# Method of Manufactured Learning for Solver-free Training of Neural Operators

arXiv: https://arxiv.org/abs/2511.12890 · HTML: https://mlsanity.com/paper/2511.12890/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 53 of its 83 references that are in the map.
- First posted on arXiv 17 Nov 2025.

## Builds on (most cited first, 30 of 53)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Learning Mesh-Based Simulation with Graph Networks](https://mlsanity.com/paper/2010.03409/) (2020)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [Learning the solution operator of parametric partial differential equations with physics-informed DeepOnets](https://mlsanity.com/paper/2103.10974/) (2021)
- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Deep learning for universal linear embeddings of nonlinear dynamics](https://mlsanity.com/paper/1712.09707/) (2017)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [Universal Differential Equations for Scientific Machine Learning](https://mlsanity.com/paper/2001.04385/) (2020)
- [Multipole Graph Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2006.09535/) (2020)
- [GNOT: A General Neural Operator Transformer for Operator Learning](https://mlsanity.com/paper/2302.14376/) (2023)
- [Transformer for Partial Differential Equations' Operator Learning](https://mlsanity.com/paper/2205.13671/) (2022)
- [Adaptive Fourier Neural Operators: Efficient Token Mixers for Transformers](https://mlsanity.com/paper/2111.13587/) (2021)
- [Neural Operators for Accelerating Scientific Simulations and Design](https://mlsanity.com/paper/2309.15325/) (2023)
- [PDE-Refiner: Achieving Accurate Long Rollouts with Neural PDE Solvers](https://mlsanity.com/paper/2308.05732/) (2023)
- [Convolutional Neural Operators for robust and accurate learning of PDEs](https://mlsanity.com/paper/2302.01178/) (2023)
- [Neural Jump Stochastic Differential Equations](https://mlsanity.com/paper/1905.10403/) (2019)
- [Physics-Informed Deep Neural Operator Networks](https://mlsanity.com/paper/2207.05748/) (2022)
- [Learning Operators with Coupled Attention](https://mlsanity.com/paper/2201.01032/) (2022)
- [Operator Learning: Algorithms and Analysis](https://mlsanity.com/paper/2402.15715/) (2024)
- [From PINNs to PIKANs: Recent Advances in Physics-Informed Machine Learning](https://mlsanity.com/paper/2410.13228/) (2024)
- [Multi-Grid Tensorized Fourier Neural Operator for High-Resolution PDEs](https://mlsanity.com/paper/2310.00120/) (2023)
- [Convergence Rates for Learning Linear Operators from Noisy Data](https://mlsanity.com/paper/2108.12515/) (2021)
- [Derivative-Informed Neural Operator: An Efficient Framework for High-Dimensional Parametric Derivative Learning](https://mlsanity.com/paper/2206.10745/) (2022)
- [Continuum Attention for Neural Operators](https://mlsanity.com/paper/2406.06486/) (2024)
- [Domain Agnostic Fourier Neural Operators](https://mlsanity.com/paper/2305.00478/) (2023)
- [Data-Efficient Operator Learning via Unsupervised Pretraining and In-Context Learning](https://mlsanity.com/paper/2402.15734/) (2024)

## Built on by (most cited first, 3 of 3)

- [Project and Generate: Divergence-Free Neural Operators for Incompressible Flows](https://mlsanity.com/paper/2603.24500/) (2026)
- [WLNO: Wavelet-Laplace Neural Operator for Solving Partial Differential Equations](https://mlsanity.com/paper/2605.24658/) (2026)
- [Quantum SEDONet: Spectrally-Embedded Quantum Deep Operator Networks for Partial Differential Equations](https://mlsanity.com/paper/2608.27626/) (2026)

## Abstract (arXiv)

Training neural operators to approximate mappings between infinite-dimensional function spaces often requires extensive datasets generated by either demanding experimental setups or computationally expensive numerical solvers. This dependence on solver-based data limits scalability and constrains exploration across physical systems. Here we introduce the Method of Manufactured Learning (MML), a solver-independent framework for training neural operators using analytically constructed, physics-consistent datasets. Inspired by the classical method of manufactured solutions, MML replaces numerical data generation with functional synthesis, i.e., smooth candidate solutions are sampled from controlled analytical spaces, and the corresponding forcing fields are derived by direct application of the governing differential operators. During inference, setting these forcing terms to zero restores the original governing equations, allowing the trained neural operator to emulate the true solution operator of the system. The framework is agnostic to network architecture and can be integrated with any operator learning paradigm. In this paper, we employ Fourier neural operator as a representative example. Across canonical benchmarks including heat, advection, Burgers, and diffusion-reaction equations. MML achieves high spectral accuracy, low residual errors, and strong generalization to unseen conditions. By reframing data generation as a process of analytical synthesis, MML offers a scalable, solver-agnostic pathway toward constructing physically grounded neural operators that retain fidelity to governing laws without reliance on expensive numerical simulations or costly experimental data for training.