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Solving Parametric PDEs with Radial Basis Functions and Deep Neural Networks
Apr 12, 2024
We propose the POD-DNN, a novel algorithm leveraging deep neural networks (DNNs) along with radial basis functions (RBFs) in the context of the proper orthogonal decomposition (POD) reduced basis method (RBM), aimed at approximating the parametric mapping of parametric partial differential equations on irregular domains. The POD-DNN algorithm capitalizes on the low-dimensional characteristics of the solution manifold for parametric equations, alongside the inherent offline-online computational strategy of RBM and DNNs. In numerical experiments, POD-DNN demonstrates significantly accelerated computation speeds during the online phase. Compared to other algorithms that utilize RBF without integrating DNNs, POD-DNN substantially improves the computational speed in the online inference process. Furthermore, under reasonable assumptions, we have rigorously derived upper bounds on the complexity of approximating parametric mappings with POD-DNN, thereby providing a theoretical analysis of the algorithm's empirical performance.
Solving PDEs on Spheres with Physics-Informed Convolutional Neural Networks
Aug 18, 2023Physics-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.
Pairwise Ranking with Gaussian Kernels
Apr 06, 2023Regularized 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.