arXiv subjects

Heng Yong

Publications and source records attributed to Heng Yong.

7 recordsLinked to original sources

arXiv2025

Gaussian process surrogate with physical law-corrected prior for multi-coupled PDEs defined on irregular geometry

Parametric partial differential equations (PDEs) serve as fundamental mathematical tools for modeling complex physical phenomena, yet repeated high-fidelity numerical simulations across parameter spaces remain computationally prohibitive. In this work, we propose a physical law-corrected prior Gaussian process (LC-prior GP) for efficient surrogate modeling of parametric PDEs. The proposed method employs proper orthogonal decomposition (POD) to represent high-dimensional discrete solutions in a low-dimensional modal coefficient space, significantly reducing the computational cost of kernel optimization compared with standard GP approaches in full-order spaces. The governing physical laws are further incorporated to construct a law-corrected prior to overcome the limitation of existing physics-informed GP methods that rely on linear operator invariance, which enables applications to nonlinear and multi-coupled PDE systems without kernel redesign. Furthermore, the radial basis function-finite difference (RBF-FD) method is adopted for generating training data, allowing flexible handling of irregular spatial domains. The resulting differentiation matrices are independent of solution fields, enabling efficient optimization in the physical correction stage without repeated assembly. The proposed framework is validated through extensive numerical experiments, including nonlinear multi-parameter systems and scenarios involving multi-coupled physical variables defined on different two-dimensional irregular domains to highlight the accuracy and efficiency compared with baseline approaches.

stat.ML↗

arXiv2025

DeePoly: A High-Order Accuracy Scientific Machine Learning Framework for Function Approximation and Solving PDEs

Recently, machine learning methods have gained significant traction in scientific computing, particularly for solving Partial Differential Equations (PDEs). However, methods based on deep neural networks (DNNs) often lack convergence guarantees and computational efficiency compared to traditional numerical schemes. This work introduces DeePoly, a novel framework that transforms the solution paradigm from pure non-convex parameter optimization to a two-stage approach: first employing a DNN to capture complex global features, followed by linear space optimization with combined DNN-extracted features (Spotter) and polynomial basis functions (Sniper). This strategic combination leverages the complementary strengths of both methods -- DNNs excel at approximating complex global features (i.e., high-gradient features) and stabilize the polynomial approximation while polynomial bases provide high-precision local corrections with convergence guarantees. Theoretical analysis and numerical experiments demonstrate that this approach significantly enhances both high-order accuracy and efficiency across diverse problem types while maintaining mesh-free and scheme-free properties. This paper also serves as a theoretical exposition for the open-source project DeePoly.

cs.AI↗

arXiv2025

A general physics-constrained method for the modelling of equation's closure terms with sparse data

Accurate modeling of closure terms is a critical challenge in engineering and scientific research, particularly when data is sparse (scarse or incomplete), making widely applicable models difficult to develop. This study proposes a novel approach for constructing closure models in such challenging scenarios. We introduce a Series-Parallel Multi-Network Architecture that integrates Physics-Informed Neural Networks (PINNs) to incorporate physical constraints and heterogeneous data from multiple initial and boundary conditions, while employing dedicated subnetworks to independently model unknown closure terms, enhancing generalizability across diverse problems. These closure models are integrated into an accurate Partial Differential Equation (PDE) solver, enabling robust solutions to complex predictive simulations in engineering applications.

cs.LG↗

arXiv2024

Numerical Approximation Capacity of Neural Networks with Bounded Parameters: Do Limits Exist, and How Can They Be Measured?

