arXiv subjects

Weibing Deng

Publications and source records attributed to Weibing Deng.

At least 19 recordsLinked to original sources

arXiv2026

High-Order Exponential Integrators with Improved Uniform Accuracy for Charged-Particle in a Perpendicular Strong Magnetic Field

This paper considers a class of charged-particle dynamics problems in which the particle is subjected to a magnetic force, with a magnetic flux density inversely proportional to a small parameter $0<\varepsilon\ll 1$, and a nonlinear electric force. The resulting highly oscillatory behavior poses significant challenges for numerical computation. To enhance the performance of exponential integrators (EIs), this paper employs a technique that linearizes the ordinary differential equation through a dimension-raising approach. Based on this technique, a new family of EIs is developed that achieves arbitrarily high order. For short-time simulations on the interval $[0,T]$, it is rigorously proved that the proposed method--which employs auxiliary polynomials of degree $k$ and a time step $\Delta t$--satisfies two distinct error bounds: $O(\varepsilon \Delta t^{k+1})$ and $O(\varepsilon^{k+2})$. The latter bound guarantees that the algorithm stays accurate even when the step size is of order $O(1)$. Furthermore, when a large step size $\varepsilon^{-1}\Delta t$ is used to simulate the long-term dynamics over $[0,\varepsilon^{-1}T]$, the numerical scheme attains a uniform convergence rate of $O(\Delta t^{k+1})$. Several numerical experiments confirm these theoretical results.

math.NA↗

arXiv2026

Adaptive Randomized Neural Networks with Locally Activation Function: Theory and Algorithm for Solving PDEs

This paper establishes an approximation theory and develop an adaptive computational framework for randomized neural networks (RaNNs). For RaNNs of the form $\sum_{i=1}^{N} W_i \sigma(A_i\cdot x+B_i)$ with hidden parameters uniformly sampled from a bounded set of scale $M$, we show that the choice $M \asymp N^{p/[2((p-1)(d+1)+p\eta)]}$ yields an expected $W^{k,p}$-approximation error of order $\mathcal{O}\left(N^{-p\eta/[2((p-1)(d+1)+p\eta)]}\right)$, where $\eta$ is associated with the regularity of the target function in a generalized Barron spectral space. This theoretical result demonstrates that lower regularity requires a larger sampling range for the hidden parameters. Motivated by the relationship between $M$ and $\eta$, we introduce an adaptive physics-informed RaNN (PIRaNN) method that which couples parameter sampling with an adaptive partition of unity via local affine scaling. Residual-based a posteriori error indicators and D\"{o}rfler marking are used to drive local refinement. Numerical experiments, including the 1D viscous Burgers' equation and 2D/3D problems with localized peaks and L-shaped corner singularities, demonstrate that our method preferentially refines regions exhibiting large gradients and singularities. Compared to standard non-adaptive RaNNs, the adaptive PIRaNN effectively captures complex local features with significantly enhanced accuracy and efficiency.

math.NA↗

arXiv2025

Error Analysis on a Novel Class of Exponential Integrators with Local Linear Extension Techniques for Highly Oscillatory ODEs

This paper investigates a class of non-autonomous highly oscillatory ordinary differential equations characterized by a linear component inversely proportional to a small parameter $\varepsilon$, with purely imaginary eigenvalues, and an $\varepsilon$-independent nonlinear part. When $0<\varepsilon\ll 1$, the rapidly oscillatory nature of the solution imposes severe constraints on step size selection and numerical accuracy, leading to considerable computational difficulties. Inspired by a linearization technique that introduces auxiliary polynomial variables, a new family of explicit exponential integrators has recently been proposed. These methods do not require the linear part to be diagonal or to have eigenvalues that are integer multiples of a fixed value - a common assumption in multiscale approaches - and they achieve arbitrarily high orders of convergence without imposing order conditions. The main contribution of this work is to provide a rigorous error analysis for this new class of methods under a bounded oscillatory energy condition. To this end, we first establish the equivalence between the high-dimensional system and the original problem using algebraic techniques. Building on these foundational results, we prove that the numerical schemes, when employing auxiliary polynomial variables of degree $k$, achieve a uniform convergence order of $O(h^{k+1})$. In particular, an improved order of $O(\varepsilon h^k)$ is attained when $h$ is larger than the scale of $\varepsilon$. These theoretical findings are further applied to second-order oscillatory systems, leading to improved uniform accuracy with respect to $\varepsilon$. Finally, numerical experiments confirm the optimality of the derived error estimates.

