mlsanityData to 7 Oct 2026

Generating synthetic data for neural operators

arXiv 2401.02398 ↗First posted 4 Jan 2024cs.LG cs.NA math.NAcited by 11

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Abstract (arXiv)

Recent advances in the literature show promising potential of deep learning methods, particularly neural operators, in obtaining numerical solutions to partial differential equations (PDEs) beyond the reach of current numerical solvers. However, existing data-driven approaches often rely on training data produced by numerical PDE solvers (e.g., finite difference or finite element methods). We introduce a "backward" data generation method that avoids solving the PDE numerically: by randomly sampling candidate solutions uj from the appropriate solution space (e.g., H01(Omega)), we compute the corresponding right-hand side fj directly from the equation by differentiation. This produces training pairs (fj, uj) by computing derivatives rather than solving a PDE numerically for each data point, enabling fast, large-scale data generation consisting of exact solutions. Experiments indicate that models trained on this synthetic data generalize well when tested on data produced by standard solvers. While the idea is simple, we hope this method will expand the potential of neural PDE solvers that do not rely on classical numerical solvers to generate their data.

Key facts. Cited by 11 papers in the map (data through 7 Oct 2026). Builds on 26 of its 41 references that are in the map. First posted on arXiv 4 Jan 2024.

Builds on

26 of 41 references.

References · 26

  1. 2020Neural Operator: Graph Kernel Network for Partial Differential Equationscited by 722
  2. 2017DGM: A deep learning algorithm for solving partial differential equationscited by 674
  3. 2017The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problemscited by 581
  4. 2021Learning the solution operator of parametric partial differential equations with physics-informed DeepOnetscited by 537
  5. 2022Fourier Neural Operator with Learned Deformations for PDEs on General Geometriescited by 416
  6. 2021Neural Operator: Learning Maps Between Function Spacescited by 413
  7. 2020Multipole Graph Neural Operator for Parametric Partial Differential Equationscited by 343
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  9. 2020Model Reduction and Neural Networks for Parametric PDEscited by 257
  10. 2021Factorized Fourier Neural Operatorscited by 237
  11. 2022U-NO: U-shaped Neural Operatorscited by 219
  12. 2023Convolutional Neural Operators for robust and accurate learning of PDEscited by 207
14 more
  1. 2023Towards Foundation Models for Scientific Machine Learning: Characterizing Scaling and Transfer Behaviorcited by 128
  2. 2022NOMAD: Nonlinear Manifold Decoders for Operator Learningcited by 81
  3. 2023Neural Inverse Operators for Solving PDE Inverse Problemscited by 61
  4. 2023LNO: Laplace Neural Operator for Solving Differential Equationscited by 47
  5. 2023A graph convolutional autoencoder approach to model order reduction for parametrized PDEscited by 44
  6. 2021Data-driven discovery of Green's functions with human-understandable deep learningcited by 43
  7. 2021A Metalearning Approach for Physics-Informed Neural Networks (PINNs): Application to Parameterized PDEscited by 31
  8. 2021MOD-Net: A Machine Learning Approach via Model-Operator-Data Network for Solving PDEscited by 24
  9. 2022Enhanced DeepONet for Modeling Partial Differential Operators Considering Multiple Input Functionscited by 11
  10. 2022LordNet: An Efficient Neural Network for Learning to Solve Parametric Partial Differential Equations without Simulated Datacited by 10
  11. 2021Bayesian neural networks for weak solution of PDEs with uncertainty quantificationcited by 8
  12. 2022Non-equispaced Fourier Neural Solvers for PDEscited by 8
  13. 2023Random Grid Neural Processes for Parametric Partial Differential Equationscited by 8
  14. 2022GCN-FFNN: A Two-Stream Deep Model for Learning Solution to Partial Differential Equationscited by 1

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Cited by year (2026 to 7 Oct)

Citing papers per year: 2024: 6; 2025: 4; 2026: 1.

11 papers.

  1. 2024Neural Operators with Localized Integral and Differential Kernelscited by 59
  2. 2024Data-Efficient Operator Learning via Unsupervised Pretraining and In-Context Learningcited by 32
  3. 2024MODNO: Multi Operator Learning With Distributed Neural Operatorscited by 14
  4. 2024Neural Green's Operators for Parametric Partial Differential Equationscited by 10
  5. 2025HyPINO: Multi-Physics Neural Operators via HyperPINNs and the Method of Manufactured Solutionscited by 8
  6. 2024Data-Efficient Inference of Neural Fluid Fields via SciML Foundation Modelcited by 4
  7. 2025Coefficient-to-Basis Network: A Fine-Tunable Operator Learning Framework for Inverse Problems with Adaptive Discretizations and Theoretical Guaranteescited by 3
  8. 2025Method of Manufactured Learning for Solver-free Training of Neural Operatorscited by 3
  9. 2024Neural Parameter Regression for Explicit Representations of PDE Solution Operatorscited by 1
  10. 2026WLNO: Wavelet-Laplace Neural Operator for Solving Partial Differential Equationscited by 1
  11. 2025Reinforcement Learning Closures for Underresolved Partial Differential Equations using Synthetic Datacited by 0