# Neural Shape Operator Surrogates -- Expression Rate Bounds

arXiv: https://arxiv.org/abs/2604.18012 · HTML: https://mlsanity.com/paper/2604.18012/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 20 of its 75 references that are in the map.
- First posted on arXiv 20 Apr 2026.

## Builds on (most cited first, 20 of 20)

- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Geometry-Informed Neural Operator for Large-Scale 3D PDEs](https://mlsanity.com/paper/2309.00583/) (2023)
- [DIVA: Domain Invariant Variational Autoencoders](https://mlsanity.com/paper/1905.10427/) (2019)
- [Physics-Informed Deep Neural Operator Networks](https://mlsanity.com/paper/2207.05748/) (2022)
- [Transolver++: An Accurate Neural Solver for PDEs on Million-Scale Geometries](https://mlsanity.com/paper/2502.02414/) (2025)
- [Operator Learning: Algorithms and Analysis](https://mlsanity.com/paper/2402.15715/) (2024)
- [Geometry Aware Operator Transformer as an Efficient and Accurate Neural Surrogate for PDEs on Arbitrary Domains](https://mlsanity.com/paper/2505.18781/) (2025)
- [RIGNO: A Graph-based framework for robust and accurate operator learning for PDEs on arbitrary domains](https://mlsanity.com/paper/2501.19205/) (2025)
- [Neural and spectral operator surrogates: unified construction and expression rate bounds](https://mlsanity.com/paper/2207.04950/) (2022)
- [DIMON: Learning Solution Operators of Partial Differential Equations on a Diffeomorphic Family of Domains](https://mlsanity.com/paper/2402.07250/) (2024)
- [Geometric Operator Learning with Optimal Transport](https://mlsanity.com/paper/2507.20065/) (2025)
- [Diffeomorphism Neural Operator for various domains and parameters of partial differential equations](https://mlsanity.com/paper/2402.12475/) (2024)
- [Deep Neural Operator Enabled Digital Twin Modeling for Additive Manufacturing](https://mlsanity.com/paper/2405.09572/) (2024)
- [Performance of Neural and Polynomial Operator Surrogates](https://mlsanity.com/paper/2604.00689/) (2026)
- [Shape-informed surrogate models based on signed distance function domain encoding](https://mlsanity.com/paper/2409.12400/) (2024)
- [Reference Neural Operators: Learning the Smooth Dependence of Solutions of PDEs on Geometric Deformations](https://mlsanity.com/paper/2405.17509/) (2024)
- [Graph Fourier Neural Kernels (G-FuNK): Learning Solutions of Nonlinear Diffusive Parametric PDEs on Multiple Domains](https://mlsanity.com/paper/2410.04655/) (2024)
- [Shape Derivative-Informed Neural Operators with Application to Risk-Averse Shape Optimization](https://mlsanity.com/paper/2603.03211/) (2026)
- [Learning solution operators of PDEs defined on varying domains via MIONet](https://mlsanity.com/paper/2402.15097/) (2024)
- [On the numerical solution of a time-dependent shape optimization problem for the heat equation](https://mlsanity.com/paper/1906.06221/) (2019)

## Built on by (most cited first, 1 of 1)

- [Geometry-physics confounding impairs PDE learning across varying domains](https://mlsanity.com/paper/2609.38623/) (2026)

## 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.