mlsanityData to 14 Oct 2026

Learning Green's functions associated with time-dependent partial differential equations

arXiv 2204.12789 ↗First posted 27 Apr 2022math.NA cs.LG cs.NAcited by 26

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Abstract (arXiv)

Neural operators are a popular technique in scientific machine learning to learn a mathematical model of the behavior of unknown physical systems from data. Neural operators are especially useful to learn solution operators associated with partial differential equations (PDEs) from pairs of forcing functions and solutions when numerical solvers are not available or the underlying physics is poorly understood. In this work, we attempt to provide theoretical foundations to understand the amount of training data needed to learn time-dependent PDEs. Given input-output pairs from a parabolic PDE in any spatial dimension n≥ 1, we derive the first theoretically rigorous scheme for learning the associated solution operator, which takes the form of a convolution with a Green's function G. Until now, rigorously learning Green's functions associated with time-dependent PDEs has been a major challenge in the field of scientific machine learning because G may not be square-integrable when n>1, and time-dependent PDEs have transient dynamics. By combining the hierarchical low-rank structure of G together with randomized numerical linear algebra, we construct an approximant to G that achieves a relative error of smashO(Gammaε-1/2ε) in the L1-norm with high probability by using at most smashO(ε^-fracn+22log(1/ε)) input-output training pairs, where Gammaε is a measure of the quality of the training dataset for learning G, and ε>0 is sufficiently small.

Key facts. Cited by 26 papers in the map (data through 14 Oct 2026). Builds on 23 of its 67 references that are in the map. First posted on arXiv 27 Apr 2022.

Builds on

23 of 67 references.

References · 23

  1. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,908
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Citing papers per year: 2022: 1; 2023: 7; 2024: 7; 2025: 5; 2026: 6.

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  3. 2023A Mathematical Guide to Operator Learningcited by 76
  4. 2023Elliptic PDE learning is provably data-efficientcited by 26
  5. 2022Minimax Optimal Kernel Operator Learning via Multilevel Trainingcited by 13
  6. 2023Interpreting and generalizing deep learning in physics-based problems with functional linear modelscited by 9
  7. 2025A Deep Learning Framework for Multi-Operator Learning: Architectures and Approximation Theorycited by 9
  8. 2024Operator learning without the adjointcited by 8
  9. 2025Data-Efficient Kernel Methods for Learning Differential Equations and Their Solution Operators: Algorithms and Error Analysiscited by 7
  10. 2023Neural Control of Parametric Solutions for High-dimensional Evolution PDEscited by 5
  11. 2023Operator learning for hyperbolic partial differential equationscited by 5
  12. 2024Latent Neural PDE Solver: a reduced-order modelling framework for partial differential equationscited by 5
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  2. 2024Structure-Preserving Operator Learningcited by 3
  3. 2026Generalization Bounds and Statistical Guarantees for Multi-Task and Multiple Operator Learning with MNO Networkscited by 3
  4. 2023Learning Green's Function Efficiently Using Low-Rank Approximationscited by 1
  5. 2025chebgreen: Learning and Interpolating Continuous Empirical Green's Functions from Datacited by 1
  6. 2025Rates and architectures for learning geometrically non-trivial operatorscited by 1
  7. 2026Optimal Multiscale Learning of Linear Operatorscited by 1
  8. 2026Intrinsic Green's Learning: Supervised Learning on Manifolds via Inverse PDEcited by 1
  9. 2024Approximation of Solution Operators for High-dimensional PDEscited by 0
  10. 2024Adaptive Error-Bounded Hierarchical Matrices for Efficient Neural Network Compressioncited by 0
  11. 2025Learning convolution operators on compact Abelian groupscited by 0
  12. 2026A transition-density-based operator learning method for Fokker-Planck equations with various initial conditionscited by 0
  13. 2026Learning PDE Dynamics between Submanifolds Using Green's Observation Operatorscited by 0
  14. 2026Green-Routed Neural Operators: Physics Determines Where the Network Readscited by 0