We consider the approximation of initial/boundary value problems involving, possibly high-dimensional, dissipative evolution partial differential equations (PDEs) using a deep neural network framework. More specifically, we first propose discrete gradient flow approximations based on non-standard Dirichlet energies for problems involving essential boundary conditions posed on bounded spatial domains. The imposition of the boundary conditions is realized weakly via non-standard functionals; the latter classically arise in the construction of Galerkin-type numerical methods and are often referred to as "Nitsche-type" methods. Moreover, inspired by the seminal work of Jordan, Kinderleher, and Otto (JKO) \cite{jko}, we consider the second class of discrete gradient flows for special classes of dissipative evolution PDE problems with non-essential boundary conditions. These JKO-type gradient flows are solved via deep neural network approximations. A key, distinct aspect of the proposed methods is that the discretization is constructed via a sequence of residual-type deep neural networks (DNN) corresponding to implicit time-stepping. As a result, a DNN represents the PDE problem solution at each time node. This approach offers several advantages in the training of each DNN. We present a series of numerical experiments which showcase the good performance of Dirichlet-type energy approximations for lower space dimensions and the excellent performance of the JKO-type energies for higher spatial dimensions.
翻译:我们考虑初步/边界价值问题的近似近似值问题,可能涉及高维、分解性演变部分差异方程式(PDEs),使用深神经网络框架。更具体地说,我们首先提出基于非标准的Drichlet能源的离散梯度流动近似值,以解决在受约束的空间域中造成的基本边界条件问题。通过非标准功能,对边界条件的强制实施不力;后一种典型的方法是在建造Galerkin类数字方法时产生的,通常被称为“Nitsche型”方法。此外,在约旦、Kinderleher和Otto(JKO)的初始工作启发下,我们考虑在非基本边界条件中特殊类别消散演变PDE问题的特殊类别下梯度流动。这些JKO型梯度流动是通过深神经网络近近近近距离的近似方法解决的。一个关键、独特的方面是,离散式的深层空间神经网络(DNNNW)的序列,与隐含时间步相对应。作为目前高空级JKO(JKO)和每个高空级系列的高级性实验方法的一种结果,这是目前高端实验的每个高端实验的优势。