  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.
