains