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Physics-informed neural networks with residual/gradient-based adaptive sampling methods for solving partial differential equations with sharp solutions

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Applied Mathematics and Mechanics(English Edition) 2023年第7期44卷 1069-1084页

作者： Zhiping MAO Xuhui MENG School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific ComputingXiamen UniversityXiamen 361005Fujian ProvinceChina Institute of Interdisciplinary Research for Mathematics and Applied Science School of Mathematics and StatisticsHuazhong University of Science and TechnologyWuhan 430074China

We consider solving the forward and inverse partial differential equations(PDEs)which have sharp solutions with physics-informed neural networks(PINNs)in this *** particular,to better capture the sharpness of the solution,we propose the adaptive sampling methods(ASMs)based on the residual and the gradient of the *** first present a residual only-based ASM denoted by ASMⅠ.In this approach,we first train the neural network using a small number of residual points and divide the computational domain into a certain number of sub-domains,then we add new residual points in the sub-domain which has the largest mean absolute value of the residual,and those points which have the largest absolute values of the residual in this sub-domain as new residual *** further develop a second type of ASM(denoted by ASMⅡ)based on both the residual and the gradient of the solution due to the fact that only the residual may not be able to efficiently capture the sharpness of the *** procedure of ASMⅡis almost the same as that of ASMⅠ,and we add new residual points which have not only large residuals but also large *** demonstrate the effectiveness of the present methods,we use both ASMⅠand ASMⅡto solve a number of PDEs,including the Burger equation,the compressible Euler equation,the Poisson equation over an Lshape domain as well as the high-dimensional Poisson *** has been shown from the numerical results that the sharp solutions can be well approximated by using either ASMⅠor ASMⅡ,and both methods deliver much more accurate solutions than the original PINNs with the same number of residual ***,the ASMⅡalgorithm has better performance in terms of accuracy,efficiency,and stability compared with the ASMⅠ*** means that the gradient of the solution improves the stability and efficiency of the adaptive sampling procedure as well as the accuracy of the ***,we also employ the similar adaptive sampling technique fo

关键词： physics-informed neural network(PINN) adaptive sampling high-dimension L-shape Poisson equation accuracy

Asymptotic stability of explicit infinite energy blowup solutions of the 3D incompressible Navier-Stokes equations

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Science China Mathematics 2023年第11期66卷 2523-2544页

作者： Fangyu Han Zhong Tan School of Mathematical Sciences Xiamen UniversityXiamen 361005China Fujian Provincial Key Laboratory on Mathematical Modeling and High-Performance Scientific Computing Xiamen UniversityXiamen UniversityXiamen 361005China Shenzhen Research Institute of Xiamen University Shenzhen 518057China

In this paper,we study the dynamical stability of a family of explicit blowup solutions of the threedimensional(3D)incompressible Navier-Stokes(NS)equations with smooth initial values,which is constructed in Guo et al.(2008).This family of solutions has finite energy in any bounded domain of R3,but unbounded energy in *** on similarity coordinates,energy estimates and the Nash-Moser-H?rmander iteration scheme,we show that these solutions are asymptotically stable in the backward light-cone of the ***,the result shows the existence of local energy blowup solutions to the 3D incompressible NS equations with growing ***,the result also shows that in the absence of physical boundaries,the viscous vanishing limit of the solutions does not satisfy the 3D incompressible Euler equations.

关键词： Navier-Stokes equations asymptotic stability blowup solution infinite energy Nash-Moser-Hormander iteration scheme zero-viscosity limit

Neural-Network-Augmented Empirical Interpolation Method for Field Reconstruction with Noise and Vibration Tolerance

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SSRN

SSRN 2024年

作者： Li, Han Gong, Helin Xu, Chuanju Paris Elite Institute of Technology Shanghai Jiao Tong University Shanghai200240 China School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical Modeling and High Performance Scientific Computing Xiamen University Xiamen361005 China

Field reconstruction methods based on machine learning often face challenges when handling unstructured data and designing complex model structures. On the other hand, Reduced Order Models (ROM) struggle with robustness in the presence of observation noise and sensor vibrations. Striking a balance between lightweight design, the ability to handle unstructured data, and robustness presents a significant challenge. In this paper, we introduce the EIM-NN algorithm, which leverages neural networks to determine the coefficients of the subspace found by the EIM algorithm. Furthermore, we present the EIM-TNN algorithm, which enhances robustness by designing a loss function incorporating Tikhonov regularization. Our neural network consists of only two fully connected layers, allowing it to handle unstructured data while maintaining a lightweight profile. Experimental results demonstrate the algorithm's ability to significantly enhance robustness against noise and sensor vibrations without compromising its accuracy in fitting the original data. Additionally, the algorithm's lightweight nature ensures that the added training time and memory requirements over EIM remain acceptable, making it adaptable to a range of industrial applications. © 2024, The Authors. All rights reserved.

