Nonlocality and Nonlinearity Implies Universality in Operator Learning
- URL: http://arxiv.org/abs/2304.13221v2
- Date: Sat, 15 Jun 2024 00:00:32 GMT
- Title: Nonlocality and Nonlinearity Implies Universality in Operator Learning
- Authors: Samuel Lanthaler, Zongyi Li, Andrew M. Stuart,
- Abstract summary: Neural operator architectures approximate operators between infinite-dimensional Banach spaces of functions.
It is clear that any general approximation of operators between spaces of functions must be both nonlocal and nonlinear.
We show how these two attributes may be combined in a simple way to deduce universal approximation.
- Score: 8.83910715280152
- License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
- 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.
Related papers
- Hilbert Neural Operator: Operator Learning in the Analytic Signal Domain [0.0]
We introduce the textbfHilbert Neural Operator (HNO), a new neural operator architecture to address some advantages.<n>HNO operates by first mapping the input signal to its analytic representation via the Hilbert transform.<n>We hypothesize that this architecture enables HNO to model operators more effectively for causal, phase-sensitive, and non-stationary systems.
arXiv Detail & Related papers (2025-08-06T21:12:15Z) - Fourier Neural Operators for Non-Markovian Processes:Approximation Theorems and Experiments [2.84475965465923]
This paper introduces an operator-based neural network, the mirror-padded neural operator (MFNO)<n>MFNO extends the standard Fourier neural operator (FNO) by incorporating mirror padding, enabling it to handle non-periodic inputs.<n>We rigorously prove that MFNOs can approximate solutions of path-dependent differential equations and transformations of fractional Brownian motions to an arbitrary degree of accuracy.
arXiv Detail & Related papers (2025-07-23T19:30:34Z) - Principled Approaches for Extending Neural Architectures to Function Spaces for Operator Learning [78.88684753303794]
Deep learning has predominantly advanced through applications in computer vision and natural language processing.<n>Neural operators are a principled way to generalize neural networks to mappings between function spaces.<n>This paper identifies and distills the key principles for constructing practical implementations of mappings between infinite-dimensional function spaces.
arXiv Detail & Related papers (2025-06-12T17:59:31Z) - Convergence analysis of wide shallow neural operators within the framework of Neural Tangent Kernel [4.313136216120379]
We conduct the convergence analysis of gradient descent for the wide shallow neural operators and physics-informed shallow neural operators within the framework of Neural Tangent Kernel (NTK)
Under the setting of over-parametrization, gradient descent can find the global minimum regardless of whether it is in continuous time or discrete time.
arXiv Detail & Related papers (2024-12-07T05:47:28Z) - DimOL: Dimensional Awareness as A New 'Dimension' in Operator Learning [63.5925701087252]
We introduce DimOL (Dimension-aware Operator Learning), drawing insights from dimensional analysis.
To implement DimOL, we propose the ProdLayer, which can be seamlessly integrated into FNO-based and Transformer-based PDE solvers.
Empirically, DimOL models achieve up to 48% performance gain within the PDE datasets.
arXiv Detail & Related papers (2024-10-08T10:48:50Z) - Linearization Turns Neural Operators into Function-Valued Gaussian Processes [23.85470417458593]
We introduce a new framework for approximate Bayesian uncertainty quantification in neural operators.
Our approach can be interpreted as a probabilistic analogue of the concept of currying from functional programming.
We showcase the efficacy of our approach through applications to different types of partial differential equations.
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) - Operator Learning: Algorithms and Analysis [8.305111048568737]
Operator learning refers to the application of ideas from machine learning to approximate operators mapping between Banach spaces of functions.
This review focuses on neural operators, built on the success of deep neural networks in the approximation of functions defined on finite dimensional Euclidean spaces.
arXiv Detail & Related papers (2024-02-24T04:40:27Z) - The Parametric Complexity of Operator Learning [6.800286371280922]
This paper aims to prove that for general classes of operators which are characterized only by their $Cr$- or Lipschitz-regularity, operator learning suffers from a curse of parametric complexity''
The second contribution of the paper is to prove that this general curse can be overcome for solution operators defined by the Hamilton-Jacobi equation.
A novel neural operator architecture is introduced, termed HJ-Net, which explicitly takes into account characteristic information of the underlying Hamiltonian system.
arXiv Detail & Related papers (2023-06-28T05:02:03Z) - 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) - Factorized Fourier Neural Operators [77.47313102926017]
The Factorized Fourier Neural Operator (F-FNO) is a learning-based method for simulating partial differential equations.
We show that our model maintains an error rate of 2% while still running an order of magnitude faster than a numerical solver.
arXiv Detail & Related papers (2021-11-27T03:34:13Z) - 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) - 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) - Provably Efficient Neural Estimation of Structural Equation Model: An
Adversarial Approach [144.21892195917758]
We study estimation in a class of generalized Structural equation models (SEMs)
We formulate the linear operator equation as a min-max game, where both players are parameterized by neural networks (NNs), and learn the parameters of these neural networks using a gradient descent.
For the first time we provide a tractable estimation procedure for SEMs based on NNs with provable convergence and without the need for sample splitting.
arXiv Detail & Related papers (2020-07-02T17:55:47Z)
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.