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Operator Learning: Algorithms and Analysis

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arxiv 2402.15715 v1 pith:T7FUZL4N submitted 2024-02-24 cs.LG cs.NAmath.NA

Nikola B. Kovachki , Samuel Lanthaler , Andrew M. Stuart This is my paper
classification cs.LGcs.NAmath.NA
keywords operatorsneurallearningapproximateapproximationfunctionsmodelsoperator
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Operator learning refers to the application of ideas from machine learning to approximate (typically nonlinear) operators mapping between Banach spaces of functions. Such operators often arise from physical models expressed in terms of partial differential equations (PDEs). In this context, such approximate operators hold great potential as efficient surrogate models to complement traditional numerical methods in many-query tasks. Being data-driven, they also enable model discovery when a mathematical description in terms of a PDE is not available. This review focuses primarily on neural operators, built on the success of deep neural networks in the approximation of functions defined on finite dimensional Euclidean spaces. Empirically, neural operators have shown success in a variety of applications, but our theoretical understanding remains incomplete. This review article summarizes recent progress and the current state of our theoretical understanding of neural operators, focusing on an approximation theoretic point of view.

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Cited by 12 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Quasi-optimal hierarchically semi-separable matrix approximation

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  3. Computational Math with Neural Networks is Hard

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    Under SETH, approximating integrals, Poisson solutions, or matrix-vector products for neural network inputs requires runtime at least accuracy^{-1+o(1)}.

  4. Probabilistic operator learning: generative modeling and uncertainty quantification for foundation models of differential equations

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    ICON is shown to compute the posterior predictive mean of differential equation solutions, and a generative extension, GenICON, provides samples from this distribution for uncertainty quantification.

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  6. Diffeomorphic Neural Operator Learning

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  7. Mondrian: Transformer Operators via Domain Decomposition

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  8. Convergent Operator-Splitting Scheme for Viscosity Solutions: A Foundation for Learning Domain-to-Solution Maps

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    A splitting FEM for constrained viscosity solutions is claimed to be convergent and to serve as a blueprint for a curse-of-dimensionality-free neural operator, but key proof steps are unsubstantiated.

  9. Graph-Based Operator Learning from Limited Data on Irregular Domains

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  11. An AI Approach for Learning the Spectrum of the Laplace-Beltrami Operator

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    A graph neural network predicts eigenvalues 2-50 of the Laplace-Beltrami operator on mechanical CAD meshes about 5 times faster than FEM and is claimed accurate on 99.3% of held-out parts.

  12. Principled Approaches for Extending Neural Architectures to Function Spaces for Operator Learning

    cs.LG 2025-06 conditional novelty 4.0 of 10

    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.

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