Journal of Computational Physics最新文献_第2页

Neural operator-based super-fidelity: A warm-start approach for accelerating steady-state simulations

IF 3.8 2区物理与天体物理 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Computational Physics

Pub Date : 2025-02-24 DOI: 10.1016/j.jcp.2025.113871

Xu-Hui Zhou , Jiequn Han , Muhammad I. Zafar , Eric M. Wolf , Christopher R. Schrock , Christopher J. Roy , Heng Xiao

Neural networks have recently emerged as powerful tools for accelerated solving of partial differential equations (PDEs) in both academic and industrial settings. However, their use as standalone surrogate models raises concerns about reliability, as solution accuracy heavily depends on data quality, volume, and training algorithms. This concern is particularly pronounced in tasks that prioritize computational precision and deterministic outcomes. In response, this study introduces “super-fidelity”, a method that employs neural networks for initial warm-starts, significantly speeding up the solution of steady-state PDEs without compromising on accuracy. Drawing from super-resolution in computer vision, super-fidelity maps solutions from low-fidelity computational models to high-fidelity ones using a vector-cloud neural network with equivariance (VCNN-e)—a neural operator that preserves physical symmetries and adapts to different spatial discretizations. We evaluated the proposed method across scenarios with varying degrees of nonlinearity, including (1) two-dimensional laminar flows around elliptical cylinders at low Reynolds numbers, exhibiting monotonic convergence, (2) two-dimensional turbulent flows over airfoils at high Reynolds numbers, characterized by oscillatory convergence, and (3) practical three-dimensional turbulent flows over a wing. The results demonstrate that our neural operator-based initialization can accelerate convergence by at least a factor of two while maintaining the same level of accuracy, outperforming traditional initialization methods using uniform fields or potential flows. The approach's robustness and scalability are confirmed across different linear equation solvers and multi-process computing configurations. Additional investigations highlight its reduced dependence on high quality of training data, and real time savings across multiple simulations, even when including the neural-network model preparation time. Our study presents a promising strategy for accelerated solving of steady-state PDEs using neural operators, ensuring high accuracy in applications where precision is of utmost importance.

{"title":"Neural operator-based super-fidelity: A warm-start approach for accelerating steady-state simulations","authors":"Xu-Hui Zhou , Jiequn Han , Muhammad I. Zafar , Eric M. Wolf , Christopher R. Schrock , Christopher J. Roy , Heng Xiao","doi":"10.1016/j.jcp.2025.113871","DOIUrl":"10.1016/j.jcp.2025.113871","url":null,"abstract":"<div><div>Neural networks have recently emerged as powerful tools for accelerated solving of partial differential equations (PDEs) in both academic and industrial settings. However, their use as standalone surrogate models raises concerns about reliability, as solution accuracy heavily depends on data quality, volume, and training algorithms. This concern is particularly pronounced in tasks that prioritize computational precision and deterministic outcomes. In response, this study introduces “super-fidelity”, a method that employs neural networks for initial warm-starts, significantly speeding up the solution of steady-state PDEs without compromising on accuracy. Drawing from super-resolution in computer vision, super-fidelity maps solutions from low-fidelity computational models to high-fidelity ones using a vector-cloud neural network with equivariance (VCNN-e)—a neural operator that preserves physical symmetries and adapts to different spatial discretizations. We evaluated the proposed method across scenarios with varying degrees of nonlinearity, including (1) two-dimensional laminar flows around elliptical cylinders at low Reynolds numbers, exhibiting monotonic convergence, (2) two-dimensional turbulent flows over airfoils at high Reynolds numbers, characterized by oscillatory convergence, and (3) practical three-dimensional turbulent flows over a wing. The results demonstrate that our neural operator-based initialization can accelerate convergence by at least a factor of two while maintaining the same level of accuracy, outperforming traditional initialization methods using uniform fields or potential flows. The approach's robustness and scalability are confirmed across different linear equation solvers and multi-process computing configurations. Additional investigations highlight its reduced dependence on high quality of training data, and real time savings across multiple simulations, even when including the neural-network model preparation time. Our study presents a promising strategy for accelerated solving of steady-state PDEs using neural operators, ensuring high accuracy in applications where precision is of utmost importance.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"529 ","pages":"Article 113871"},"PeriodicalIF":3.8,"publicationDate":"2025-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143510540","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