math.NA↗

arXiv2025

A Stabilized Unfitted Space-time Finite Element Method for Parabolic Problems on Moving Domains

This paper presents a space-time finite element method (FEM) based on an unfitted mesh for solving parabolic problems on moving domains. Unlike other unfitted space-time finite element approaches that commonly employ the discontinuous Galerkin (DG) method for time-stepping, the proposed method employs a fully coupled space-time discretization. To stabilize the time-advection term, the streamline upwind Petrov-Galerkin (SUPG) scheme is applied in the temporal direction. A ghost penalty stabilization term is further incorporated to mitigate the small cut issue, thereby ensuring the well-conditioning of the stiffness matrix. Moreover, an a priori error estimate is derived in a discrete energy norm, which achieves an optimal convergence rate with respect to the mesh size. In particular, a space-time Poincare-Friedrichs inequality is established to support the condition number analysis. Several numerical examples are provided to validate the theoretical findings.

math.NA↗

arXiv2025

A Two-stage Adaptive Lifting PINN Framework for Solving Viscous Approximations to Hyperbolic Conservation Laws

Training physics informed neural networks PINNs for hyperbolic conservation laws near the inviscid limit presents considerable difficulties because strong form residuals become ill posed at shock discontinuities, while small viscosity regularization introduces narrow boundary layers that exacerbate spectral bias. To address these issues this paper proposes a novel two stage adaptive lifting PINN, a lifting based framework designed to mitigate such challenges without requiring a priori knowledge of the interface geometry. The key idea is to augment the physical coordinates by introducing a learned auxiliary field generated through r adaptive coordinate transformations. Theoretically we first derive an a posteriori L2 error estimate to quantify how training difficulty depends on viscosity. Secondly we provide a statistical interpretation revealing that embedded sampling induces variance reduction analogous to importance sampling. Finally we perform an NTK and gradient flow analysis, demonstrating that input augmentation improves conditioning and accelerates residual decay. Supported by these insights our numerical experiments show accelerated and more stable convergence as well as accurate reconstructions near discontinuities.

math.NA↗

arXiv2025

Extended Interface Physics-Informed Neural Networks Method for Moving Interface Problems

Physics-informed neural networks (PINNs) have emerged as an effective class of mesh-free methods for solving partial differential equations (PDEs), particularly on complex geometries. In this paper, we introduce an Extended Interface Physics-Informed Neural Network (XI-PINN) framework designed to solve parabolic moving interface problems. The proposed method employs a level set function--which can be either analytically prescribed or learned via a neural network--to capture the moving interface. Furthermore, we establish an a priori error analysis for the XI-PINN method and derive error bounds for the approximation. Extensive numerical experiments are provided to validate the accuracy and robustness of the framework, and its applicability is further demonstrated by solving the Oseen equations.

math.NA↗

arXiv2025

Siamese Neural Network for Label-Efficient Critical Phenomena Prediction in 3D Percolation Models

Predicting critical phenomena from limited labeled data remains a challenging task in statistical physics. As percolation theory provides a canonical model for phase transitions with well-established critical exponents, it serves as an ideal benchmark for validating new machine learning frameworks. Here, we introduce a label-efficient learning framework based on a Siamese Neural Network (SNN) to identify phase transitions in three-dimensional site and bond percolation models. Using only 22 labeled probability points drawn entirely from non-critical regions, the method locates percolation thresholds with percent-level accuracy and yields estimates of the critical exponent $\nu$ consistent with literature values within statistical uncertainty. Analysis of the learned representations clarifies what the network learns: although trained solely on binary similarity labels, the network autonomously converges to a statistic that coincides quantitatively with the normalized largest-cluster size $S_{max}/L^3$ ($r > 0.99$), the finite-size order parameter of percolation. This underlies the framework's most distinctive capability -- a model trained solely on simple cubic lattices identifies the phase transition in face-centered cubic lattices without retraining. The framework thus offers a complementary route to criticality detection in settings where no quantitative order parameter is explicitly defined or labeled data is scarce.

