{"title":"Computing ground states of Bose-Einstein condensation by normalized deep neural network","authors":"Weizhu Bao , Zhipeng Chang , Xiaofei Zhao","doi":"10.1016/j.jcp.2024.113486","DOIUrl":"10.1016/j.jcp.2024.113486","url":null,"abstract":"<div><div>We propose a normalized deep neural network (norm-DNN) for computing ground states of Bose-Einstein condensation (BEC) via the minimization of the Gross-Pitaevskii energy functional under unitary mass normalization. Compared with the traditional deep neural network for solving partial differential equations, two additional layers are added in training our norm-DNN for solving this kind of unitary constraint minimization problems: (i) a normalization layer is introduced to enforce the unitary mass normalization, and (ii) a shift layer is added to guide the training to non-negative ground state. The proposed norm-DNN gives rise to an efficient unsupervised approach for learning ground states of BEC. Systematical investigations are first carried out through extensive numerical experiments for computing ground states of BEC in one dimension. Extensions to high dimensions and multi-component are then studied in details. The results demonstrate the effectiveness and efficiency of norm-DNN for learning ground states of BEC. Finally, we extend the norm-DNN for computing the first excited states of BEC and discuss parameter generalization issues as well as compare with some existing machine learning methods for computing ground states of BEC in the literature.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"520 ","pages":"Article 113486"},"PeriodicalIF":3.8,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142425886","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}
{"title":"A projection-based time-segmented reduced order model for fluid-structure interactions","authors":"Qijia Zhai , Shiquan Zhang , Pengtao Sun , Xiaoping Xie","doi":"10.1016/j.jcp.2024.113481","DOIUrl":"10.1016/j.jcp.2024.113481","url":null,"abstract":"<div><div>In this paper, a type of novel projection-based, time-segmented reduced order model (ROM) is proposed for dynamic fluid-structure interaction (FSI) problems based upon the arbitrary Lagrangian–Eulerian (ALE)-finite element method (FEM) in a monolithic frame, where spatially, each variable is separated from others in terms of their attribution (fluid/structure), category (velocity/pressure) and component (two/three dimension) while temporally, the proper orthogonal decomposition (POD) bases are constructed in some deliberately partitioned time segments tailored through extensive numerical trials. By the combination of spatial and temporal decompositions, the developed ROM approach enables prolonged simulations under prescribed accuracy thresholds. Numerical experiments are carried out to compare numerical performances of the proposed ROM with corresponding full-order model (FOM) by solving a two-dimensional FSI benchmark problem that involves a vibrating elastic beam in the fluid, where the performance of offline ROM on perturbed physical parameters in the online phase is investigated as well. Extensive numerical results demonstrate that the proposed ROM has a comparable accuracy to while much higher efficiency than the FOM. The developed ROM approach is dimension-independent and can be seamlessly extended to solve high dimensional FSI problems.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"520 ","pages":"Article 113481"},"PeriodicalIF":3.8,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142425953","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}
Cheng Liu , Yiding Hu , Ruoqing Gao , Changhong Hu
{"title":"Robust treatment for the coarse/fine interface of adaptive mesh in the simulation of two-phase flow","authors":"Cheng Liu , Yiding Hu , Ruoqing Gao , Changhong Hu","doi":"10.1016/j.jcp.2024.113485","DOIUrl":"10.1016/j.jcp.2024.113485","url":null,"abstract":"<div><div>We propose a numerical framework for simulating two-phase flow considering gravity and surface tension, and implement it with adaptive mesh. The framework employs a pressure compensation method and a jump model, ensuring accurate modeling of gravity and surface tension when the interface crosses between adaptive meshes of different refinement levels. Additional treatments have been developed to accommodate existing interface capturing and curvature estimation approaches for the adaptive mesh. The method is straightforward to implement and significantly reduces the unnecessary refinement around the free surface. Numerical validations demonstrate the robustness of the treatment for coarse/fine interfaces, with overall accuracy converging and comparable to results obtained with a uniform mesh. Furthermore, the numerical approach accurately reproduces the physics of bubble rising and jet capillary breakup.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"520 ","pages":"Article 113485"},"PeriodicalIF":3.8,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142425962","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}