关键词： Model structures

Stability and Error Analysis of the Linear, Decoupled Imex Schemes for Sixth-Order Cahn-Hilliard-Type Equations

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SSRN

SSRN 2024年

作者： Zheng, Nan Shen, Jie Department of Applied Mathematics The Hong Kong Polytechnic University Hung Hom Kowloon Hong Kong School of Mathematical Sciences Fujian Provincial Key Laboratory on Mathematical Modeling and High Performance Scientific Computing Xiamen University Fujian Xiamen361005 China School of Mathematical Science Eastern Institute of Technology Ningbo China

In this paper, we develop efficient implicit-explicit (IMEX) schemes for solving sixth-order Cahn-Hilliard-type equations based on the generalized scalar auxiliary variable (GSAV) approach. These novel schemes provide several remarkable advantages: (i) they are linear and only require solving one elliptic equation with constant coefficients at each time step;(ii) they are unconditionally energy stable and yield a uniform bound for the numerical solution. We also establish rigorous error estimates of up to fifth-order for these schemes, and present various numerical experiments to validate the stability and accuracy of the proposed schemes. © 2024, The Authors. All rights reserved.

关键词： Error analysis

Numerical analysis of a semi-implicit Euler scheme for the Keller-Segel model

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arXiv 2025年

作者： Huang, Xueling Goubet, Olivier Shen, Jie School of Mathematical Sciences Fujian Provincial Key Laboratory on Mathematical Modeling and High Performance Scientific Computing Xiamen University Fujian Xiamen361005 China Laboratoire Paul Painlevé CNRS UMR 8524 et équipe projet INRIA PARADYSE Université de Lille Lille59000 France School of Mathematical Science Eastern Institute of Technology Zhejiang Ningbo315200 China

We study the properties of a semi-implicit Euler scheme that is widely used in time discretization of Keller-Segel equations both in the parabolic-elliptic form and the parabolic-parabolic form. We prove that this linear, decoupled, first-order scheme preserves unconditionally the important properties of Keller-Segel equations at the semi-discrete level, including the mass conservation and positivity preserving of the cell density, and the energy dissipation. We also establish optimal error estimates in Lp-norm (1 Copyright © 2025, The Authors. All rights reserved.

关键词： Euler equations

AN EFFICIENT SPECTRAL TRUST-REGION DEFLATION METHOD FOR MULTIPLE SOLUTIONS

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arXiv 2023年

作者： Li, Lin Wang, Li-Lian Li, Huiyuan School of Mathematics and Physics Hunan Key Laboratory for Mathematical Modeling and Scientific Computing University of South China Hengyang421001 China Division of Mathematical Sciences School of Physical and Mathematical Sciences Nanyang Technological University 637371 Singapore State Key Laboratory of Computer Science Laboratory of Parallel Computing Institute of Software Chinese Academy of Sciences Beijing100190 China

It is quite common that a nonlinear partial differential equation (PDE) admits multiple distinct solutions and each solution may carry a unique physical meaning. One typical approach for finding multiple solutions is to use the Newton method with different initial guesses that ideally fall into the basins of attraction confining the solutions. In this paper, we propose a fast and accurate numerical method for multiple solutions comprised of three ingredients: (i) a well-designed spectral-Galerkin discretization of the underlying PDE leading to a nonlinear algebraic system (NLAS) with multiple solutions;(ii) an effective deflation technique to eliminate a known (founded) solution from the other unknown solutions leading to deflated NLAS;and (iii) a viable nonlinear least-squares and trust-region (LSTR) method for solving the NLAS and the deflated NLAS to find the multiple solutions sequentially one by one. We demonstrate through ample examples of differential equations and comparison with relevant existing approaches that the spectral LSTR-Deflation method has the merits: (i) it is quite flexible in choosing initial values, even starting from the same initial guess for finding all multiple solutions;(ii) it guarantees high-order accuracy;and (iii) it is quite fast to locate multiple distinct solutions and explore new solutions which are not reported in literature. Copyright © 2023, The Authors. All rights reserved.

