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.

我们提出了一种基于有限维控制的方法来近似解决高维演化型偏微分方程的解算符。通过使用通用的降阶模型，例如深度神经网络，我们将模型参数的演化与相应函数空间中的轨迹连接起来。利用神经常微分方程的计算技术，我们学习参数空间上的控制，从而使受控轨迹与 PDE 的解非常接近。对于一类二阶非线性 PDE，我们验证了近似精确度。对几个高维 PDE，包括解决 Hamilton-Jacobi-Bellman 方程的真实应用，我们展示了所提方法的准确性和效率。