Approximation of Solution Operators for High-dimensional PDEs
This work addresses the computational challenge of solving high-dimensional PDEs, which is critical for applications like optimal control and finance, but it appears incremental as it builds on existing neural ODE and reduced-order modeling techniques.
The authors tackled the problem of approximating solution operators for high-dimensional evolutional PDEs by proposing a control-based method that uses neural ODEs to learn parameter trajectories, achieving demonstrated accuracy and efficiency in numerical experiments including Hamilton-Jacobi-Bellman equations.
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.