In this work, we present a hybrid numerical method for solving evolution partial differential equations (PDEs) by merging the time finite element method with deep neural networks. In contrast to the conventional deep learning-based formulation where the neural network is defined on a spatiotemporal domain, our methodology utilizes finite element basis functions in the time direction where the space-dependent coefficients are defined as the output of a neural network. We then apply the Galerkin or collocation projection in the time direction to obtain a system of PDEs for the space-dependent coefficients which is approximated in the framework of PINN. The advantages of such a hybrid formulation are twofold: statistical errors are avoided for the integral in the time direction, and the neural network's output can be regarded as a set of reduced spatial basis functions. To further alleviate the difficulties from high dimensionality and low regularity, we have developed an adaptive sampling strategy that refines the training set. More specifically, we use an explicit density model to approximate the distribution induced by the PDE residual and then augment the training set with new time-dependent random samples given by the learned density model. The effectiveness and efficiency of our proposed method have been demonstrated through a series of numerical experiments.

本研究提出了一种混合数值方法，通过将时间有限元法与深度神经网络相结合，解决演变偏微分方程的问题。与传统的方法不同，我们的创新在于采用有限元素基函数处理时间方向，从而有效避免了正则性差和维度高带来的困难，显著提高了计算的有效性和效率。

一种混合有限元-深度神经网络方法用于时间相关的偏微分方程