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

## TL;DR

This paper introduces a variational solve-training framework for neural networks to efficiently approximate low-dimensional solution maps of physical models, avoiding costly data labeling and demonstrating effectiveness on various PDEs.

## Contribution

It presents a novel variational training method that directly leverages physical model equations, bypassing traditional supervised data preparation.

## Key findings

- Effective approximation of solution maps for elliptic equations
- Successful application to Schrödinger equations
- Reduced data preparation costs

## Abstract

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.