标题： 求解偏微分方程的无监督网络架构搜索方法 显示英文标题

标题： An Unsupervised Network Architecture Search Method for Solving Partial Differential Equations

摘要： 求解偏微分方程（PDEs）在科学和工程应用中必不可少。最近，深度学习方法被广泛用于求解高维问题，其中物理信息神经网络（PINN）是一种。通常，深度学习方法有三个主要组成部分：神经网络、损失函数和优化器。虽然损失函数的构建根植于解空间的定义，但如何选择最优的神经网络则有些凭经验，还有很大的改进空间。在PINN框架下，我们提出了一种用于求解PDEs的无监督网络架构搜索方法，称为PINN-DARTS，该方法应用可微分架构搜索（DARTS）在一个给定的神经网络集合中找到最优的网络架构结构。在这个集合中，每层的层数和每层中的神经元数量都可以变化。在搜索阶段，网络参数和架构参数同时更新，因此运行时间接近预先确定网络结构的PINN的运行时间。与现有工作不同的是，我们的方法是无监督的，并且完全基于PDE残差，没有任何先验解决方案的使用。PINN-DARTS输出最优的网络结构以及相关的数值解。PINN-DARTS在几个基准PDE上进行了验证，包括椭圆型、抛物型、波动型和Burgers方程。与传统架构搜索方法相比，PINN-DARTS实现了显著更高的架构精度。另一个有趣的观察是，解的复杂性和PDE类型对最优网络架构有显著影响。我们的研究表明，从一层到另一层宽度不均的架构可能在不同的解复杂性和不同的PDE类型中表现出优越的性能。 显示英文摘要

摘要： Solving partial differential equations (PDEs) has been indispensable in scientific and engineering applications. Recently, deep learning methods have been widely used to solve high-dimensional problems, one of which is the physics-informed neural network (PINN). Typically, a deep learning method has three main components: a neural network, a loss function, and an optimizer. While the construction of the loss function is rooted in the definition of solution space, how to choose a optimal neural network is somewhat ad hoc, leaving much room for improvement. In the framework of PINN, we propose an unsupervised network architecture search method for solving PDEs, termed PINN-DARTS, which applies the differentiable architecture search (DARTS) to find the optimal network architecture structure in a given set of neural networks. In this set, the number of layers and the number of neurons in each layer can change. In the searching phase, both network and architecture parameters are updated simultaneously, so the running time is close to that of PINN with a pre-determined network structure. Unlike available works, our approach is unsupervised and purely based on the PDE residual without any prior usage of solutions. PINN-DARTS outputs the optimal network structure as well as the associated numerical solution. The performance of PINN-DARTS is verified on several benchmark PDEs, including elliptic, parabolic, wave, and Burgers' equations. Compared to traditional architecture search methods, PINN-DARTS achieves significantly higher architectural accuracy. Another interesting observation is that both the solution complexity and the PDE type have a prominent impact on the optimal network architecture. Our study suggests that architectures with uneven widths from layer to layer may have superior performance across different solution complexities and different PDE types.