cond-mat.dis-nn↗

arXiv2025

Identifying Ising and percolation phase transitions based on KAN method

Modern machine learning, grounded in the Universal Approximation Theorem, has achieved significant success in the study of phase transitions in both equilibrium and non-equilibrium systems. However, identifying the critical points of percolation models using raw configurations remains a challenging and intriguing problem. This paper proposes the use of the Kolmogorov-Arnold Network, which is based on the Kolmogorov-Arnold Representation Theorem, to input raw configurations into a learning model. The results demonstrate that the KAN can indeed predict the critical points of percolation models. Further observation reveals that, apart from models associated with the density of occupied points, KAN is also capable of effectively achieving phase classification for models where the sole alteration pertains to the orientation of spins, resulting in an order parameter that manifests as an external magnetic flux, such as the Ising model.

cond-mat.stat-mech↗

arXiv2024

q Index Degree Distribution in Random Networks via Superstatistics

In this study, we employ a superstatistical approach to construct q exponential and q Maxwell Boltzmann complex networks, generalizing the concept of scale free networks. By adjusting the crossover parameter {\lambda}, we control the degree of the q exponential plateau at low node degrees, allowing a smooth transition to pure power law degree distributions. Similarly, the parameter b modulates the q Maxwell Boltzmann curvature, facilitating a shift toward pure power law networks. This framework introduces a novel perspective for constructing and analyzing scale free networks. Our results show that these additional degrees of freedom significantly enhance the flexibility of both network types in terms of topological and transport properties, including clustering coefficients, small world characteristics, and resilience to attacks. Future research will focus on exploring the dynamic properties of these networks, offering promising directions for further investigation.

physics.soc-ph↗

arXiv2024

R-adaptive DeepONet: Learning Solution Operators for PDEs with Discontinuous Solutions Using an R-adaptive Strategy

DeepONet has recently been proposed as a representative framework for learning nonlinear mappings between function spaces. However, when it comes to approximating solution operators of partial differential equations (PDEs) with discontinuous solutions, DeepONet poses a foundational approximation lower bound due to its linear reconstruction property. Inspired by the moving mesh (R-adaptive) method, we propose an R-adaptive DeepONet method, which contains the following components: (1) the output data representation is transformed from the physical domain to the computational domain using the equidistribution principle; (2) the maps from input parameters to the solution and the coordinate transformation function over the computational domain are learned using DeepONets separately; (3) the solution over the physical domain is obtained via post-processing methods such as the (linear) interpolation method. Additionally, we introduce a solution-dependent weighting strategy in the training process to reduce the final error. We establish an upper bound for the reconstruction error based on piecewise linear interpolation and show that the introduced R-adaptive DeepONet can reduce this bound. Moreover, for two prototypical PDEs with sharp gradients or discontinuities, we prove that the approximation error decays at a superlinear rate with respect to the trunk basis size, unlike the linear decay observed in vanilla DeepONets. Therefore, the R-adaptive DeepONet overcomes the limitations of DeepONet, and can reduce the approximation error for problems with discontinuous solutions. Numerical experiments on PDEs with discontinuous solutions, including the linear advection equation, the Burgers' equation with low viscosity, and the compressible Euler equations of gas dynamics, are conducted to verify the advantages of the R-adaptive DeepONet over available variants of DeepONet.

math.NA↗

arXiv2024

FINER++: Building a Family of Variable-periodic Functions for Activating Implicit Neural Representation

