REVIEW 13 cited by
Nonlocality and Nonlinearity Implies Universality in Operator Learning
Not yet reviewed by Pith; the record is open.
This paper has not been read by Pith yet. Machine review is queued; the pith claim, tier, and objections will appear here once it completes.
SPECIMEN: schema-true, not a live event
T0 review · schema-true
One-sentence machine reading of the paper's core claim.
pith:XXXXXXXX · record.json · timestamp
Signed reviews
read the original abstract
Neural operator architectures approximate operators between infinite-dimensional Banach spaces of functions. They are gaining increased attention in computational science and engineering, due to their potential both to accelerate traditional numerical methods and to enable data-driven discovery. As the field is in its infancy basic questions about minimal requirements for universal approximation remain open. It is clear that any general approximation of operators between spaces of functions must be both nonlocal and nonlinear. In this paper we describe how these two attributes may be combined in a simple way to deduce universal approximation. In so doing we unify the analysis of a wide range of neural operator architectures and open up consideration of new ones. A popular variant of neural operators is the Fourier neural operator (FNO). Previous analysis proving universal operator approximation theorems for FNOs resorts to use of an unbounded number of Fourier modes, relying on intuition from traditional analysis of spectral methods. The present work challenges this point of view: (i) the work reduces FNO to its core essence, resulting in a minimal architecture termed the ``averaging neural operator'' (ANO); and (ii) analysis of the ANO shows that even this minimal ANO architecture benefits from universal approximation. This result is obtained based on only a spatial average as its only nonlocal ingredient (corresponding to retaining only a \emph{single} Fourier mode in the special case of the FNO). The analysis paves the way for a more systematic exploration of nonlocality, both through the development of new operator learning architectures and the analysis of existing and new architectures. Numerical results are presented which give insight into complexity issues related to the roles of channel width (embedding dimension) and number of Fourier modes.
Forward citations
Cited by 13 Pith papers
-
Can neural operators always be continuously discretized?
Neural operators that are diffeomorphisms generally cannot be continuously discretized, but strongly monotone neural operators can, and bilipschitz neural operators decompose into strongly monotone layers plus a singl...
-
Adaptive Physics Transformer with Fused Global-Local Attention for Subsurface Energy Systems
APT, a mesh-agnostic neural operator fusing graph-based local features with global attention, is claimed to be the first architecture trained directly on adaptive-mesh-refinement simulations and outperforms state-of-t...
-
Self-composing neural operators for high-frequency and multiscale PDE surrogates
Self-composing neural operators that reuse one backbone block and are trained with a depth-increasing curriculum sharply reduce prediction error on Darcy flow and ultrasound tomography benchmarks while using far fewer...
-
Delay-adaptive Control of Nonlinear Systems with Approximate Neural Operator Predictors
A neural operator approximate predictor with online delay adaptation yields semi-global practical stability for nonlinear systems with unknown constant input delay.
-
A Learning-based Domain Decomposition Method
A single pretrained neural operator can act as a reusable local solver inside additive Schwarz iterations, solving elliptic PDEs with random microstructures on large, non-convex domains.
-
Learning High-dimensional Ionic Model Dynamics Using Fourier Neural Operators
A single Fourier Neural Operator can learn the full state dynamics of stiff ionic models up to 41 variables with roughly 2% relative L2 test error.
-
Point Cloud Neural Operator for Parametric PDEs on Complex and Variable Geometries
A point cloud neural operator combining Fourier integral and least-squares gradient layers approximates PDE solution maps on variable geometries with reported test errors around 0.17 percent to 7 percent.
-
Neural Operators for Predictor Feedback Control of Nonlinear Delay Systems
Neural operators can approximate the predictor mapping in nonlinear delay systems, and under a uniform error bound the closed loop is semiglobal practical stable.
-
LazyDINO: Fast, scalable, and efficiently amortized Bayesian inversion via structure-exploiting and surrogate-driven measure transport
A new amortized Bayesian inversion method trains a derivative-informed neural surrogate of the parameter-to-observable map and then uses it to optimize a lazy transport map in a low-dimensional latent space.
-
Diffeomorphic Neural Operator Learning
A neural operator that evolves fields by composing learned diffeomorphisms, enforcing relabeling symmetry and targeting conservative, non-diffusive turbulent forecasts.
-
Approximation Rates in Fr\'echet Metrics: Barron Spaces, Paley-Wiener Spaces, and Fourier Multipliers
Two theorems give sufficient shallow-network width to reach a prescribed error in a Fréchet metric of semi-norms, applied to exponential spectral Barron, Gelfand-Shilov, and bandlimited (Paley-Wiener type) symbol classes.
-
Principled Approaches for Extending Neural Architectures to Function Spaces for Operator Learning
A practical recipe to convert common neural architectures into discretization-agnostic neural operators, validated by Navier-Stokes experiments showing cross-resolution generalization of FNO-style models.
-
Bulk-boundary decomposition of neural networks
The paper reframes SGD training of deep networks as a local Lagrangian with data confined to the boundaries, but the advertised energy continuity equation is absent from the body.
Discussion (0). Continue with ORCID to comment.