mlsanityData to 14 Oct 2026

A Mathematical Analysis of Neural Operator Behaviors

arXiv 2410.21481 ↗First posted 28 Oct 2024math.NA cs.LG cs.NAcited by 8

Not on the map. Why?

Abstract (arXiv)

Neural operators have emerged as transformative tools for learning mappings between infinite-dimensional function spaces, offering useful applications in solving complex partial differential equations (PDEs). This paper presents a rigorous mathematical framework for analyzing the behaviors of neural operators, with a focus on their stability, convergence, clustering dynamics, universality, and generalization error. By proposing a list of novel theorems, we provide stability bounds in Sobolev spaces and demonstrate clustering in function space via gradient flow interpretation, guiding neural operator design and optimization. Based on these theoretical gurantees, we aim to offer clear and unified guidance in a single setting for the future design of neural operator-based methods.

Key facts. Cited by 8 papers in the map (data through 14 Oct 2026). Builds on 5 of its 23 references that are in the map. First posted on arXiv 28 Oct 2024.

Builds on

5 of 23 references.

References · 5

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
  2. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 2,018
  3. 2021Neural Operator: Learning Maps Between Function Spacescited by 414
  4. 2014Identifying and attacking the saddle point problem in high-dimensional non-convex optimizationcited by 289
  5. 2020Machine Learning and Computational Mathematicscited by 22

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Cited by year (2026 to 14 Oct)

Citing papers per year: 2024: 1; 2025: 4; 2026: 3.

8 papers.

  1. 2025Stochastic Operator Network: A Stochastic Maximum Principle Based Approach to Operator Learningcited by 4
  2. 2025Split Conformal Prediction in the Function Space with Neural Operatorscited by 4
  3. 2026A Score Filter Enhanced Data Assimilation Framework for Data-Driven Dynamical Systemscited by 3
  4. 2025Maximal Update Parametrization and Zero-Shot Hyperparameter Transfer for Fourier Neural Operatorscited by 1
  5. 2025ATOM: A Pretrained Neural Operator for Multitask Molecular Dynamicscited by 1
  6. 2024How Analysis Can Teach Us the Optimal Way to Design Neural Operatorscited by 0
  7. 2026Extending Neural Operators: Robust Handling of Functions Beyond the Training Setcited by 0
  8. 2026Quantitative Sobolev Approximation Bounds for Neural Operators with Empirical Validation on Burgers Equationcited by 0