  We develop a novel computational framework to approximate solution operators
of evolution partial differential equations (PDEs). By employing a general
nonlinear reduced-order model, such as a deep neural network, to approximate
the solution of a given PDE, we realize that the evolution of the model
parameter is a control problem in the parameter space. Based on this
observation, we propose to approximate the solution operator of the PDE by
learning the control vector field in the parameter space. From any initial
value, this control field can steer the parameter to generate a trajectory such
that the corresponding reduced-order model solves the PDE. This allows for
substantially reduced computational cost to solve the evolution PDE with
arbitrary initial conditions. We also develop comprehensive error analysis for
the proposed method when solving a large class of semilinear parabolic PDEs.
Numerical experiments on different high-dimensional evolution PDEs with various
initial conditions demonstrate the promising results of the proposed method.
