Neural Shape Operator Surrogates -- Expression Rate Bounds

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.

• The paper establishes rigorous error and expression rate bounds for neural and spectral operator surrogates using shape holomorphy of the solution maps.
• It introduces an affine-parametric framework with PCA and frame systems to transform complex shape dependencies into analytically tractable latent parameters.
• The results provide quantitative guarantees for surrogate accuracy, supporting domain-invariant operator learning and advancing scientific machine learning applications.

Neural Shape Operator Surrogates: Expression Rate Bounds

Mathematical Framework for Shape-to-Solution Operator Learning

The paper develops a rigorous mathematical foundation for operator surrogates—neural and spectral—for solution maps of partial differential and boundary integral equations (PDEs/BIEs) across parametric families of domains. Central to the analysis is the concept of diffeomorphic domain parametrization, encoding admissible shapes via affine-parametric deformations anchored to a reference domain. The abstract hypotheses, verified for broad classes of elliptic and parabolic PDEs—linear and nonlinear—establish uniform stability, existence, and holomorphy of parametric solution families on the reference domain.

Shape encoding is realized through principal-component (PCA) bases or frame systems, yielding latent shape coordinates that define the deformation vector field. The pullback of the PDE to the latent domain transforms the shape dependence problem into analytic dependence w.r.t. latent parameters, producing holomorphic parametric PDEs. Sufficient conditions for uniform well-posedness are provided, ensuring bounded inverse operators and stability constants independent of the shape.

Analyticity and Shape Holomorphy of Solution Maps

An essential innovation is the establishment of quantified shape holomorphy for a suite of model operator equations. The paper proves that, under affine-parametric domain deformation, solution maps admit analytic dependence on shape parameters. For numerous PDEs—including Poisson, heterogeneous media, linear elasticity, stationary Navier-Stokes, and boundary integral formulations for wave scattering—the paper rigorously derives estimates on the high-order derivatives of the solution with respect to latent domain parameters. The key analytic bound is

$$\|\partial_\mathbf{y}^\alpha \hat{u}_\mathbf{y}\|_{H^1(D_0)} \lesssim |\alpha|! \rho^{|\alpha|}$$

for multi-index $\alpha$, with $\rho>0$ dependent on deformation regularity, affording exponential decay in the coordinate dimension. This strongly supports the existence of low-complexity surrogate models in the latent shape space.

The analysis systematically extends to Gevrey-class regularity and non-affine shape encodings, covering nonlinear phenomena and composite materials, and includes parabolic evolution problems with analytic dependence in space-time Bochner settings.

Surrogate Operator Architectures and Expression Rate Guarantees

The principal technical result is the constructive proof of existence and approximation rate bounds for shape-to-solution neural and spectral operator surrogates. Neural operator surrogates are architected as explicit shape encoder/solution decoder compositions, leveraging frames, Riesz bases, and PCA encodings. The foundational theorem asserts:

• For a holomorphic shape-to-solution map and shape parameter regularity $s>1$, surrogates with $N$ parameters achieve uniform worst-case error bounds of $O(N^{-\min\{s-1, t\}+\delta})$, where $t$ is a solution smoothness index and $\delta$ arbitrarily small.

Spectral surrogate results parallel these bounds, with constructive interpolating polynomial architectures achievable. The mean-square error bounds are improved for Riesz basis shape encodings, achieving rate $O(N^{-\min\{s-1/2, t\}+\delta})$ for shape distributions. The proofs rely on holomorphic coordinate-to-solution mapping results and optimal approximation theory in Hilbert settings [HSZ].

The encoding/decoding formalism encompasses practical constructions: PCA, wavelets, B-splines, finite-element multiresolution frames, and admits general frame-based systems. The approach enables seamless shape-transfer learning, supporting operator surrogate generalization across variable domains without retraining.

Implications and Potential Directions in Scientific Machine Learning

These rigorous expression rate bounds mathematically justify empirical successes observed in operator learning for domain-invariant PDE solvers, including neural operators that generalize across shape ensembles [LiHuangFNOGeo23, yin2024dimonlearningsolutionoperators]. The formalism addresses domain parametrization, encoding regularity, and operator surrogate design, encompassing recent architectures such as Shape-DINO, shape-FNO, and domain-invariant autoencoders.

Practically, the results provide quantitative guarantees for surrogate accuracy and generalization in engineering simulation, uncertainty quantification, shape optimization, and digital twin modeling [gong2026shapederivativeinformedneuraloperators, liu2024deepneuraloperatorenabled]. The analytic dependence results support transfer learning and rapid design workflows.

Theoretically, these findings open avenues for extending operator surrogate analysis to point-cloud and mesh-free geometries (measure-transport formalism [li2025geometricoperatorlearningoptimal]), nonlinear encoders, and further regularity regimes (e.g., log-affine or variational autoencoders).

Conclusion

The paper establishes a rigorous operator learning framework for shape-parametric PDEs and BIEs, proving uniform error and expression rate bounds for both neural and spectral operator surrogates. By leveraging shape holomorphy, affine-parametric encoding, and analytic solution map regularity, it delivers a comprehensive mathematical justification for the generalization and efficiency of operator networks across variable domains. These results underpin much of the recent algorithmic innovation in shape-agnostic scientific machine learning and provide a foundation for continued advancements in surrogate modeling, transfer learning, and domain-invariant operator approximation (2604.18012).