关键词： Newton-Raphson method

Benchmark modeling and 3D applications of solidification and macro-segregation based on an operator-splitting and fully decoupled scheme with term-wise matrix assembly

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arXiv 2023年

作者： Feng, Xiaoyu Chen, Huangxin Yu, Bo Sun, Shuyu Thuwal23955-6900 Saudi Arabia School of Mathematical Sciences Fujian Provincial Key Laboratory on Mathematical Modeling and High Performance Scientific Computing Xiamen University Xiamen361005 China School of Mechanical Engineering Beijing Institute of Petrochemical Technology Beijing102617 China

The solidification and macro-segregation problem involving unsteady multiphysics and multi-phase fields is typically a complex process with mass, momentum, heat, and species transfers among solid, mushy, and liquid phase regions. The quantitative prediction of phase change, chemical heterogeneities, and multi-phase and multi-component flows plays critical roles in many natural scenarios and industrial applications that involve many disciplines, like material, energy, and even planet science. In view of this, some scholars and research institutions have called for more contributors to join the benchmark analysis of solidification and segregation problems. Our work proposes an operator-splitting and matrix-based method to avoid non-linear systems. Also, the combination of vectorization and forward equation-based matrix assembly techniques enhances the implementability of extensions of 3D applications. Lastly, the novel scheme is well validated through a bunch of 2D and 3D benchmark cases. The numerical results also illustrate that this method can ensure accurate prediction and adequately capture the physical details of phenomena caused by the solutally and thermally driven flow, which include channel segregation, the formation of freckles, edge effect, aspect ratio effect, and 3D effect. Copyright © 2023, The Authors. All rights reserved.

关键词： Solidification

Benchmark modeling and 3d Applications of Solidification and Macro-Segregation Based on an Operator-Splitting and Fully Decoupled Scheme with Term-Wise Matrix Assembly

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SSRN

SSRN 2023年

The solidification and macro-segregation problem involving unsteady multi-physics and multi-phase fields is typically a complex process with mass, momentum, heat, and constituent transfers among solid, mushy, and liquid phase regions. The quantitative prediction of phase change, chemical heterogeneities, and multi-phase and multi-component flows plays critical roles in many natural scenarios and industrial applications that involve many disciplines, like material, energy, and even planet science. In view of this, some scholars and research institutions have called for more contributors to join the benchmark analysis of solidification and segregation problems. Our work proposes fully decoupled and iterative schemes based on operator-splitting and matrix-based ideas to avoid solving non-linear systems directly. Also, the combination of direct vectorization and equation-based forward matrix assembly techniques enhances the implementability and efficiency of extensions to 3D applications. Lastly, the novel scheme is well validated through a bunch of 2D and 3D benchmark cases. The numerical results also illustrate that this method can ensure accurate prediction and efficient computation. It adequately captures the physical details of phenomena caused by the solutally and thermally driven flow, which include channel segregation, the formation of freckles, edge effect, aspect ratio effect, and 3D effect. © 2023, The Authors. All rights reserved.

关键词： Solidification

Physics-informed neural networks with residual/gradient-based adaptive sampling methods for solving PDEs with sharp solutions

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arXiv 2023年

作者： Mao, Zhiping Meng, Xuhui School of Mathematical Sciences Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing Xiamen University Xiamen China Institute of Interdisciplinary Research for Mathematics and Applied Science School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China

We consider solving the forward and inverse PDEs which have sharp solutions using physics-informed neural networks (PINNs) in this work. In particular, to better capture the sharpness of the solution, we propose adaptive sampling methods (ASMs) based on the residual and the gradient of the solution. We first present a residual only based ASM algorithm denoted by ASM I. In this approach, we first train the neural network by using a small number of residual points and divide the computational domain into a certain number of sub-domains, we then add new residual points in the sub-domain which has the largest mean absolute value of the residual, and those points which have largest absolute values of the residual in this sub-domain will be added as new residual points. We further develop a second type of ASM algorithm (denoted by ASM II) based on both the residual and the gradient of the solution due to the fact that only the residual may be not able to efficiently capture the sharpness of the solution. The procedure of ASM II is almost the same as that of ASM I except that in ASM II, we add new residual points which not only have large residual but also large gradient. To demonstrate the effectiveness of the present methods, we employ both ASM I and ASM II to solve a number of PDEs, including Burger equation, compressible Euler equation, Poisson equation over an L-shape domain as well as high-dimensional Poisson equation. It has been shown from the numerical results that the sharp solutions can be well approximated by using either ASM I or ASM II algorithm, and both methods deliver much more accurate solution than original PINNs with the same number of residual points. Moreover, the ASM II algorithm has better performance in terms of accuracy, efficiency and stability compared with the ASM I algorithm. This means that the gradient of the solution improves the stability and efficiency of the adaptive sampling procedure as well as the accuracy of the solution. Furthermore

关键词： Poisson equation