The Universal Approximation Theorem posits that neural networks can theoretically possess unlimited approximation capacity with a suitable activation function and a freely chosen or trained set of parameters. However, a more practical scenario arises when these neural parameters, especially the nonlinear weights and biases, are bounded. This leads us to question: \textbf{Does the approximation capacity of a neural network remain universal, or does it have a limit when the parameters are practically bounded? And if it has a limit, how can it be measured?} Our theoretical study indicates that while universal approximation is theoretically feasible, in practical numerical scenarios, Deep Neural Networks (DNNs) with any analytic activation functions (such as Tanh and Sigmoid) can only be approximated by a finite-dimensional vector space under a bounded nonlinear parameter space (NP space), whether in a continuous or discrete sense. Based on this study, we introduce the concepts of \textit{$\epsilon$ outer measure} and \textit{Numerical Span Dimension (NSdim)} to quantify the approximation capacity limit of a family of networks both theoretically and practically. Furthermore, drawing on our new theoretical study and adopting a fresh perspective, we strive to understand the relationship between back-propagation neural networks and random parameter networks (such as the Extreme Learning Machine (ELM)) with both finite and infinite width. We also aim to provide fresh insights into regularization, the trade-off between width and depth, parameter space, width redundancy, condensation, and other related important issues.

cs.LG↗

arXiv2022

Discontinuity Computing using Physics-Informed Neural Network

Simulating discontinuities is a long standing problem especially for shock waves with strong nonlinear feather. Despite being a promising method, the recently developed physics-informed neural network (PINN) is still weak for calculating discontinuities compared with traditional shock-capturing methods. In this paper, we intend to improve the shock-capturing ability of the PINN. The primary strategy of this work is to weaken the expression of the network near discontinuities by adding a gradient-weight into the governing equations locally at each residual point. This strategy allows the network to focus on training smooth parts of the solutions. Then, automatically affected by the compressible property near shock waves, a sharp discontinuity appears with wrong inside shock transition-points compressed into well-trained smooth regions as passive particles. We study the solutions of one-dimensional Burgers equation and one- and two-dimensional Euler equations. Compared with the traditional high-order WENO-Z method in numerical examples, the proposed method can substantially improve discontinuity computing.

physics.flu-dyn↗

arXiv2022

A weighted first-order formulation for solving anisotropic diffusion equations with deep neural networks

In this paper, a new weighted first-order formulation is proposed for solving the anisotropic diffusion equations with deep neural networks. For many numerical schemes, the accurate approximation of anisotropic heat flux is crucial for the overall accuracy. In this work, the heat flux is firstly decomposed into two components along the two eigenvectors of the diffusion tensor, thus the anisotropic heat flux approximation is converted into the approximation of two isotropic components. Moreover, to handle the possible jump of the diffusion tensor across the interface, the weighted first-order formulation is obtained by multiplying this first-order formulation by a weighted function. By the decaying property of the weighted function, the weighted first-order formulation is always well-defined in the pointwise way. Finally, the weighted first-order formulation is solved with deep neural network approximation. Compared to the neural network approximation with the original second-order elliptic formulation, the proposed method can significantly improve the accuracy, especially for the discontinuous anisotropic diffusion problems.

math.NA↗

arXiv2017

An Adaptive Characteristic-wise Reconstruction WENOZ scheme for Gas Dynamic Euler Equations

Due to its excellent shock-capturing capability and high resolution, the WENO scheme family has been widely used in varieties of compressive flow simulation. However, for problems containing strong shocks and contact discontinuities, such as the Lax shock tube problem, the WENO scheme still produces numerical oscillations. To avoid such numerical oscillations, the characteristic-wise construction method should be applied. Compared to component-wise reconstruction, characteristic-wise reconstruction leads to much more computational cost and thus is not suite for large scale simulation such as direct numeric simulation of turbulence. In this paper, an adaptive characteristic-wise reconstruction WENO scheme, i.e. the AdaWENO scheme, is proposed to improve the computational efficiency of the characteristic-wise reconstruction method. The new scheme performs characteristic-wise reconstruction near discontinuities while switching to component-wise reconstruction for smooth regions. Meanwhile, a new calculation strategy for the WENO smoothness indicators is implemented to reduce over-all computational cost. Several one dimensional and two dimensional numerical tests are performed to validate and evaluate the AdaWENO scheme. Numerical results show that AdaWENO maintains essentially non-oscillatory flow field near discontinuities as the characteristic-wise reconstruction method. Besieds, compared to the component-wise reconstruction, AdaWENO is about 40\% faster which indicates its excellent efficiency.

physics.flu-dyn↗