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Operator Learning: A Statistical Perspective

Unique Subedi, Ambuj Tewari

TL;DR

Operator learning frames mappings between function spaces as a function-to-function regression problem to build fast surrogates for PDE solution operators and data-driven simulators. It surveys linear, neural, and RKHS-based operator classes (e.g., DeepONet, Fourier Neural Operator), data-generation strategies via Gaussian-process inputs, and multiple error sources (approximation, truncation, discretization, statistical) while highlighting PDE-informed architectures. The article outlines active data collection and uncertainty quantification as crucial future directions, arguing that leveraging PDE structure can yield data-efficient, scalable, and reliable surrogates. By bridging functional data analysis concepts with modern operator-learning architectures, the work lays a foundation for a general statistical theory and practice that scales to complex scientific problems.

Abstract

Operator learning has emerged as a powerful tool in scientific computing for approximating mappings between infinite-dimensional function spaces. A primary application of operator learning is the development of surrogate models for the solution operators of partial differential equations (PDEs). These methods can also be used to develop black-box simulators to model system behavior from experimental data, even without a known mathematical model. In this article, we begin by formalizing operator learning as a function-to-function regression problem and review some recent developments in the field. We also discuss PDE-specific operator learning, outlining strategies for incorporating physical and mathematical constraints into architecture design and training processes. Finally, we end by highlighting key future directions such as active data collection and the development of rigorous uncertainty quantification frameworks.

Operator Learning: A Statistical Perspective

TL;DR

Operator learning frames mappings between function spaces as a function-to-function regression problem to build fast surrogates for PDE solution operators and data-driven simulators. It surveys linear, neural, and RKHS-based operator classes (e.g., DeepONet, Fourier Neural Operator), data-generation strategies via Gaussian-process inputs, and multiple error sources (approximation, truncation, discretization, statistical) while highlighting PDE-informed architectures. The article outlines active data collection and uncertainty quantification as crucial future directions, arguing that leveraging PDE structure can yield data-efficient, scalable, and reliable surrogates. By bridging functional data analysis concepts with modern operator-learning architectures, the work lays a foundation for a general statistical theory and practice that scales to complex scientific problems.

Abstract

Operator learning has emerged as a powerful tool in scientific computing for approximating mappings between infinite-dimensional function spaces. A primary application of operator learning is the development of surrogate models for the solution operators of partial differential equations (PDEs). These methods can also be used to develop black-box simulators to model system behavior from experimental data, even without a known mathematical model. In this article, we begin by formalizing operator learning as a function-to-function regression problem and review some recent developments in the field. We also discuss PDE-specific operator learning, outlining strategies for incorporating physical and mathematical constraints into architecture design and training processes. Finally, we end by highlighting key future directions such as active data collection and the development of rigorous uncertainty quantification frameworks.

Paper Structure

This paper contains 28 sections, 28 equations, 2 figures.

Table of Contents

  1. INTRODUCTION
  2. A STATISTICAL FRAMEWORK FOR OPERATOR LEARNING
  3. Loss Functions
  4. Distribution Families
  5. OPERATOR CLASSES
  6. Linear Operators
  7. Neural Operators
  8. Fourier Neural Operators (FNO)
  9. DeepOnet
  10. RKHS and Random Features
  11. DATA GENERATION, ESTIMATION, AND EVALUATION
  12. Data Generation and Sampling
  13. Estimation
  14. Evaluation & Out-of-Distribution Generalization
  15. ERROR ANALYSIS AND CONVERGENCE RATES
  16. ...and 13 more sections

Figures (2)

  • Figure 1: Input function, ground truth solution of the heat equation, and the predicted solution by a Fourier Neural Operator. Here, we use $\mathcal{X} = [0,1)^2, \tau=0.05$, and $T=1$. The corresponding code is available in this https://github.com/unique-subedi/ARSIA-review/blob/main/ARSIA_review_OL_statistical.ipynb.
  • Figure 2: Samples from $\operatorname{GP}(0, (-\nabla^2 + \mathbf{I})^{-\gamma} )$ for different values of $\gamma$ on the domain $[0,1)^2$, illustrating how $\gamma$ determines the smoothness of the generated functions.