A polytopal discrete de Rham complex on manifolds, with application to the Maxwell equations

IF 3.8 2区物理与天体物理 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Computational Physics

Pub Date : 2025-02-24 DOI: 10.1016/j.jcp.2025.113886

Jérôme Droniou , Marien Hanot , Todd Oliynyk

We design in this work a discrete de Rham complex on manifolds. This complex, written in the framework of exterior calculus, has the same cohomology as the continuous de Rham complex, is of arbitrary order of accuracy and, in principle, can be designed on meshes made of generic elements (that is, elements whose boundary is the union of an arbitrary number of curved edges/faces). Notions of local (full and trimmed) polynomial spaces are developed, with compatibility requirements between polynomials on mesh entities of various dimensions. We give explicit constructions of such polynomials in 2D, for some meshes made of curved triangles or quadrangles (such meshes are easy to design in many cases, starting from a few charts describing the manifold). The discrete de Rham complex is then used to set up a scheme for the Maxwell equations on a 2D manifold without boundary, and we show that a natural discrete version of the constraint linking the electric field and the electric charge density is satisfied. Numerical examples are provided on the sphere and the torus, based on bespoke analytical solutions and meshes on each of these manifolds.

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引用次数: 0

Data-enabled reduction of the time complexity of iterative solvers 通过数据降低迭代求解器的时间复杂度

IF 3.8 2区物理与天体物理 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Computational Physics

Pub Date : 2025-02-21 DOI: 10.1016/j.jcp.2025.113859

Yuanwei Bin , Xiang I.A. Yang , Samuel J. Grauer , Robert F. Kunz

In the field of scientific computing, complex matrices arise from Laplace, Burgers, Kuramoto-Sivashinsky, and Allen-Cahn equations that are not necessarily symmetric positive definite. Computational fluid dynamics, in particular, often deals with pressure Poisson equation. For iterative solvers, time complexity is one of the most critical properties, if not the most critical. Its notation is

O (N^{α})

with N denoting the size of the discretized system and α the scaling exponent. This property indicates how an iterative method's performance scales with the size of the discretized system. Due to the large size of systems in today's scientific computing, methods with lower time complexity are almost always preferred over those with higher time complexity, regardless of the prefactor. This emphasis on time complexity reveals a significant gap in the literature: although the integration of data-enabled methodologies in scientific computing has led to the developments of convergence accelerators and the observation of a speedup of O(10) or so, the reported reductions in cost predominantly concern the prefactor rather than the time complexity. This paper aims to explore reduction in time complexity. The accelerator developed in this paper involves projecting the intermediate solution, which is otherwise only used to assess the residual in the baseline iterative method, onto a low-dimensional Hilbert subspace and directly solving the discretized system there. The solver alternates between the baseline iterative method and the accelerator. Our scaling analysis, which is usually not possible for data-based methods, shows a

O (N_{i})

reduction in the time complexity for

N_{i}^{d}

-sized problems in d-dimensional space. Here,

N_{i}

is the number of grids in each dimension, and the system size is

N = N_{i}^{d}

. Consolidated by tests up to 10⁹ degrees of freedom, the present method is shown to offer increasingly more acceleration as the problem size increases, up to 200 times speedup for systems of size 10⁹. Moreover, we demonstrate that the accelerator remains effective for highly nonlinear equations and unstructured grids, yielding similar speedup as for Poisson equation.

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引用次数: 0

A bound- and positivity-preserving path-conservative discontinuous Galerkin method for compressible two-medium flows

IF 3.8 2区物理与天体物理 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Computational Physics

Pub Date : 2025-02-20 DOI: 10.1016/j.jcp.2025.113867

Haiyun Wang , Hongqiang Zhu , Zhen Gao

This work presents a high-order path-conservative Runge-Kutta discontinuous Galerkin method to simulate compressible two-medium flows by solving a γ-based model with the stiffened equation of state. The main contributions are as follows. Firstly, the path-conservative discontinuous Galerkin method is used to solve the γ-based model and is able to preserve uniform velocity and pressure fields around an isolated material interface. Secondly, a conservative-variables-based affine-invariant weighted essentially non-oscillatory limiter is employed to suppress nonlinear instability in the vicinity of discontinuities. Furthermore, an adaptive local Lax-Friedrichs numerical flux is adopted to improve the numerical resolutions. Last but not least, a bound- and positivity-preserving limiting strategy with strict theoretical analysis is developed for the stiffened equation of state to avoid the occurrence of inadmissible solutions while improving the robustness of the simulations. The h-adaptive Cartesian mesh is used for the numerical experiments to verify the validity of proposed method, and the numerical results of various one- and two-dimensional benchmark test cases demonstrate that the proposed method can efficiently and accurately handle complex two-phase flow problems involving pronounced interface deformations and large pressure ratios up to 10⁵.

