Infinite Neural Operators: Gaussian processes on functions

• URL: http://arxiv.org/abs/2510.16675v1
• Date: Sun, 19 Oct 2025 00:35:43 GMT
• Title: Infinite Neural Operators: Gaussian processes on functions
• Authors: Daniel Augusto de Souza, Yuchen Zhu, Harry Jake Cunningham, Yuri Saporito, Diego Mesquita, Marc Peter Deisenroth,
• Abstract summary: In this work, we extend this connection to neural operators (NOs), a class of models designed to learn mappings between function spaces.<n>We show conditions for when arbitrary-depth NOs with Gaussian-distributed convolution kernels converge to function-valued GPs.<n>We compute the posteriors of these GPs in regression scenarios, including PDE solution operators.
• Score: 18.723789296695937
• License: http://creativecommons.org/licenses/by-nc-nd/4.0/
• Abstract: A variety of infinitely wide neural architectures (e.g., dense NNs, CNNs, and transformers) induce Gaussian process (GP) priors over their outputs. These relationships provide both an accurate characterization of the prior predictive distribution and enable the use of GP machinery to improve the uncertainty quantification of deep neural networks. In this work, we extend this connection to neural operators (NOs), a class of models designed to learn mappings between function spaces. Specifically, we show conditions for when arbitrary-depth NOs with Gaussian-distributed convolution kernels converge to function-valued GPs. Based on this result, we show how to compute the covariance functions of these NO-GPs for two NO parametrizations, including the popular Fourier neural operator (FNO). With this, we compute the posteriors of these GPs in regression scenarios, including PDE solution operators. This work is an important step towards uncovering the inductive biases of current FNO architectures and opens a path to incorporate novel inductive biases for use in kernel-based operator learning methods.

Related papers

• Optimal Convergence Rates for Neural Operators [2.9388890036358104]
  We provide bounds on the number of hidden neurons and the number of second-stage samples necessary for generalization.<n>A key application of neural operators is learning surrogate maps for the solution operators of partial differential equations.
  arXiv  Detail & Related papers  (2024-12-23T12:31:38Z)
• Linearization Turns Neural Operators into Function-Valued Gaussian Processes [23.85470417458593]
  We introduce LUNO, a novel framework for approximate Bayesian uncertainty quantification in trained neural operators.<n>Our approach leverages model linearization to push (Gaussian) weight-space uncertainty forward to the neural operator's predictions.<n>We show that this can be interpreted as a probabilistic version of the concept of currying from functional programming, yielding a function-valued (Gaussian) random process belief.
  arXiv  Detail & Related papers  (2024-06-07T16:43:54Z)
• Neural Operators with Localized Integral and Differential Kernels [77.76991758980003]
  We present a principled approach to operator learning that can capture local features under two frameworks. We prove that we obtain differential operators under an appropriate scaling of the kernel values of CNNs. To obtain local integral operators, we utilize suitable basis representations for the kernels based on discrete-continuous convolutions.
  arXiv  Detail & Related papers  (2024-02-26T18:59:31Z)
• Linear Time GPs for Inferring Latent Trajectories from Neural Spike Trains [7.936841911281107]
  We propose cvHM, a general inference framework for latent GP models leveraging Hida-Mat'ern kernels and conjugate variational inference (CVI) We are able to perform variational inference of latent neural trajectories with linear time complexity for arbitrary likelihoods.
  arXiv  Detail & Related papers  (2023-06-01T16:31:36Z)
• Promises and Pitfalls of the Linearized Laplace in Bayesian Optimization [73.80101701431103]
  The linearized-Laplace approximation (LLA) has been shown to be effective and efficient in constructing Bayesian neural networks. We study the usefulness of the LLA in Bayesian optimization and highlight its strong performance and flexibility.
  arXiv  Detail & Related papers  (2023-04-17T14:23:43Z)
• Resolution-Invariant Image Classification based on Fourier Neural Operators [1.3190581566723918]
  We investigate the use of generalization Neural Operators (FNOs) for image classification in comparison to standard Convolutional Neural Networks (CNNs) We derive the FNO architecture as an example for continuous and Fr'echet-differentiable neural operators on Lebesgue spaces.
  arXiv  Detail & Related papers  (2023-04-02T10:23:36Z)
• Non-Gaussian Gaussian Processes for Few-Shot Regression [71.33730039795921]
  We propose an invertible ODE-based mapping that operates on each component of the random variable vectors and shares the parameters across all of them. NGGPs outperform the competing state-of-the-art approaches on a diversified set of benchmarks and applications.
  arXiv  Detail & Related papers  (2021-10-26T10:45:25Z)
• Neural Operator: Learning Maps Between Function Spaces [75.93843876663128]
  We propose a generalization of neural networks to learn operators, termed neural operators, that map between infinite dimensional function spaces. We prove a universal approximation theorem for our proposed neural operator, showing that it can approximate any given nonlinear continuous operator. An important application for neural operators is learning surrogate maps for the solution operators of partial differential equations.
  arXiv  Detail & Related papers  (2021-08-19T03:56:49Z)
• Deep Neural Networks as Point Estimates for Deep Gaussian Processes [44.585609003513625]
  We propose a sparse variational approximation for DGPs for which the approximate posterior mean has the same mathematical structure as a Deep Neural Network (DNN) We make the forward pass through a DGP equivalent to a ReLU DNN by finding an interdomain transformation that represents the GP posterior mean as a sum of ReLU basis functions. Experiments demonstrate improved accuracy and faster training compared to current DGP methods, while retaining favourable predictive uncertainties.
  arXiv  Detail & Related papers  (2021-05-10T16:55:17Z)
• Double-descent curves in neural networks: a new perspective using Gaussian processes [9.153116600213641]
  Double-descent curves in neural networks describe the phenomenon that the generalisation error initially descends with increasing parameters, then grows after reaching an optimal number of parameters. We use techniques from random matrix theory to characterize the spectral distribution of the empirical feature covariance matrix as a width-dependent of the spectrum of the neural network Gaussian process kernel.
  arXiv  Detail & Related papers  (2021-02-14T20:31:49Z)
• Fourier Neural Operator for Parametric Partial Differential Equations [57.90284928158383]
  We formulate a new neural operator by parameterizing the integral kernel directly in Fourier space. We perform experiments on Burgers' equation, Darcy flow, and Navier-Stokes equation. It is up to three orders of magnitude faster compared to traditional PDE solvers.
  arXiv  Detail & Related papers  (2020-10-18T00:34:21Z)
• SLEIPNIR: Deterministic and Provably Accurate Feature Expansion for Gaussian Process Regression with Derivatives [86.01677297601624]
  We propose a novel approach for scaling GP regression with derivatives based on quadrature Fourier features. We prove deterministic, non-asymptotic and exponentially fast decaying error bounds which apply for both the approximated kernel as well as the approximated posterior.
  arXiv  Detail & Related papers  (2020-03-05T14:33:20Z)

This list is automatically generated from the titles and abstracts of the papers in this site.

This site does not guarantee the quality of this site (including all information) and is not responsible for any consequences.