Variational training of neural network approximations of solution maps for physical models
This addresses the challenge of data preparation in physics-informed machine learning, though it appears incremental as it builds on variational methods for neural networks.
The paper tackles the problem of training neural networks to approximate solution maps for physical models without expensive labeled data, achieving effective representation for linear and nonlinear elliptic and Schrödinger equations.
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 Schrödinger equations.