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引用次数: 0

A class of high-order physics-preserving schemes for thermodynamically consistent model of incompressible and immiscible two-phase flow in porous media

IF 3.8 2区物理与天体物理 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Computational Physics

Pub Date : 2025-02-20 DOI: 10.1016/j.jcp.2025.113864

Xiaoli Li , Yujing Yan , Huangxin Chen

In this paper, we construct several high-order and physics-preserving numerical schemes for thermodynamically consistent model of incompressible and immiscible two-phase flow in porous media based on the modified generalized scalar auxiliary variable (mGSAV) approach with new relaxation and the Lagrange multiplier (LM) method with the well-known Karush-Kuhn-Tucker (KKT) conditions. We construct high-order implicit-explicit BDF-k schemes with first to fifth orders for the system with homogeneous injection/production rate and boundary condition. Due to the fact that high-order BDF-k schemes with

k = 2, 3, 4, 5

are difficult to preserve mass conservation for both phases for the system with inhomogeneous injection/production rate and boundary condition, the first- and second-order schemes are proposed based on the backward Euler and Crank-Nicolson discretizations for the mass conservation constraint equation. The constructed schemes only need to solve one linear system and a nonlinear algebraic equation with negligible computational cost at each time step. We also prove that the proposed schemes are energy stable, mass-conservative and bounds-preserving for each phase without any restrictions of time step size. Finally, various interesting numerical examples are presented to verify the accuracy and efficiency of the proposed schemes.

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引用次数: 0

Novel and general discontinuity-removing PINNs for elliptic interface problems

IF 3.8 2区物理与天体物理 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Computational Physics

Pub Date : 2025-02-20 DOI: 10.1016/j.jcp.2025.113861

Haolong Fan , Zhijun Tan

This paper proposes a novel and general framework of the discontinuity-removing physics-informed neural networks (DR-PINNs) for addressing elliptic interface problems. In the DR-PINNs, the solution is split into a smooth component and a non-smooth component, each represented by a separate network surrogate that can be trained either independently or together. The decoupling strategy involves training the two components sequentially. The first network handles the non-smooth part and pre-learns partial or full jumps to assist the second network in learning the complementary PDE conditions. Three decoupling strategies of handling interface problems are built by removing some jumps and incorporating cusp-capturing techniques. On the other hand, the decoupled approaches rely heavily on the cusp-enforced level-set function and are less efficient due to the need for two separate training stages. To overcome these limitations, a novel DR-PINN coupled approach is proposed in this work, where both components learn complementary conditions simultaneously in an integrated single network, eliminating the need for cusp-enforced level-set functions. Furthermore, the stability and accuracy of training are enhanced by an innovative architecture of the lightweight feedforward neural network (FNN) and a powerful geodesic acceleration Levenberg-Marquardt (gd-LM) optimizer. Several numerical experiments illustrate the effectiveness and great potential of the proposed method, with accuracy outperforming most deep neural network approaches and achieving the state-of-the-art results.

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引用次数: 0

A fast neural hybrid Newton solver adapted to implicit methods for nonlinear dynamics

IF 3.8 2区物理与天体物理 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Computational Physics

Pub Date : 2025-02-19 DOI: 10.1016/j.jcp.2025.113869

Tianyu Jin , Georg Maierhofer , Katharina Schratz , Yang Xiang

The use of implicit time-stepping schemes for the numerical approximation of solutions to stiff nonlinear time-evolution equations brings well-known advantages including, typically, better stability behaviour and corresponding support of larger time steps, and better structure preservation properties. However, this comes at the price of having to solve a nonlinear equation at every time step of the numerical scheme. In this work, we propose a novel deep learning based hybrid Newton's method to accelerate this solution of the nonlinear time step system for stiff time-evolution nonlinear equations. We propose a targeted learning strategy which facilitates robust unsupervised learning in an offline phase and provides a highly efficient initialisation for the Newton iteration leading to consistent acceleration of Newton's method. A quantifiable rate of improvement in Newton's method achieved by improved initialisation is provided and we analyse the upper bound of the generalisation error of our unsupervised learning strategy. These theoretical results are supported by extensive numerical results, demonstrating the efficiency of our proposed neural hybrid solver both in one- and two-dimensional cases.

