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

•A novel unsupervised solve-training framework is proposed to train neural network representation of solution maps.•Solve-training framework achieves effective representation of the solution map adapted to the input data distribution.•Solve-training framework efficiently obtains the solution maps of linear and nonlinear elliptic equations.•Solve-training framework efficiently learns maps from potential to ground states for linear and nonlinear Schrodinger operators. 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 trains 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 solution maps for linear and nonlinear elliptic equations, and maps from potentials to ground states of linear and nonlinear Schrödinger equations.