mlsanityData to 14 Oct 2026

Operator learning for hyperbolic partial differential equations

arXiv 2312.17489 ↗First posted 29 Dec 2023math.NA cs.LG cs.NAcited by 5

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Abstract (arXiv)

We construct the first rigorously justified probabilistic algorithm for recovering the solution operator of a hyperbolic partial differential equation (PDE) in two variables from input-output training pairs. The primary challenge of recovering the solution operator of hyperbolic PDEs is the presence of characteristics, along which the associated Green's function is discontinuous. Therefore, a central component of our algorithm is a rank detection scheme that identifies the approximate location of the characteristics. By combining the randomized singular value decomposition with an adaptive hierarchical partition of the domain, we construct an approximant to the solution operator using O(Psiε-1ε-7log(Xiε-1ε-1)) input-output pairs with relative error O(Xiε-1ε) in the operator norm as ε→0, with high probability. Here, Psiε represents the existence of degenerate singular values of the solution operator, and Xiε measures the quality of the training data. Our assumptions on the regularity of the coefficients of the hyperbolic PDE are relatively weak given that hyperbolic PDEs do not have the ``instantaneous smoothing effect'' of elliptic and parabolic PDEs, and our recovery rate improves as the regularity of the coefficients increases.

Key facts. Cited by 5 papers in the map (data through 14 Oct 2026). Builds on 28 of its 100 references that are in the map. First posted on arXiv 29 Dec 2023.

Builds on

28 of 100 references.

References · 28

  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. 2021Characterizing possible failure modes in physics-informed neural networkscited by 643
  4. 2019DeepXDE: A deep learning library for solving differential equationscited by 619
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  6. 2020Multipole Graph Neural Operator for Parametric Partial Differential Equationscited by 349
  7. 2018Deep learning to represent sub-grid processes in climate modelscited by 156
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  11. 2020DeepGreen: Deep Learning of Green's Functions for Nonlinear Boundary Value Problemscited by 62
  12. 2022Neural Galerkin Schemes with Active Learning for High-Dimensional Evolution Equationscited by 56
16 more
  1. 2021Convergence Rates for Learning Linear Operators from Noisy Datacited by 55
  2. 2021Data-driven discovery of Green's functions with human-understandable deep learningcited by 44
  3. 2018SwitchNet: a neural network model for forward and inverse scattering problemscited by 35
  4. 2021Learning elliptic partial differential equations with randomized linear algebracited by 29
  5. 2022Learning Green's functions associated with time-dependent partial differential equationscited by 26
  6. 2023Elliptic PDE learning is provably data-efficientcited by 26
  7. 2017Compression, inversion, and approximate PCA of dense kernel matrices at near-linear computational complexitycited by 24
  8. 2019Solving Electrical Impedance Tomography with Deep Learningcited by 18
  9. 2023Deep Operator Learning Lessens the Curse of Dimensionality for PDEscited by 18
  10. 2023Randomized Sparse Neural Galerkin Schemes for Solving Evolution Equations with Deep Networkscited by 18
  11. 2019Meta-learning Pseudo-differential Operators with Deep Neural Networkscited by 17
  12. 2023Evolve Smoothly, Fit Consistently: Learning Smooth Latent Dynamics For Advection-Dominated Systemscited by 14
  13. 2023On the Limitations of Physics-informed Deep Learning: Illustrations Using First Order Hyperbolic Conservation Law-based Traffic Flow Modelscited by 14
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  15. 2023Learning-based solutions to nonlinear hyperbolic PDEs: Empirical insights on generalization errorscited by 7
  16. 2022Model Reduction for Nonlinear Systems by Balanced Truncation of State and Gradient Covariancecited by 6

Built on by

Cited by year (2026 to 14 Oct)

Citing papers per year: 2024: 1; 2025: 2; 2026: 2.

5 papers.

  1. 2024Operator Learning: Algorithms and Analysiscited by 85
  2. 2025Data-Efficient Kernel Methods for Learning Differential Equations and Their Solution Operators: Algorithms and Error Analysiscited by 7
  3. 2026Supervised and Unsupervised Neural Network Solver for First Order Hyperbolic Nonlinear PDEscited by 1
  4. 2025Query Efficient Structured Matrix Learningcited by 0
  5. 2026From Spectral Methods to Sample Complexity Bounds for Fourier Neural Operatorscited by 0