Deep neural networks (DNNs) recently emerged as a promising tool for analyzing and solving complex differential equations arising in science and engineering applications. Alternative to traditional numerical schemes, learning-based solvers utilize the representation power of DNNs to approximate the input-output relations in an automated manner. However, the lack of physics-in-the-loop often makes it difficult to construct a neural network solver that simultaneously achieves high accuracy, low computational burden, and interpretability. In this work, focusing on a class of evolutionary PDEs characterized by having decomposable operators, we show that the classical ``operator splitting'' numerical scheme of solving these equations can be exploited to design neural network architectures. This gives rise to a learning-based PDE solver, which we name Deep Operator-Splitting Network (DOSnet). Such non-black-box network design is constructed from the physical rules and operators governing the underlying dynamics contains learnable parameters, and is thus more flexible than the standard operator splitting scheme. Once trained, it enables the fast solution of the same type of PDEs. To validate the special structure inside DOSnet, we take the linear PDEs as the benchmark and give the mathematical explanation for the weight behavior. Furthermore, to demonstrate the advantages of our new AI-enhanced PDE solver, we train and validate it on several types of operator-decomposable differential equations. We also apply DOSnet to nonlinear Schr\"odinger equations (NLSE) which have important applications in the signal processing for modern optical fiber transmission systems, and experimental results show that our model has better accuracy and lower computational complexity than numerical schemes and the baseline DNNs.
翻译:深神经网络(DNN)最近成为分析和解决科学和工程应用中产生的复杂差异方程式的一个很有希望的工具。除了传统的数字计划外,学习型的解决方案利用DNN的演示力来以自动化的方式接近输入输出关系。然而,缺乏“在环形中物理学”往往使得难以构建一个同时达到高精确度、低计算负担和可解释性的神经网络求解器。在这项工作中,侧重于以可调解的操作员为特征的进化式PDE为特征的一类进化式PDE,我们展示了传统的“Operator 分裂” 解决这些方程式的数字计划可以用来设计神经网络结构。这产生了一个基于学习型的 PDE 求求解器,我们命名了深操作员- Spliting 网络(DOSnet ) 。这种非黑信箱网络设计是根据物理规则构建的,管理基本动态的操作员含有可学习的参数,因此比标准操作员的分解方案更灵活。我们训练过它能快速解决PDE 的解算数字应用程序的快速解算法非数字应用方法。 测试了我们内部的数学运算法的运算法,也显示了了我们内部的内数的内测算法的运算法的内测算法, 显示了我们测算法的运的内数的内测算法的内数法的基数变法的基数结构的变法, 。