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引用次数: 0

Wave propagation modeling using machine learning-based finite difference scheme

IF 3.8 2区物理与天体物理 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Computational Physics

Pub Date : 2025-02-19 DOI: 10.1016/j.jcp.2025.113870

Duofa Ji , Chenxi Li , Changhai Zhai

The staggered-grid finite-difference (SGFD) method is essential in wave forward modeling, waveform inversion, and seismic imaging. However, the numerical dispersion that can lead to reduced accuracy in simulations may arise from either coarse spatial discretization or a suboptimal SGFD scheme. Given the high computational cost associated with finer spatial steps, employing the optimal SGFD scheme offers a feasible and effective approach for dispersion suppression. However, the commonly used SGFD schemes are limited by a narrow maximum wavenumber range, reducing their dispersion suppression efficacy. To address this issue, a machine learning-based SGFD scheme is presented in this study. A composite objective function that combines the sum of the absolute error and the maximum absolute error is proposed, aiming to broaden the maximum wavenumber range while minimizing the cumulative error. A physics-consistent neural network is constructed by specifying weights, biases, activation functions, layer connections, and loss function, enabling the back-propagation of the proposed objective function within the machine learning framework to yield globally optimal SGFD coefficients.

{"title":"Wave propagation modeling using machine learning-based finite difference scheme","authors":"Duofa Ji , Chenxi Li , Changhai Zhai","doi":"10.1016/j.jcp.2025.113870","DOIUrl":"10.1016/j.jcp.2025.113870","url":null,"abstract":"<div><div>The staggered-grid finite-difference (SGFD) method is essential in wave forward modeling, waveform inversion, and seismic imaging. However, the numerical dispersion that can lead to reduced accuracy in simulations may arise from either coarse spatial discretization or a suboptimal SGFD scheme. Given the high computational cost associated with finer spatial steps, employing the optimal SGFD scheme offers a feasible and effective approach for dispersion suppression. However, the commonly used SGFD schemes are limited by a narrow maximum wavenumber range, reducing their dispersion suppression efficacy. To address this issue, a machine learning-based SGFD scheme is presented in this study. A composite objective function that combines the sum of the absolute error and the maximum absolute error is proposed, aiming to broaden the maximum wavenumber range while minimizing the cumulative error. A physics-consistent neural network is constructed by specifying weights, biases, activation functions, layer connections, and loss function, enabling the back-propagation of the proposed objective function within the machine learning framework to yield globally optimal SGFD coefficients.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"529 ","pages":"Article 113870"},"PeriodicalIF":3.8,"publicationDate":"2025-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143471641","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

An improved SPH method for compressible flows with high density ratios

IF 3.8 2区物理与天体物理 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Computational Physics

Pub Date : 2025-02-19 DOI: 10.1016/j.jcp.2025.113868

Wenbo Fan , Delong Xiao , Jun Liu

An improved Smoothed Particle Hydrodynamics (SPH) method is proposed for supersonic compressible flows with high density ratios. Based on the Reproducing Kernel Particle Method (RKPM), the interaction forces are adjusted by incorporating a pressure correction factor that accounts for the sound speed in the momentum and thermal energy equations. The modification enhances the accuracy and stability of the SPH method when calculating supersonic compressible flows with high density ratios while ensuring conservation. Numerical results demonstrate the capability of the improved SPH method to accurately capture shock waves and rarefaction waves in benchmark problems. Furthermore, compared with the RKPM, the improved SPH method exhibits reduced uncertainty at contact discontinuities in one dimensional strong shock problems with high density ratios. The method also mitigates unphysical oscillations in physical quantities and imposes lower requirements on the time step. Additionally, when simulating two dimensional compressible flow problems with high density ratios, it can effectively suppress tensile and compressive instabilities.

