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Implementing deep neural networks for learning the solution maps ofparametric partial differential equations (PDEs) turns out to be more efficientthan using many conventional numerical methods. However, limited theoreticalanalyses have been conducted on this approach. In this study, we investigatethe expressive power of deep rectified quadratic unit (ReQU) neural networksfor approximating the solution maps of parametric PDEs. The proposed approachis motivated by the recent important work of G. Kutyniok, P. Petersen, M.Raslan and R. Schneider (Gitta Kutyniok, Philipp Petersen, Mones Raslan, andReinhold Schneider. A theoretical analysis of deep neural networks andparametric pdes. Constructive Approximation, pages 1-53, 2021), which uses deeprectified linear unit (ReLU) neural networks for solving parametric PDEs. Incontrast to the previously established complexity-bound$\mathcal{O}\left(d^3\log_{2}^{q}(1/ \epsilon) \right)$ for ReLU neuralnetworks, we derive an upper bound $\mathcal{O}\left(d^3\log_{2}^{q}\log_{2}(1/\epsilon) \right)$ on the size of the deep ReQU neural network required toachieve accuracy $\epsilon>0$, where $d$ is the dimension of reduced basisrepresenting the solutions. Our method takes full advantage of the inherentlow-dimensionality of the solution manifolds and better approximationperformance of deep ReQU neural networks. Numerical experiments are performedto verify our theoretical result.

Solving parametric partial differential equations with deep rectified quadratic unit neural networks