# Mathematical artificial data for operator learning

arXiv: https://arxiv.org/abs/2507.06752 · HTML: https://mlsanity.com/paper/2507.06752/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 23 references that are in the map.
- First posted on arXiv 9 Jul 2025.

## Builds on (most cited first, 8 of 8)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Gaussian Error Linear Units (GELUs)](https://mlsanity.com/paper/1606.08415/) (2016)
- [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)
- [Neural Operator: Graph Kernel Network for Partial Differential Equations](https://mlsanity.com/paper/2003.03485/) (2020)
- [MIONet: Learning multiple-input operators via tensor product](https://mlsanity.com/paper/2202.06137/) (2022)
- [Deep Learning Methods for Reynolds-Averaged Navier-Stokes Simulations of Airfoil Flows](https://mlsanity.com/paper/1810.08217/) (2018)
- [Deep neural network solution of the electronic Schr"odinger equation](https://mlsanity.com/paper/1909.08423/) (2019)

## Built on by (most cited first, 2 of 2)

- [Operator learning on domain boundary through combining fundamental solution-based artificial data and boundary integral techniques](https://mlsanity.com/paper/2601.11222/) (2026)
- [Linear-Nonlinear Fusion Neural Operator for Partial Differential Equations](https://mlsanity.com/paper/2603.24143/) (2026)

## Abstract (arXiv)

Machine learning has emerged as a transformative tool for solving differential equations (DEs), yet prevailing methodologies remain constrained by dual limitations: data-driven methods demand costly labeled datasets while model-driven techniques face efficiency-accuracy trade-offs. We present the Mathematical Artificial Data (MAD) framework, a new paradigm that integrates physical laws with data-driven learning to facilitate large-scale operator discovery. By exploiting DEs' intrinsic mathematical structure to generate physics-embedded analytical solutions and associated synthetic data, MAD fundamentally eliminates dependence on experimental or simulated training data. This enables computationally efficient operator learning across multi-parameter systems while maintaining mathematical rigor. Through numerical demonstrations spanning 2D parametric problems where both the boundary values and source term are functions, we showcase MAD's generalizability and superior efficiency/accuracy across various DE scenarios. This physics-embedded-data-driven framework and its capacity to handle complex parameter spaces gives it the potential to become a universal paradigm for physics-informed machine intelligence in scientific computing.