Operator Learning: A Statistical Perspective

• URL: http://arxiv.org/abs/2504.03503v1
• Date: Fri, 04 Apr 2025 14:58:45 GMT
• Title: Operator Learning: A Statistical Perspective
• Authors: Unique Subedi, Ambuj Tewari,
• Abstract summary: Operator learning has emerged as a powerful tool in scientific computing for approximating mappings between infinite-dimensional function spaces.<n>We begin by formalizing operator learning as a function-to-function regression problem and review some recent developments in the field.<n>We also discuss strategies for incorporating physical and mathematical constraints into architecture design and training processes.
• Score: 17.98959620987217
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

Related papers

• MathFimer: Enhancing Mathematical Reasoning by Expanding Reasoning Steps through Fill-in-the-Middle Task [49.355810887265925]
  We introduce MathFimer, a novel framework for mathematical reasoning step expansion. We develop a specialized model, MathFimer-7B, on our carefully curated NuminaMath-FIM dataset. We then apply these models to enhance existing mathematical reasoning datasets by inserting detailed intermediate steps into their solution chains.
  arXiv  Detail & Related papers  (2025-02-17T11:22:24Z)
• DimOL: Dimensional Awareness as A New 'Dimension' in Operator Learning [60.58067866537143]
  We introduce DimOL (Dimension-aware Operator Learning), drawing insights from dimensional analysis.<n>To implement DimOL, we propose the ProdLayer, which can be seamlessly integrated into FNO-based and Transformer-based PDE solvers.<n> Empirically, DimOL models achieve up to 48% performance gain within the PDE datasets.
  arXiv  Detail & Related papers  (2024-10-08T10:48:50Z)
• DeltaPhi: Learning Physical Trajectory Residual for PDE Solving [54.13671100638092]
  We propose and formulate the Physical Trajectory Residual Learning (DeltaPhi) We learn the surrogate model for the residual operator mapping based on existing neural operator networks. We conclude that, compared to direct learning, physical residual learning is preferred for PDE solving.
  arXiv  Detail & Related papers  (2024-06-14T07:45:07Z)
• Towards a Foundation Model for Partial Differential Equations: Multi-Operator Learning and Extrapolation [4.286691905364396]
  We introduce a multi-modal foundation model for scientific problems, named PROSE-PDE.<n>Our model is a multi-operator learning approach which can predict future states of systems while concurrently learning the underlying governing equations of the physical system.
  arXiv  Detail & Related papers  (2024-04-18T17:34:20Z)
• 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)
• Parametric Learning of Time-Advancement Operators for Unstable Flame Evolution [0.0]
  This study investigates the application of machine learning to learn time-advancement operators for parametric partial differential equations (PDEs) Our focus is on extending existing operator learning methods to handle additional inputs representing PDE parameters. The goal is to create a unified learning approach that accurately predicts short-term solutions and provides robust long-term statistics.
  arXiv  Detail & Related papers  (2024-02-14T18:12:42Z)
• PICL: Physics Informed Contrastive Learning for Partial Differential Equations [7.136205674624813]
  We develop a novel contrastive pretraining framework that improves neural operator generalization across multiple governing equations simultaneously. A combination of physics-informed system evolution and latent-space model output are anchored to input data and used in our distance function. We find that physics-informed contrastive pretraining improves accuracy for the Fourier Neural Operator in fixed-future and autoregressive rollout tasks for the 1D and 2D Heat, Burgers', and linear advection equations.
  arXiv  Detail & Related papers  (2024-01-29T17:32:22Z)
• Discovering Interpretable Physical Models using Symbolic Regression and Discrete Exterior Calculus [55.2480439325792]
  We propose a framework that combines Symbolic Regression (SR) and Discrete Exterior Calculus (DEC) for the automated discovery of physical models. DEC provides building blocks for the discrete analogue of field theories, which are beyond the state-of-the-art applications of SR to physical problems. We prove the effectiveness of our methodology by re-discovering three models of Continuum Physics from synthetic experimental data.
  arXiv  Detail & Related papers  (2023-10-10T13:23:05Z)
• Neural Operators for Accelerating Scientific Simulations and Design [85.89660065887956]
  An AI framework, known as Neural Operators, presents a principled framework for learning mappings between functions defined on continuous domains. Neural Operators can augment or even replace existing simulators in many applications, such as computational fluid dynamics, weather forecasting, and material modeling.
  arXiv  Detail & Related papers  (2023-09-27T00:12:07Z)
• Self-Supervised Learning with Lie Symmetries for Partial Differential Equations [25.584036829191902]
  We learn general-purpose representations of PDEs by implementing joint embedding methods for self-supervised learning (SSL) Our representation outperforms baseline approaches to invariant tasks, such as regressing the coefficients of a PDE, while also improving the time-stepping performance of neural solvers. We hope that our proposed methodology will prove useful in the eventual development of general-purpose foundation models for PDEs.
  arXiv  Detail & Related papers  (2023-07-11T16:52:22Z)
• Advancing Reacting Flow Simulations with Data-Driven Models [50.9598607067535]
  Key to effective use of machine learning tools in multi-physics problems is to couple them to physical and computer models. The present chapter reviews some of the open opportunities for the application of data-driven reduced-order modeling of combustion systems.
  arXiv  Detail & Related papers  (2022-09-05T16:48:34Z)

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.