论文标题
用深层整流的二次单元神经网络求解参数偏微分方程
Solving parametric partial differential equations with deep rectified quadratic unit neural networks
论文作者
论文摘要
实施深层神经网络来学习参数部分微分方程（PDE）的解决方案图比使用许多常规数值方法更有效。但是，对这种方法进行了有限的理论分析。在这项研究中，我们研究了深层二次单元（requ）神经网络的表达能力，以近似参数PDE的溶液图。拟议的方法是由G. Kutyniok，P。Petersen，M。Raslan和R. Schneider（Gitta Kutyniok，Philipp Petersen，Mones Raslan和Reinhold Schneider。对深度神经网络和参数的理论分析）的理论分析的持续性分析。用于求解参数PDE的神经网络。与先前确定的复杂性 - $ \ MATHCAL {o} \ left（d^3 \ log_ {2}^{q}（1/ε）（1/ε）\ right）$用于relu Neural网络，我们得出了一个上限的$ \ \ \ \ \ \ \米卡{O} \ right）$在实现准确性$ε> 0 $所需的深层神经网络的大小上，其中$ d $是代表解决方案的减少基础的维度。我们的方法充分利用了解决方案歧管的固有低维度和深层requen Necter网络的更好近似性能。进行数值实验以验证我们的理论结果。
Implementing deep neural networks for learning the solution maps of parametric partial differential equations (PDEs) turns out to be more efficient than using many conventional numerical methods. However, limited theoretical analyses have been conducted on this approach. In this study, we investigate the expressive power of deep rectified quadratic unit (ReQU) neural networks for approximating the solution maps of parametric PDEs. The proposed approach is motivated by the recent important work of G. Kutyniok, P. Petersen, M. Raslan and R. Schneider (Gitta Kutyniok, Philipp Petersen, Mones Raslan, and Reinhold Schneider. A theoretical analysis of deep neural networks and parametric pdes. Constructive Approximation, pages 1-53, 2021), which uses deep rectified linear unit (ReLU) neural networks for solving parametric PDEs. In contrast to the previously established complexity-bound $\mathcal{O}\left(d^3\log_{2}^{q}(1/ ε) \right)$ for ReLU neural networks, we derive an upper bound $\mathcal{O}\left(d^3\log_{2}^{q}\log_{2}(1/ ε) \right)$ on the size of the deep ReQU neural network required to achieve accuracy $ε>0$, where $d$ is the dimension of reduced basis representing the solutions. Our method takes full advantage of the inherent low-dimensionality of the solution manifolds and better approximation performance of deep ReQU neural networks. Numerical experiments are performed to verify our theoretical result.
