arXiv subjects

Wenzhi Xu

Publications and source records attributed to Wenzhi Xu.

4 recordsLinked to original sources

arXiv2026

An improved PINN framework integrating localized collocation scheme and PIKF

We propose a localized physics-informed kernel function neural network (LPIKFNN), which is an improved physics-informed neural network (PINN) based on physics-informed kernel function (PIKF). In the LPIKFNN framework, the localized collocation scheme discretizes the physical quantities within the local domain, where the physical field is represented as a linear combination of PIKFs. Based on this representation, the multilayer perceptron is trained to iteratively learn the physical quantities. To overcome the computational challenges of conventional PINN in higher-order derivative and high wavenumber problems, the LPIKFNN constructs the loss function using the PIKF and a localized collocation scheme rather than relying on automatic differentiation. As a result, the costly derivative evaluations required to enforce governing equations during iterative training are eliminated, leading to significantly improved computational efficiency and training performance. Moreover, incorporating PIKFs into the loss function enables the proposed LPIKFNN to significantly improve computational accuracy in high-wavenumber problems characterized by highly oscillatory physical fields. To overcome the computational bottleneck of the physics-informed kernel function neural network (PIKFNN) in heterogeneous problems, the LPIKFNN introduces a localized collocation scheme that removes reliance on global PIKFs, enabling accurate predictions where global PIKFs are unavailable. The feasibility and accuracy of the proposed LPIKFNN are demonstrated through a series of benchmark studies, including high wavenumber problems, higher-order derivative problems, nonlinear problems, heterogeneous problems, and potential-based inverse electromyography. The numerical predictions obtained by LPIKFNN show excellent agreement with available analytical solutions and experimental measurements.

cs.CE↗

arXiv2026

HiQ-Lip: A Hierarchical Quantum-Classical Method for Global Lipschitz Constant Estimation of ReLU Networks

Estimating the global Lipschitz constant of neural networks is crucial for understanding and improving their robustness and generalization capabilities. However, precise calculations are NP-hard, and current semidefinite programming (SDP) methods face challenges such as high memory usage and slow processing speeds. In this paper, we propose HiQ-Lip, a hybrid quantum-classical hierarchical method that leverages quantum computing to estimate the global Lipschitz constant. We tackle the estimation by converting it into a Quadratic Unconstrained Binary Optimization problem and implement a multilevel graph coarsening and refinement strategy to adapt to the constraints of contemporary quantum hardware. Our experimental evaluations on fully connected neural networks demonstrate that HiQ-Lip not only provides estimates comparable to state-of-the-art methods but also significantly accelerates the computation process. In specific tests involving two-layer neural networks with 256 hidden neurons, HiQ-Lip doubles the solving speed and offers more accurate upper bounds than the existing best method, LiPopt. These findings highlight the promising utility of small-scale quantum devices in advancing the estimation of neural network robustness.

cs.LG↗

arXiv2023

FEM-PIKFNNs for underwater acoustic propagation induced by structural vibrations in different ocean environments

In this paper, a novel hybrid method based on the finite element method (FEM) and physics-informed kernel function neural networks (PIKFNNs) is proposed and applied to the prediction of underwater acoustic propagation induced by structural vibrations in the unbounded ocean, deep ocean and shallow ocean. In the hybrid method, PIKFNNs are a class of improved shallow physics-informed neural networks (PINNs) that replace the activation functions in PINNs with the physics-informed kernel functions (PIKFs), thereby integrating prior physical information into the neural network model. Moreover, this neural network circumvents the step of embedding the governing equations into the loss function in PINNs, and requires only training on boundary data. By using the Green's functions as the PIKFs and the structural-acoustic coupling response information obtained from the FEM as boundary training data, the PIKFNNs can inherently capture the Sommerfeld radiation condition at infinity, which is naturally suitable for predicting ocean acoustic propagation. Numerical experiments demonstrate the accuracy and feasibility of the FEM-PIKFNNs in comparison with the true solutions and finite element results.

math.NA↗

arXiv2023

Physics-Informed Kernel Function Neural Networks for Solving Partial Differential Equations

This paper proposed a novel radial basis function neural network (RBFNN) to solve various partial differential equations (PDEs). In the proposed RBF neural networks, the physics-informed kernel functions (PIKFs), which are derived according to the governing equations of the considered PDEs, are used to be the activation functions instead of the traditional RBFs. Similar to the well-known physics-informed neural networks (PINNs), the proposed physics-informed kernel function neural networks (PIKFNNs) also include the physical information of the considered PDEs in the neural network. The difference is that the PINNs put this physical information in the loss function, and the proposed PIKFNNs put the physical information of the considered governing equations in the activation functions. By using the derived physics-informed kernel functions satisfying the considered governing equations of homogeneous, nonhomogeneous, transient PDEs as the activation functions, only the boundary/initial data are required to train the neural network. Finally, the feasibility and accuracy of the proposed PIKFNNs are validated by several benchmark examples referred to high-wavenumber wave propagation problem, infinite domain problem, nonhomogeneous problem, long-time evolution problem, inverse problem, spatial structural derivative diffusion model, and so on.

math.NA↗