Chenyu Zeng

Guanhang Lei, Zhen Lei, Lei Shi et al.

Physics-informed neural networks (PINNs) have been demonstrated to be efficient in solving partial differential equations (PDEs) from a variety of experimental perspectives. Some recent studies have also proposed PINN algorithms for PDEs on surfaces, including spheres. However, theoretical understanding of the numerical performance of PINNs, especially PINNs on surfaces or manifolds, is still lacking. In this paper, we establish rigorous analysis of the physics-informed convolutional neural network (PICNN) for solving PDEs on the sphere. By using and improving the latest approximation results of deep convolutional neural networks and spherical harmonic analysis, we prove an upper bound for the approximation error with respect to the Sobolev norm. Subsequently, we integrate this with innovative localization complexity analysis to establish fast convergence rates for PICNN. Our theoretical results are also confirmed and supplemented by our experiments. In light of these findings, we explore potential strategies for circumventing the curse of dimensionality that arises when solving high-dimensional PDEs.

Zhen Lei, Lei Shi, Chenyu Zeng

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.

Guanhang Lei, Zhen Lei, Lei Shi et al.

In this paper, we consider approximating the parameter-to-solution maps of parametric partial differential equations (PPDEs) using deep neural networks (DNNs). We propose an efficient approach combining reduced collocation methods (RCMs) and DNNs. In the approximation analysis section, we rigorously derive sharp upper bounds on the complexity of the neural networks. These bounds only depend on the reduced basis dimension rather than the high-fidelity discretization dimension, thereby theoretically guaranteeing the computational efficiency of our approach. In numerical experiments, we implement the RCM using radial basis function finite differences (RBF-FD) and proper orthogonal decomposition (POD), and propose the POD-DNN algorithm. We consider various types of PPDEs and compare the accuracy and efficiency of different solvers. The POD-DNN has demonstrated significantly accelerated inference speeds compared with conventional numerical methods owing to the offline-online computation strategy. Furthermore, by employing the reduced basis methods (RBMs), it also outperforms standard DNNs in computational efficiency while maintaining comparable accuracy.