Implicit Neural Representation (INR), which utilizes a neural network to map coordinate inputs to corresponding attributes, is causing a revolution in the field of signal processing. However, current INR techniques suffer from the "frequency"-specified spectral bias and capacity-convergence gap, resulting in imperfect performance when representing complex signals with multiple "frequencies". We have identified that both of these two characteristics could be handled by increasing the utilization of definition domain in current activation functions, for which we propose the FINER++ framework by extending existing periodic/non-periodic activation functions to variable-periodic ones. By initializing the bias of the neural network with different ranges, sub-functions with various frequencies in the variable-periodic function are selected for activation. Consequently, the supported frequency set can be flexibly tuned, leading to improved performance in signal representation. We demonstrate the generalization and capabilities of FINER++ with different activation function backbones (Sine, Gauss. and Wavelet) and various tasks (2D image fitting, 3D signed distance field representation, 5D neural radiance fields optimization and streamable INR transmission), and we show that it improves existing INRs. Project page: {https://liuzhen0212.github.io/finerpp/}

cs.CV↗

arXiv2024

XI-DeepONet: An operator learning method for elliptic interface problems

Scientific computing has been an indispensable tool in applied sciences and engineering, where traditional numerical methods are often employed due to their superior accuracy guarantees. However, these methods often encounter challenges when dealing with problems involving complex geometries. Machine learning-based methods, on the other hand, are mesh-free, thus providing a promising alternative. In particular, operator learning methods have been proposed to learn the mapping from the input space to the solution space, enabling rapid inference of solutions to partial differential equations (PDEs) once trained. In this work, we address the parametric elliptic interface problem. Building upon the deep operator network (DeepONet), we propose an extended interface deep operator network (XI-DeepONet). XI-DeepONet exhibits three unique features: (1) The interface geometry is incorporated into the neural network as an additional input, enabling the network to infer solutions for new interface geometries once trained; (2) The level set function associated with the interface geometry is treated as the input, on which the solution mapping is continuous and can be effectively approximated by the deep operator network; (3) The network can be trained without any input-output data pairs, thus completely avoiding the need for meshes of any kind, directly or indirectly. We conduct a comprehensive series of numerical experiments to demonstrate the accuracy and robustness of the proposed method.

math.NA↗

arXiv2024

How to quantify an examination? Evidence from physics examinations via complex networks

Given the untapped potential for continuous improvement of examinations, quantitative investigations of examinations could guide efforts to considerably improve learning efficiency and evaluation and thus greatly help both learners and educators. However, there is a general lack of quantitative methods for investigating examinations. To address this gap, we propose a new metric via complex networks; i.e., the knowledge point network (KPN) of an examination is constructed by representing the knowledge points (concepts, laws, etc.) as nodes and adding links when these points appear in the same question. Then, the topological quantities of KPNs, such as degree, centrality, and community, can be employed to systematically explore the structural properties and evolution of examinations. In this work, 35 physics examinations from the NCEE examination spanning from 2006 to 2020 were investigated as an evidence. We found that the constructed KPNs are scale-free networks that show strong assortativity and small-world effects in most cases. The communities within the KPNs are obvious, and the key nodes are mainly related to mechanics and electromagnetism. Different question types are related to specific knowledge points, leading to noticeable structural variations in KPNs. Moreover, changes in the KPN topology between examinations administered in different years may offer insights guiding college entrance examination reforms. Based on topological quantities such as the average degree, network density, average clustering coefficient, and network transitivity, the Fd is proposed to evaluate examination difficulty. All the above results show that our approach can comprehensively quantify the knowledge structures and examination characteristics. These networks may elucidate comprehensive examination knowledge graphs for educators and guide improvements in teaching.

physics.soc-ph↗

arXiv2023

FINER: Flexible spectral-bias tuning in Implicit NEural Representation by Variable-periodic Activation Functions

Implicit Neural Representation (INR), which utilizes a neural network to map coordinate inputs to corresponding attributes, is causing a revolution in the field of signal processing. However, current INR techniques suffer from a restricted capability to tune their supported frequency set, resulting in imperfect performance when representing complex signals with multiple frequencies. We have identified that this frequency-related problem can be greatly alleviated by introducing variable-periodic activation functions, for which we propose FINER. By initializing the bias of the neural network within different ranges, sub-functions with various frequencies in the variable-periodic function are selected for activation. Consequently, the supported frequency set of FINER can be flexibly tuned, leading to improved performance in signal representation. We demonstrate the capabilities of FINER in the contexts of 2D image fitting, 3D signed distance field representation, and 5D neural radiance fields optimization, and we show that it outperforms existing INRs.

