# Variational training of neural network approximations of solution maps for physical models

arXiv: https://arxiv.org/abs/1905.02789 · HTML: https://mlsanity.com/paper/1905.02789/

## Key facts

- Cited by 8 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 7 May 2019.

## Built on by (most cited first, 8 of 8)

- [Physics-Informed Neural Operator for Learning Partial Differential Equations](https://mlsanity.com/paper/2111.03794/) (2021)
- [Operator Learning Using Random Features: A Tool for Scientific Computing](https://mlsanity.com/paper/2408.06526/) (2024)
- [Coupling parameter and particle dynamics for adaptive sampling in Neural Galerkin schemes](https://mlsanity.com/paper/2306.15630/) (2023)
- [AMS-Net: Adaptive Multiscale Sparse Neural Network with Interpretable Basis Expansion for Multiphase Flow Problems](https://mlsanity.com/paper/2207.11735/) (2022)
- [On the choice of the non-trainable internal weights in random feature maps](https://mlsanity.com/paper/2408.03626/) (2024)
- [Cell-average based neural network method for hyperbolic and parabolic partial differential equations](https://mlsanity.com/paper/2107.00813/) (2021)
- [Accuracy and Architecture Studies of Residual Neural Network solving Ordinary Differential Equations](https://mlsanity.com/paper/2101.03583/) (2021)
- [Windowed Fourier Propagator: A Frequency-Local Neural Operator for Wave Equations in Inhomogeneous Media](https://mlsanity.com/paper/2603.14289/) (2026)

## Abstract (arXiv)

A novel solve-training framework is proposed to train neural network in representing low dimensional solution maps of physical models. Solve-training framework uses the neural network as the ansatz of the solution map and train the network variationally via loss functions from the underlying physical models. Solve-training framework avoids expensive data preparation in the traditional supervised training procedure, which prepares labels for input data, and still achieves effective representation of the solution map adapted to the input data distribution. The efficiency of solve-training framework is demonstrated through obtaining solutions maps for linear and nonlinear elliptic equations, and maps from potentials to ground states of linear and nonlinear Schr"odinger equations.