Neural Shape Operator Surrogates -- Expression Rate Bounds
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Abstract (arXiv)
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 Dref. The pullback of the PDE to Dref via affine-parametric shape encoding produces a collection of holomorphic parametric PDEs on Dref. Sufficient conditions for (uniformly with respect to the parameter) well-posedness are given, implying existence, uniqueness and stability of parametric solution families on Dref. 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.
Key facts. Cited by 1 paper in the map (data through 7 Oct 2026). Builds on 20 of its 75 references that are in the map. First posted on arXiv 20 Apr 2026.
Builds on
20 of 75 references.
References · 20
- 2022Fourier Neural Operator with Learned Deformations for PDEs on General Geometriescited by 416
- 2023Geometry-Informed Neural Operator for Large-Scale 3D PDEscited by 293
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- 2022Physics-Informed Deep Neural Operator Networkscited by 110
- 2025Transolver++: An Accurate Neural Solver for PDEs on Million-Scale Geometriescited by 89
- 2024Operator Learning: Algorithms and Analysiscited by 84
- 2025Geometry Aware Operator Transformer as an Efficient and Accurate Neural Surrogate for PDEs on Arbitrary Domainscited by 56
- 2025RIGNO: A Graph-based framework for robust and accurate operator learning for PDEs on arbitrary domainscited by 22
- 2022Neural and spectral operator surrogates: unified construction and expression rate boundscited by 21
- 2024Diffeomorphism Neural Operator for various domains and parameters of partial differential equationscited by 15
- 2025Geometric Operator Learning with Optimal Transportcited by 15
- 2024DIMON: Learning Solution Operators of Partial Differential Equations on a Diffeomorphic Family of Domainscited by 14
8 more
- 2024Deep Neural Operator Enabled Digital Twin Modeling for Additive Manufacturingcited by 14
- 2024Shape-informed surrogate models based on signed distance function domain encodingcited by 6
- 2024Reference Neural Operators: Learning the Smooth Dependence of Solutions of PDEs on Geometric Deformationscited by 5
- 2024Graph Fourier Neural Kernels (G-FuNK): Learning Solutions of Nonlinear Diffusive Parametric PDEs on Multiple Domainscited by 5
- 2026Shape Derivative-Informed Neural Operators with Application to Risk-Averse Shape Optimizationcited by 5
- 2026Performance of Neural and Polynomial Operator Surrogatescited by 5
- 2024Learning solution operators of PDEs defined on varying domains via MIONetcited by 3
- 2019On the numerical solution of a time-dependent shape optimization problem for the heat equationcited by 1
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Citing papers per year: 2026: 1.
1 paper.