# Generating synthetic data for neural operators

arXiv: https://arxiv.org/abs/2401.02398 · HTML: https://mlsanity.com/paper/2401.02398/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 26 of its 41 references that are in the map.
- First posted on arXiv 4 Jan 2024.

## Builds on (most cited first, 26 of 26)

- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [DGM: A deep learning algorithm for solving partial differential equations](https://mlsanity.com/paper/1708.07469/) (2017)
- [The Deep Ritz method: A deep learning-based numerical algorithm for solving variational problems](https://mlsanity.com/paper/1710.00211/) (2017)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [Neural Operator: Learning Maps Between Function Spaces](https://mlsanity.com/paper/2108.08481/) (2021)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [PDE-Net: Learning PDEs from Data](https://mlsanity.com/paper/1710.09668/) (2017)
- [Model Reduction and Neural Networks for Parametric PDEs](https://mlsanity.com/paper/2005.03180/) (2020)
- [Factorized Fourier Neural Operators](https://mlsanity.com/paper/2111.13802/) (2021)
- [U-NO: U-shaped Neural Operators](https://mlsanity.com/paper/2204.11127/) (2022)
- [Convolutional Neural Operators for robust and accurate learning of PDEs](https://mlsanity.com/paper/2302.01178/) (2023)
- [Towards Foundation Models for Scientific Machine Learning: Characterizing Scaling and Transfer Behavior](https://mlsanity.com/paper/2306.00258/) (2023)
- [NOMAD: Nonlinear Manifold Decoders for Operator Learning](https://mlsanity.com/paper/2206.03551/) (2022)
- [Neural Inverse Operators for Solving PDE Inverse Problems](https://mlsanity.com/paper/2301.11167/) (2023)
- [LNO: Laplace Neural Operator for Solving Differential Equations](https://mlsanity.com/paper/2303.10528/) (2023)
- [Data-driven discovery of Green's functions with human-understandable deep learning](https://mlsanity.com/paper/2105.00266/) (2021)
- [A graph convolutional autoencoder approach to model order reduction for parametrized PDEs](https://mlsanity.com/paper/2305.08573/) (2023)
- [A Metalearning Approach for Physics-Informed Neural Networks (PINNs): Application to Parameterized PDEs](https://mlsanity.com/paper/2110.13361/) (2021)
- [MOD-Net: A Machine Learning Approach via Model-Operator-Data Network for Solving PDEs](https://mlsanity.com/paper/2107.03673/) (2021)
- [Enhanced DeepONet for Modeling Partial Differential Operators Considering Multiple Input Functions](https://mlsanity.com/paper/2202.08942/) (2022)
- [LordNet: An Efficient Neural Network for Learning to Solve Parametric Partial Differential Equations without Simulated Data](https://mlsanity.com/paper/2206.09418/) (2022)
- [Bayesian neural networks for weak solution of PDEs with uncertainty quantification](https://mlsanity.com/paper/2101.04879/) (2021)
- [Non-equispaced Fourier Neural Solvers for PDEs](https://mlsanity.com/paper/2212.04689/) (2022)
- [Random Grid Neural Processes for Parametric Partial Differential Equations](https://mlsanity.com/paper/2301.11040/) (2023)
- [GCN-FFNN: A Two-Stream Deep Model for Learning Solution to Partial Differential Equations](https://mlsanity.com/paper/2204.13744/) (2022)

## Built on by (most cited first, 11 of 11)

- [Neural Operators with Localized Integral and Differential Kernels](https://mlsanity.com/paper/2402.16845/) (2024)
- [Data-Efficient Operator Learning via Unsupervised Pretraining and In-Context Learning](https://mlsanity.com/paper/2402.15734/) (2024)
- [MODNO: Multi Operator Learning With Distributed Neural Operators](https://mlsanity.com/paper/2404.02892/) (2024)
- [Neural Green's Operators for Parametric Partial Differential Equations](https://mlsanity.com/paper/2406.01857/) (2024)
- [HyPINO: Multi-Physics Neural Operators via HyperPINNs and the Method of Manufactured Solutions](https://mlsanity.com/paper/2509.05117/) (2025)
- [Data-Efficient Inference of Neural Fluid Fields via SciML Foundation Model](https://mlsanity.com/paper/2412.13897/) (2024)
- [Coefficient-to-Basis Network: A Fine-Tunable Operator Learning Framework for Inverse Problems with Adaptive Discretizations and Theoretical Guarantees](https://mlsanity.com/paper/2503.08642/) (2025)
- [Method of Manufactured Learning for Solver-free Training of Neural Operators](https://mlsanity.com/paper/2511.12890/) (2025)
- [WLNO: Wavelet-Laplace Neural Operator for Solving Partial Differential Equations](https://mlsanity.com/paper/2605.24658/) (2026)
- [Neural Parameter Regression for Explicit Representations of PDE Solution Operators](https://mlsanity.com/paper/2403.12764/) (2024)
- [Reinforcement Learning Closures for Underresolved Partial Differential Equations using Synthetic Data](https://mlsanity.com/paper/2505.11308/) (2025)

## 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.