{"title":"Neural dynamical operator: Continuous spatial-temporal model with gradient-based and derivative-free optimization methods","authors":"Chuanqi Chen, Jin-Long Wu","doi":"10.1016/j.jcp.2024.113480","DOIUrl":"10.1016/j.jcp.2024.113480","url":null,"abstract":"<div><div>Data-driven modeling techniques have been explored in the spatial-temporal modeling of complex dynamical systems for many engineering applications. However, a systematic approach is still lacking to leverage the information from different types of data, e.g., with different spatial and temporal resolutions, and the combined use of short-term trajectories and long-term statistics. In this work, we build on the recent progress of neural operator and present a data-driven modeling framework called neural dynamical operator that is continuous in both space and time. A key feature of the neural dynamical operator is the resolution-invariance with respect to both spatial and temporal discretizations, without demanding abundant training data in different temporal resolutions. To improve the long-term performance of the calibrated model, we further propose a hybrid optimization scheme that leverages both gradient-based and derivative-free optimization methods and efficiently trains on both short-term time series and long-term statistics. We investigate the performance of the neural dynamical operator with three numerical examples, including the viscous Burgers' equation, the Navier–Stokes equations, and the Kuramoto–Sivashinsky equation. The results confirm the resolution-invariance of the proposed modeling framework and also demonstrate stable long-term simulations with only short-term time series data. In addition, we show that the proposed model can better predict long-term statistics via the hybrid optimization scheme with a combined use of short-term and long-term data.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"520 ","pages":"Article 113480"},"PeriodicalIF":3.8,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142425960","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}
{"title":"Framework of acoustic analysis and shape optimization for three-dimensional doubly periodic multilayered structures","authors":"Fuhang Jiang , Toru Takahashi , Changjun Zheng , Toshiro Matsumoto , Haibo Chen","doi":"10.1016/j.jcp.2024.113483","DOIUrl":"10.1016/j.jcp.2024.113483","url":null,"abstract":"<div><div>In this research, a framework of acoustic analysis and shape optimization, based on isogeometric boundary element method (IGA-BEM), is proposed for three-dimensional doubly periodic multilayered structures. The study addresses a gap in the literature by focusing on the shape optimization of such structures, which has not been extensively explored previously. The interface between different acoustic media is an infinite doubly periodic surface, which can be constructed by an open non-uniform rational B-splines. A periodic IGA-BEM is developed for the sound field analysis of the doubly periodic multilayered structure, in which the Ewald method is used to accelerate the calculation of periodic Green function. Furthermore, the shape derivative of the doubly periodic multiple boundaries is derived by imposing boundary perturbation and using the adjoint variable method. The control points of the NURBS surfaces are defined as the shape design variables, and all shape sensitivities can be quickly calculated by discretizing the shape derivative formula. Finally, in according with shape sensitivities, the corresponding shape optimization problem is solved by the method of moving asymptotes, so that the optimized shape design can be obtained. A series of numerical examples validates the accuracy and applicability of the proposed approaches.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"520 ","pages":"Article 113483"},"PeriodicalIF":3.8,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142425610","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}
{"title":"An augmented streamline upwind/Petrov-Galerkin method for the time-spectral convection-diffusion equation","authors":"Mahdi Esmaily, Dongjie Jia","doi":"10.1016/j.jcp.2024.113484","DOIUrl":"10.1016/j.jcp.2024.113484","url":null,"abstract":"<div><div>Discretizing a solution in the spectral rather than time domain presents a significant advantage in solving transport problems encountered in fields like cardiorespiratory modeling, where the flow varies smoothly and periodically in time. To solve the system expressed in the frequency domain, one may rely on the classical time domain upwind techniques, such as the streamline upwind/Petrov-Galerkin (SUPG). While these classical methods successfully remove spurious oscillations in the solution in convection dominated flows, their accuracy deteriorates in a time-spectral setting as the element Womersley number approaches one. To overcome this limitation, this study introduces a new stabilized method, which we call augmented SUPG (ASU). The ASU is a consistent weighted residual method with two complex-valued stabilization parameters that act independently on the source and convective trial functions. Through a series of test cases, the superior accuracy of the ASU in comparison to four classical methods is shown across a wide range of flow conditions.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"520 ","pages":"Article 113484"},"PeriodicalIF":3.8,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142425958","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}
Ange Pacifique Ishimwe , Eric Deleersnijder , Vincent Legat , Jonathan Lambrechts
{"title":"A multi-scale IMEX second order Runge-Kutta method for 3D hydrodynamic ocean models","authors":"Ange Pacifique Ishimwe , Eric Deleersnijder , Vincent Legat , Jonathan Lambrechts","doi":"10.1016/j.jcp.2024.113482","DOIUrl":"10.1016/j.jcp.2024.113482","url":null,"abstract":"<div><div>Understanding complex physical phenomena often involves dealing with partial differential equations (PDEs) where different phenomena exhibit distinct timescales. Fast terms, associated with short characteristic times, coexist with slower ones requiring relatively longer time steps for resolution. The challenge becomes more manageable when, despite the varying characteristic times of fast and slow terms, the computational cost associated with faster terms is significantly lower than that of slower terms. Additionally, slower terms can also exhibit two distinct longer characteristic times, adding complexity to the system and resulting in a total of three characteristic timescales. In this paper, an innovative split second-order IMEX (IMplicit-EXplicit) temporal scheme is introduced to address this temporal complexity. It is used to solve the primitive equation ocean model. Extremely short times are handled explicitly with small time steps, while