This paper develops high-order accurate, well-balanced (WB), and positivity-preserving (PP) finite volume schemes for shallow water equations on adaptive moving structured meshes. The mesh movement poses new challenges in maintaining the WB property, which not only depends on the balance between flux gradients and source terms but is also affected by the mesh movement. To address these complexities, the WB property in curvilinear coordinates is decomposed into flux source balance and mesh movement balance. The flux source balance is achieved by suitable decomposition of the source terms, the numerical fluxes based on hydrostatic reconstruction, and appropriate discretization of the geometric conservation laws (GCLs). Concurrently, the mesh movement balance is maintained by integrating additional schemes to update the bottom topography during mesh adjustments. The proposed schemes are rigorously proven to maintain the WB property by using the discrete GCLs and these two balances. We provide rigorous analyses of the PP property under a sufficient condition enforced by a PP limiter. Due to the involvement of mesh metrics and movement, the analyses are nontrivial, while some standard techniques, such as splitting high-order schemes into convex combinations of formally first-order PP schemes, are not directly applicable. Various numerical examples validate the high-order accuracy, high efficiency, WB, and PP properties of the proposed schemes.
{"title":"High-order accurate structure-preserving finite volume schemes on adaptive moving meshes for shallow water equations: Well-balancedness and positivity","authors":"Zhihao Zhang, Huazhong Tang, Kailiang Wu","doi":"arxiv-2409.09600","DOIUrl":"https://doi.org/arxiv-2409.09600","url":null,"abstract":"This paper develops high-order accurate, well-balanced (WB), and\u0000positivity-preserving (PP) finite volume schemes for shallow water equations on\u0000adaptive moving structured meshes. The mesh movement poses new challenges in\u0000maintaining the WB property, which not only depends on the balance between flux\u0000gradients and source terms but is also affected by the mesh movement. To\u0000address these complexities, the WB property in curvilinear coordinates is\u0000decomposed into flux source balance and mesh movement balance. The flux source\u0000balance is achieved by suitable decomposition of the source terms, the\u0000numerical fluxes based on hydrostatic reconstruction, and appropriate\u0000discretization of the geometric conservation laws (GCLs). Concurrently, the\u0000mesh movement balance is maintained by integrating additional schemes to update\u0000the bottom topography during mesh adjustments. The proposed schemes are\u0000rigorously proven to maintain the WB property by using the discrete GCLs and\u0000these two balances. We provide rigorous analyses of the PP property under a\u0000sufficient condition enforced by a PP limiter. Due to the involvement of mesh\u0000metrics and movement, the analyses are nontrivial, while some standard\u0000techniques, such as splitting high-order schemes into convex combinations of\u0000formally first-order PP schemes, are not directly applicable. Various numerical\u0000examples validate the high-order accuracy, high efficiency, WB, and PP\u0000properties of the proposed schemes.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"215O 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259105","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Chengxi Ye, Grace Chu, Yanfeng Liu, Yichi Zhang, Lukasz Lew, Andrew Howard
The discontinuous operations inherent in quantization and sparsification introduce obstacles to backpropagation. This is particularly challenging when training deep neural networks in ultra-low precision and sparse regimes. We propose a novel, robust, and universal solution: a denoising affine transform that stabilizes training under these challenging conditions. By formulating quantization and sparsification as perturbations during training, we derive a perturbation-resilient approach based on ridge regression. Our solution employs a piecewise constant backbone model to ensure a performance lower bound and features an inherent noise reduction mechanism to mitigate perturbation-induced corruption. This formulation allows existing models to be trained at arbitrarily low precision and sparsity levels with off-the-shelf recipes. Furthermore, our method provides a novel perspective on training temporal binary neural networks, contributing to ongoing efforts to narrow the gap between artificial and biological neural networks.
{"title":"Robust Training of Neural Networks at Arbitrary Precision and Sparsity","authors":"Chengxi Ye, Grace Chu, Yanfeng Liu, Yichi Zhang, Lukasz Lew, Andrew Howard","doi":"arxiv-2409.09245","DOIUrl":"https://doi.org/arxiv-2409.09245","url":null,"abstract":"The discontinuous operations inherent in quantization and sparsification\u0000introduce obstacles to backpropagation. This is particularly challenging when\u0000training deep neural networks in ultra-low precision and sparse regimes. We\u0000propose a novel, robust, and universal solution: a denoising affine transform\u0000that stabilizes training under these challenging conditions. By formulating\u0000quantization and sparsification as perturbations during training, we derive a\u0000perturbation-resilient approach based on ridge regression. Our solution employs\u0000a piecewise constant backbone model to ensure a performance lower bound and\u0000features an inherent noise reduction mechanism to mitigate perturbation-induced\u0000corruption. This formulation allows existing models to be trained at\u0000arbitrarily low precision and sparsity levels with off-the-shelf recipes.\u0000Furthermore, our method provides a novel perspective on training temporal\u0000binary neural networks, contributing to ongoing efforts to narrow the gap\u0000between artificial and biological neural networks.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"82 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259156","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the asymptotic error between the finite element solutions of nonlocal models with a bounded interaction neighborhood and the exact solution of the limiting local model. The limit corresponds to the case when the horizon parameter, the radius of the spherical nonlocal interaction neighborhood of the nonlocal model, and the mesh size simultaneously approach zero. Two important cases are discussed: one involving the original nonlocal models and the other for nonlocal models with polygonal approximations of the nonlocal interaction neighborhood. Results of numerical experiments are also reported to substantiate the theoretical studies.
