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.

翻译：本文提出了一种基于有限维控制的方法，用于逼近发展型偏微分方程（PDEs）的解算子，特别是在高维情形下。通过采用通用降阶模型（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相联系。借助神经常微分方程这一计算技术，我们学习参数空间上的控制，使得从任意初始起点出发，受控轨迹都能紧密逼近偏微分方程的解。对于一般类别的二阶非线性偏微分方程，我们论证了该逼近方法的精度。文中给出了多个高维偏微分方程的数值结果，包括求解Hamilton-Jacobi-Bellman方程的实际应用案例，充分展示了所提方法的准确性与高效性。