longer timescales are managed explicitly and semi-implicitly using larger time steps. The decision to solve a portion of the slower terms semi-implicitly is due to the fact that it does not significantly increase the total computational cost, allowing for greater flexibility in the time step without imposing a substantial burden on the overall computational efficiency. This strategy enables efficient management of the various temporal scales present in the equations, thereby optimizing computational resources. The proposed scheme is applied to solve 3D hydrodynamics equations encompassing three time scale: fast terms representing wave phenomena, slow terms describing horizontal aspects and stiff terms for vertical ones. Furthermore, the scheme is designed to respect crucial physical properties, namely global and local conservation. The obtained results on different test cases demonstrate the robustness and efficiency of the IMEX approach in simulating these complex systems.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"520 ","pages":"Article 113482"},"PeriodicalIF":3.8,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142425955","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}
{"title":"An analysis and solution of ill-conditioning in physics-informed neural networks","authors":"Wenbo Cao , Weiwei Zhang","doi":"10.1016/j.jcp.2024.113494","DOIUrl":"10.1016/j.jcp.2024.113494","url":null,"abstract":"<div><div>Physics-informed neural networks (PINNs) have recently emerged as a novel and popular approach for solving forward and inverse problems involving partial differential equations (PDEs). However, ensuring stable training and obtaining accurate results remain challenging in many scenarios, often attributed to the ill-conditioning of PINNs. Despite this, a deeper analysis is still lacking, which hampers progress and application of PINNs in complex engineering problems. Drawing inspiration from the ill-conditioning analysis in traditional numerical methods, we establish a strong connection between the ill-conditioning of PINNs and the Jacobian matrix of the PDE system. Specifically, for any given PDE system, we construct a controlled system that allows for the adjustment of the Jacobian matrix's condition number while retaining the same solution as the original system. Our numerical experiments show that as the condition number of the Jacobian matrix decreases, PINNs exhibit faster convergence and higher accuracy. Building upon this principle and the extension of controlled systems, we propose a general approach to mitigate the ill-conditioning in PINNs, leading to successful simulations of three-dimensional flow around the M6 wing at a Reynolds number of 5,000. To the best of our knowledge, this is the first time that PINNs have successfully simulated such complex systems, offering a promising new technique for addressing industrial complexity problems. Our findings also provide valuable insights to guide the future development of PINNs.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"520 ","pages":"Article 113494"},"PeriodicalIF":3.8,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142433050","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}
{"title":"A method for numerical simulation of shock waves in rarefied gas mixtures based on direct solution of the Boltzmann kinetic equation","authors":"S.S. Sitnikov , F.G. Tcheremissine","doi":"10.1016/j.jcp.2024.113463","DOIUrl":"10.1016/j.jcp.2024.113463","url":null,"abstract":"<div><div>The paper proposes an approach to numerical simulation of shock waves in rarefied gas mixtures on the basis of direct solution of the Boltzmann kinetic equation. Software for simulating the gas flows was developed. The structure of a shock wave in a binary gas mixture was computed with an accuracy controlled by the computational parameters. The computations were performed for various molecular masses ratios and Mach numbers. The total accuracy of at least 1.4% for the local values of the molecular densities and temperatures of the mixture components was achieved. Numerical simulation of a shock wave propagation through a periodically perforated surface was performed. The distributions of the macroscopic characteristics of the mixture components at various points in time were obtained. Unsteady areas of strong separation of the gas mixture components were discovered.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"520 ","pages":"Article 113463"},"PeriodicalIF":3.8,"publicationDate":"2024-10-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142425952","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}
{"title":"A learning based numerical method for Helmholtz equations with high frequency","authors":"Yu Chen , Jin Cheng , Tingyue Li , Yun Miao","doi":"10.1016/j.jcp.2024.113478","DOIUrl":"10.1016/j.jcp.2024.113478","url":null,"abstract":"<div><div>High-frequency issues have been remarkable challenges in numerical methods for partial differential equations. In this paper, a learning based numerical method (LbNM) is proposed for Helmholtz equation with high frequency. The main novelty is using Tikhonov regularization to stably learn the solution operator by utilizing relevant information especially the fundamental solutions. Then applying the solution operator to a new boundary input could quickly update the solution. Based on the method of fundamental solutions and the quantitative Runge approximation, we give the error estimate. This indicates interpretability and generalizability of the present method. Numerical results validate the error analysis and demonstrate the high-precision and high-efficiency features.</div></div>","PeriodicalId":352,"journal":{"name":"Journal of Computational Physics","volume":"520 ","pages":"Article 113478"},"PeriodicalIF":3.8,"publicationDate":"2024-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142425951","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}