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

• URL: http://arxiv.org/abs/2506.10973v1
• Date: Thu, 12 Jun 2025 17:59:31 GMT
• Title: Principled Approaches for Extending Neural Architectures to Function Spaces for Operator Learning
• Authors: Julius Berner, Miguel Liu-Schiaffini, Jean Kossaifi, Valentin Duruisseaux, Boris Bonev, Kamyar Azizzadenesheli, Anima Anandkumar,
• Abstract summary: 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.
• Score: 78.88684753303794
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: A wide range of scientific problems, such as those described by continuous-time dynamical systems and partial differential equations (PDEs), are naturally formulated on function spaces. While function spaces are typically infinite-dimensional, deep learning has predominantly advanced through applications in computer vision and natural language processing that focus on mappings between finite-dimensional spaces. Such fundamental disparities in the nature of the data have limited neural networks from achieving a comparable level of success in scientific applications as seen in other fields. Neural operators are a principled way to generalize neural networks to mappings between function spaces, offering a pathway to replicate deep learning's transformative impact on scientific problems. For instance, neural operators can learn solution operators for entire classes of PDEs, e.g., physical systems with different boundary conditions, coefficient functions, and geometries. A key factor in deep learning's success has been the careful engineering of neural architectures through extensive empirical testing. Translating these neural architectures into neural operators allows operator learning to enjoy these same empirical optimizations. However, prior neural operator architectures have often been introduced as standalone models, not directly derived as extensions of existing neural network architectures. In this paper, we identify and distill the key principles for constructing practical implementations of mappings between infinite-dimensional function spaces. Using these principles, we propose a recipe for converting several popular neural architectures into neural operators with minimal modifications. This paper aims to guide practitioners through this process and details the steps to make neural operators work in practice. Our code can be found at https://github.com/neuraloperator/NNs-to-NOs

Related papers

• 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)
• Embedding Capabilities of Neural ODEs [0.0]
  We study input-output relations of neural ODEs using dynamical systems theory. We prove several results about the exact embedding of maps in different neural ODE architectures in low and high dimension.
  arXiv  Detail & Related papers  (2023-08-02T15:16:34Z)
• Permutation Equivariant Neural Functionals [92.0667671999604]
  This work studies the design of neural networks that can process the weights or gradients of other neural networks. We focus on the permutation symmetries that arise in the weights of deep feedforward networks because hidden layer neurons have no inherent order. In our experiments, we find that permutation equivariant neural functionals are effective on a diverse set of tasks.
  arXiv  Detail & Related papers  (2023-02-27T18:52:38Z)
• Efficient, probabilistic analysis of combinatorial neural codes [0.0]
  neural networks encode inputs in the form of combinations of individual neurons' activities. These neural codes present a computational challenge due to their high dimensionality and often large volumes of data. We apply methods previously applied to small examples and apply them to large neural codes generated by experiments.
  arXiv  Detail & Related papers  (2022-10-19T11:58:26Z)
• 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)
• A deep learning theory for neural networks grounded in physics [2.132096006921048]
  We argue that building large, fast and efficient neural networks on neuromorphic architectures requires rethinking the algorithms to implement and train them. Our framework applies to a very broad class of models, namely systems whose state or dynamics are described by variational equations.
  arXiv  Detail & Related papers  (2021-03-18T02:12:48Z)
• 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)
• Exploiting Heterogeneity in Operational Neural Networks by Synaptic Plasticity [87.32169414230822]
  Recently proposed network model, Operational Neural Networks (ONNs), can generalize the conventional Convolutional Neural Networks (CNNs) In this study the focus is drawn on searching the best-possible operator set(s) for the hidden neurons of the network based on the Synaptic Plasticity paradigm that poses the essential learning theory in biological neurons. Experimental results over highly challenging problems demonstrate that the elite ONNs even with few neurons and layers can achieve a superior learning performance than GIS-based ONNs.
  arXiv  Detail & Related papers  (2020-08-21T19:03:23Z)

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.