cs.CV↗

arXiv2023

Applications of Domain Adversarial Neural Network in phase transition of 3D Potts model

Machine learning techniques exhibit significant performance in discriminating different phases of matter and provide a new avenue for studying phase transitions. We investigate the phase transitions of three dimensional $q$-state Potts model on cubic lattice by using a transfer learning approach, Domain Adversarial Neural Network (DANN). With the unique neural network architecture, it could evaluate the high-temperature (disordered) and low-temperature (ordered) phases, and identify the first and second order phase transitions. Meanwhile, by training the DANN with a few labeled configurations, the critical points for $q=2,3,4$ and $5$ can be predicted with high accuracy, which are consistent with those of the Monte Carlo simulations. These findings would promote us to learn and explore the properties of phase transitions in high-dimensional systems.

physics.comp-ph↗

arXiv2023

Identifying percolation phase transitions with unsupervised learning based on largest clusters

The application of machine learning in the study of phase transitions has achieved remarkable success in both equilibrium and non-equilibrium systems. It is widely recognized that unsupervised learning can retrieve phase transition information through hidden variables. However, using unsupervised methods to identify the critical point of percolation models has remained an intriguing challenge. This paper suggests that, by inputting the largest cluster rather than the original configuration into the learning model, unsupervised learning can indeed predict the critical point of the percolation model. Furthermore, we observe that when the largest cluster configuration is randomly shuffled-altering the positions of occupied sites or bonds-there is no significant difference in the output compared to learning the largest cluster configuration directly. This finding suggests a more general principle: unsupervised learning primarily captures particle density, or more specifically, occupied site density. However, shuffling does impact the formation of the largest cluster, which is directly related to phase transitions. As randomness increases, we observe that the correlation length tends to decrease, providing direct evidence of this relationship. We also propose a method called Fake Finite Size Scaling (FFSS) to calculate the critical value, which improves the accuracy of fitting to a great extent.

cond-mat.stat-mech↗

arXiv2022

Study of phase transition of Potts model with Domain Adversarial Neural Network

A transfer learning method, Domain Adversarial Neural Network (DANN), is introduced to study the phase transition of two-dimensional q-state Potts model. With the DANN, we only need to choose a few labeled configurations automatically as input data, then the critical points can be obtained after training the algorithm. By an additional iterative process, the critical points can be captured to comparable accuracy to Monte Carlo simulations as we demonstrate it for q = 3, 4, 5, 7 and 10. The type of phase transition (first or second-order) is also determined at the same time. Meanwhile, for the second-order phase transition at q=3, we can calculate the critical exponent $\nu$ by data collapse. Furthermore, compared to the traditional supervised learning, we found the DANN to be more accurate with lower cost.

cond-mat.stat-mech↗

arXiv2022

A combined multiscale finite element method based on the LOD technique for the multiscale elliptic problems with singularities

In this paper, we construct a combined multiscale finite element method (MsFEM) using the Local Orthogonal Decomposition (LOD) technique to solve the multiscale problems which may have singularities in some special portions of the computational domain. For example, in the simulation of steady flow transporting through highly heterogeneous porous media driven by extraction wells, the singularities lie in the near-well regions. The basic idea of the combined method is to utilize the traditional finite element method (FEM) directly on a fine mesh of the problematic part of the domain and using the LOD-based MsFEM on a coarse mesh of the other part. The key point is how to define local correctors for the basis functions of the elements near the coarse and fine mesh interface, which require meticulous treatment. The proposed method takes advantages of the traditional FEM and the LOD-based MsFEM, which uses much less DOFs than the standard FEM and may be more accurate than the LOD-based MsFEM for problems with singularities. The error analysis is carried out for highly varying coefficients, without any assumptions on scale separation or periodicity. {Numerical examples with periodic and random highly oscillating coefficients}, as well as the multiscale problems on the L-shaped domain, and multiscale problems with high-contrast channels or well-singularities are presented to demonstrate the efficiency and accuracy of the proposed method.

math.NA↗