{"title":"An improved SPH method for compressible flows with high density ratios","authors":"Wenbo Fan , Delong Xiao , Jun Liu","doi":"10.1016/j.jcp.2025.113868","DOIUrl":"10.1016/j.jcp.2025.113868","url":null,"abstract":"<div><div>An improved Smoothed Particle Hydrodynamics (SPH) method is proposed for supersonic compressible flows with high density ratios. Based on the Reproducing Kernel Particle Method (RKPM), the interaction forces are adjusted by incorporating a pressure correction factor that accounts for the sound speed in the momentum and thermal energy equations. The modification enhances the accuracy and stability of the SPH method when calculating supersonic compressible flows with high density ratios while ensuring conservation. Numerical results demonstrate the capability of the improved SPH method to accurately capture shock waves and rarefaction waves in benchmark problems. Furthermore, compared with the RKPM, the improved SPH method exhibits reduced uncertainty at contact discontinuities in one dimensional strong shock problems with high density ratios. The method also mitigates unphysical oscillations in physical quantities and imposes lower requirements on the time step. Additionally, when simulating two dimensional compressible flow problems with high density ratios, it can effectively suppress tensile and compressive instabilities.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"529 ","pages":"Article 113868"},"PeriodicalIF":3.8,"publicationDate":"2025-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143464539","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0

Local conservation of energy in fully implicit PIC algorithms

IF 3.8 2区物理与天体物理 Q2 COMPUTER SCIENCE, INTERDISCIPLINARY APPLICATIONS

Journal of Computational Physics

Pub Date : 2025-02-18 DOI: 10.1016/j.jcp.2025.113862

L. Chacón, G. Chen

We consider the issue of strict, fully discrete local energy conservation for a whole class of fully implicit local-charge- and global-energy-conserving particle-in-cell (PIC) algorithms. Earlier studies demonstrated these algorithms feature strict global energy conservation. However, whether a local energy conservation theorem exists (in which the local energy update is governed by a flux balance equation at every mesh cell) for these schemes is unclear. In this study, we show that a local energy conservation theorem indeed exists. We begin our analysis with the 1D electrostatic PIC model without orbit-averaging, and then generalize our conclusions to account for orbit averaging, multiple dimensions, and electromagnetic models (Darwin). In all cases, a temporally, spatially, and particle-discrete local energy conservation theorem is shown to exist, proving that these formulations (as originally proposed in the literature), in addition to being locally charge conserving and globally energy conserving, are strictly locally energy conserving as well. In contrast to earlier proofs of local conservation in the literature [1], which only considered continuum time, our result is valid for the fully implicit time-discrete version of all models considered, including important features such as orbit averaging. We demonstrate the local-energy-conservation property numerically with a paradigmatic numerical example.

{"title":"Local conservation of energy in fully implicit PIC algorithms","authors":"L. Chacón, G. Chen","doi":"10.1016/j.jcp.2025.113862","DOIUrl":"10.1016/j.jcp.2025.113862","url":null,"abstract":"<div><div>We consider the issue of strict, fully discrete <em>local</em> energy conservation for a whole class of fully implicit local-charge- and global-energy-conserving particle-in-cell (PIC) algorithms. Earlier studies demonstrated these algorithms feature strict global energy conservation. However, whether a local energy conservation theorem exists (in which the local energy update is governed by a flux balance equation at every mesh cell) for these schemes is unclear. In this study, we show that a local energy conservation theorem indeed exists. We begin our analysis with the 1D electrostatic PIC model without orbit-averaging, and then generalize our conclusions to account for orbit averaging, multiple dimensions, and electromagnetic models (Darwin). In all cases, a temporally, spatially, and particle-discrete local energy conservation theorem is shown to exist, proving that these formulations (as originally proposed in the literature), in addition to being locally charge conserving and globally energy conserving, are strictly locally energy conserving as well. In contrast to earlier proofs of local conservation in the literature <span><span>[1]</span></span>, which only considered continuum time, our result is valid for the fully implicit time-discrete version of all models considered, including important features such as orbit averaging. We demonstrate the local-energy-conservation property numerically with a paradigmatic numerical example.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"529 ","pages":"Article 113862"},"PeriodicalIF":3.8,"publicationDate":"2025-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143452951","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

引用次数: 0