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.

翻译：我们提出了一种新型计算框架，用于逼近发展型偏微分方程的解算子。通过采用通用非线性降阶模型（如深度神经网络）来逼近给定偏微分方程的解，我们意识到模型参数的演化本质上是一个参数空间中的控制问题。基于这一观察，我们提出通过学习参数空间中的控制向量场来逼近偏微分方程的解算子。该控制场能从任意初始值出发，引导参数生成轨迹，使对应的降阶模型能够求解该偏微分方程。这一方法能够显著降低求解任意初始条件的发展型偏微分方程的计算成本。针对一大类半线性抛物型偏微分方程，我们建立了该方法全面的误差分析理论。在不同初始条件下的高维发展型偏微分方程数值实验中，该方法展现出优异的效果。