Guanhang Lei

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.

Guanhang Lei, Lei Shi

Regularized pairwise ranking with Gaussian kernels is one of the cutting-edge learning algorithms. Despite a wide range of applications, a rigorous theoretical demonstration still lacks to support the performance of such ranking estimators. This work aims to fill this gap by developing novel oracle inequalities for regularized pairwise ranking. With the help of these oracle inequalities, we derive fast learning rates of Gaussian ranking estimators under a general box-counting dimension assumption on the input domain combined with the noise conditions or the standard smoothness condition. Our theoretical analysis improves the existing estimates and shows that a low intrinsic dimension of input space can help the rates circumvent the curse of dimensionality.

Guanhang Lei, Zhen Lei, Lei Shi

Neural operators have shown promise in solving many types of Partial Differential Equations (PDEs). They are significantly faster compared to traditional numerical solvers once they have been trained with a certain amount of observed data. However, their numerical performance in solving time-dependent PDEs, particularly in long-time prediction of dynamic systems, still needs improvement. In this paper, we focus on solving the long-time integration of nonlinear wave equations via neural operators by replacing the initial condition with the prediction in a recurrent manner. Given limited observed temporal trajectory data, we utilize some intrinsic features of these nonlinear wave equations, such as conservation laws and well-posedness, to improve the algorithm design and reduce accumulated error. Our numerical experiments examine these improvements in the Korteweg-de Vries (KdV) equation, the sine-Gordon equation, and the Klein-Gordon wave equation on the irregular domain.

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.