mlsanityData to 7 Oct 2026

Nonlinear Reconstruction for Operator Learning of PDEs with Discontinuities

arXiv 2210.01074 ↗First posted 3 Oct 2022cs.LG cs.NA math.NAcited by 29

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Abstract (arXiv)

A large class of hyperbolic and advection-dominated PDEs can have solutions with discontinuities. This paper investigates, both theoretically and empirically, the operator learning of PDEs with discontinuous solutions. We rigorously prove, in terms of lower approximation bounds, that methods which entail a linear reconstruction step (e.g. DeepONet or PCA-Net) fail to efficiently approximate the solution operator of such PDEs. In contrast, we show that certain methods employing a non-linear reconstruction mechanism can overcome these fundamental lower bounds and approximate the underlying operator efficiently. The latter class includes Fourier Neural Operators and a novel extension of DeepONet termed shift-DeepONet. Our theoretical findings are confirmed by empirical results for advection equation, inviscid Burgers' equation and compressible Euler equations of aerodynamics.

Key facts. Cited by 29 papers in the map (data through 7 Oct 2026). Builds on 20 of its 47 references that are in the map. First posted on arXiv 3 Oct 2022.

Builds on

20 of 47 references.

References · 20

  1. 2015Deep Residual Learning for Image Recognitioncited by 51,809
  2. 2014Fully Convolutional Networks for Semantic Segmentationcited by 4,845
  3. 2020Fourier Neural Operator for Parametric Partial Differential Equationscited by 2,806
  4. 2019DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operatorscited by 1,965
  5. 2022FourCastNet: A Global Data-driven High-resolution Weather Model using Adaptive Fourier Neural Operatorscited by 748
  6. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 722
  7. 2016Error bounds for approximations with deep ReLU networkscited by 567
  8. 2021Physics-Informed Neural Operator for Learning Partial Differential Equationscited by 516
  9. 2021Neural Operator: Learning Maps Between Function Spacescited by 413
  10. 2020Multipole Graph Neural Operator for Parametric Partial Differential Equationscited by 343
  11. 2020Model Reduction and Neural Networks for Parametric PDEscited by 257
  12. 2019Deep Neural Network Approximation Theorycited by 92
8 more
  1. 2022Learning Operators with Coupled Attentioncited by 88
  2. 2022Physics-informed neural networks for inverse problems in supersonic flowscited by 75
  3. 2022Generic bounds on the approximation error for physics-informed (and) operator learningcited by 65
  4. 2018Exponential Convergence of the Deep Neural Network Approximation for Analytic Functionscited by 38
  5. 2019Deep learning observables in computational fluid dynamicscited by 38
  6. 2022wPINNs: Weak Physics informed neural networks for approximating entropy solutions of hyperbolic conservation lawscited by 29
  7. 2020Iterative Surrogate Model Optimization (ISMO): An active learning algorithm for PDE constrained optimization with deep neural networkscited by 26
  8. 2022Error analysis for deep neural network approximations of parametric hyperbolic conservation lawscited by 2

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2023: 8; 2024: 9; 2025: 3; 2026: 9.

29 papers.

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  1. 2023Neural Operators for Accelerating Scientific Simulations and Designcited by 218
  2. 2023Convolutional Neural Operators for robust and accurate learning of PDEscited by 207
  3. 2024Operator Learning: Algorithms and Analysiscited by 84
  4. 2023Operator learning with PCA-Net: upper and lower complexity boundscited by 44
  5. 2023Nonlocality and Nonlinearity Implies Universality in Operator Learningcited by 43
  6. 2023Out-of-distributional risk bounds for neural operators with applications to the Helmholtz equationcited by 32
  7. 2024Recent Advances on Machine Learning for Computational Fluid Dynamics: A Surveycited by 27
  8. 2024Active Learning for Neural PDE Solverscited by 22
  9. 2023Beyond Regular Grids: Fourier-Based Neural Operators on Arbitrary Domainscited by 20
  10. 2024Positional Knowledge is All You Need: Position-induced Transformer (PiT) for Operator Learningcited by 17
  11. 2024MODNO: Multi Operator Learning With Distributed Neural Operatorscited by 14
  12. 2024Spectral-Refiner: Accurate Fine-Tuning of Spatiotemporal Fourier Neural Operator for Turbulent Flowscited by 10
17 more
  1. 2025Generating Full-field Evolution of Physical Dynamics from Irregular Sparse Observationscited by 8
  2. 2025FEDONet : Fourier-Embedded DeepONet for Spectrally Accurate Operator Learningcited by 7
  3. 2024An Overview on Machine Learning Methods for Partial Differential Equations: from Physics Informed Neural Networks to Deep Operator Learningcited by 6
  4. 2024Neural Operators Can Play Dynamic Stackelberg Gamescited by 5
  5. 2024Advancing Generalization in PINNs through Latent-Space Representationscited by 5
  6. 2025Bubbleformer: Forecasting Boiling with Transformerscited by 5
  7. 2026Diagnosing Failure Modes of Neural Operators Across Diverse PDE Familiescited by 5
  8. 2023Algorithmically Designed Artificial Neural Networks (ADANNs): Higher order deep operator learning for parametric partial differential equationscited by 1
  9. 2023An Operator Learning Framework for Spatiotemporal Super-resolution of Scientific Simulationscited by 0
  10. 2026Supervised and Unsupervised Neural Network Solver for First Order Hyperbolic Nonlinear PDEscited by 0
  11. 2026Parameterized Temporal Extrapolation of PDEs via Disentangled Latent Dynamics Manifold Fusioncited by 0
  12. 2026Multi-Head Residual-Gated DeepONet for Coherent Nonlinear Wave Dynamicscited by 0
  13. 2026Smooth Piecewise Cutting for Neural Operator to Handle Discontinuities and Sharp Transitionscited by 0
  14. 2026IV-Net: A neural network for elliptic PDEs with random and highly varying coefficientscited by 0
  15. 2026Feature Interaction Modeling for Neural Operatorscited by 0
  16. 2026MiNO: Cotangent-bundle propagator learning for PDEscited by 0
  17. 2026Selective boundary condition reduction via learned error gatingcited by 0