Approximation of Solution Operators for High-dimensional PDEs
Nathan Gaby Xiaojing Ye
We propose a finite-dimensional control-based method to approximate solution
operators for evolutional partial differential equations (PDEs), particularly
in high-dimensions. By employing a general reduced-order model, such as a deep
neural network, we connect the evolution of the model parameters with
trajectories in a corresponding function space. Using the computational
technique of neural ordinary differential equation, we learn the control over
the parameter space such that from any initial starting point, the controlled
trajectories closely approximate the solutions to the PDE. Approximation
accuracy is justified for a general class of second-order nonlinear PDEs.
Numerical results are presented for several high-dimensional PDEs, including
real-world applications to solving Hamilton-Jacobi-Bellman equations. These are
demonstrated to show the accuracy and efficiency of the proposed method.