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：我们提出了一种新颖的计算框架，用于近似演化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）来近似给定偏微分方程的解，我们发现模型参数的演化实际上是参数空间中的一个控制问题。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近似偏微分方程的解算子。该控制场能够从任意初始值出发，引导参数生成一条轨迹，使得对应的降阶模型求解该偏微分方程。这使得在任意初始条件下求解演化偏微分方程的计算成本大幅降低。我们还针对一大类半线性抛物型偏微分方程，对所提方法进行了全面的误差分析。在不同初始条件的高维演化偏微分方程上的数值实验表明，该方法取得了令人满意的结果。