Neural Shape Operator Surrogates -- Expression Rate Bounds

TL;DR

This paper establishes error bounds and convergence rates for neural and spectral operator surrogates approximating solution maps of PDEs on shape families, supporting neural operators' ability to generalize across shapes.

Contribution

It provides theoretical guarantees for neural and spectral surrogates of shape-to-solution maps, including error bounds and convergence rates, for parametric PDEs on shape families.

Findings

Neural and spectral surrogates can approximate shape-to-solution maps with guaranteed error bounds.

Holomorphy of parametric PDEs leads to finite-parametric approximation convergence.

Results support neural operators' ability to generalize across shape families.

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.