{"title":"Error estimates of finite element methods for nonlocal problems using exact or approximated interaction neighborhoods","authors":"Qiang Du, Hehu Xie, Xiaobo Yin, Jiwei Zhang","doi":"arxiv-2409.09270","DOIUrl":"https://doi.org/arxiv-2409.09270","url":null,"abstract":"We study the asymptotic error between the finite element solutions of\u0000nonlocal models with a bounded interaction neighborhood and the exact solution\u0000of the limiting local model. The limit corresponds to the case when the horizon\u0000parameter, the radius of the spherical nonlocal interaction neighborhood of the\u0000nonlocal model, and the mesh size simultaneously approach zero. Two important\u0000cases are discussed: one involving the original nonlocal models and the other\u0000for nonlocal models with polygonal approximations of the nonlocal interaction\u0000neighborhood. Results of numerical experiments are also reported to\u0000substantiate the theoretical studies.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"4 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259149","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The block tensor of trifocal tensors provides crucial geometric information on the three-view geometry of a scene. The underlying synchronization problem seeks to recover camera poses (locations and orientations up to a global transformation) from the block trifocal tensor. We establish an explicit Tucker factorization of this tensor, revealing a low multilinear rank of $(6,4,4)$ independent of the number of cameras under appropriate scaling conditions. We prove that this rank constraint provides sufficient information for camera recovery in the noiseless case. The constraint motivates a synchronization algorithm based on the higher-order singular value decomposition of the block trifocal tensor. Experimental comparisons with state-of-the-art global synchronization methods on real datasets demonstrate the potential of this algorithm for significantly improving location estimation accuracy. Overall this work suggests that higher-order interactions in synchronization problems can be exploited to improve performance, beyond the usual pairwise-based approaches.
{"title":"Tensor-Based Synchronization and the Low-Rankness of the Block Trifocal Tensor","authors":"Daniel Miao, Gilad Lerman, Joe Kileel","doi":"arxiv-2409.09313","DOIUrl":"https://doi.org/arxiv-2409.09313","url":null,"abstract":"The block tensor of trifocal tensors provides crucial geometric information\u0000on the three-view geometry of a scene. The underlying synchronization problem\u0000seeks to recover camera poses (locations and orientations up to a global\u0000transformation) from the block trifocal tensor. We establish an explicit Tucker\u0000factorization of this tensor, revealing a low multilinear rank of $(6,4,4)$\u0000independent of the number of cameras under appropriate scaling conditions. We\u0000prove that this rank constraint provides sufficient information for camera\u0000recovery in the noiseless case. The constraint motivates a synchronization\u0000algorithm based on the higher-order singular value decomposition of the block\u0000trifocal tensor. Experimental comparisons with state-of-the-art global\u0000synchronization methods on real datasets demonstrate the potential of this\u0000algorithm for significantly improving location estimation accuracy. Overall\u0000this work suggests that higher-order interactions in synchronization problems\u0000can be exploited to improve performance, beyond the usual pairwise-based\u0000approaches.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259155","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is devoted to the convergence theory of two-grid methods for symmetric positive semidefinite linear systems, with particular focus on the singular case. In the case where the Moore--Penrose inverse of coarse-grid matrix is used as a coarse solver, we derive a succinct identity for characterizing the convergence factor of two-grid methods. More generally, we present some convergence estimates for two-grid methods with approximate coarse solvers, including both linear and general cases. A key feature of our analysis is that it does not require any additional assumptions on the system matrix, especially on its null space.
{"title":"Two-grid convergence theory for symmetric positive semidefinite linear systems","authors":"Xuefeng Xu","doi":"arxiv-2409.09442","DOIUrl":"https://doi.org/arxiv-2409.09442","url":null,"abstract":"This paper is devoted to the convergence theory of two-grid methods for\u0000symmetric positive semidefinite linear systems, with particular focus on the\u0000singular case. In the case where the Moore--Penrose inverse of coarse-grid\u0000matrix is used as a coarse solver, we derive a succinct identity for\u0000characterizing the convergence factor of two-grid methods. More generally, we\u0000present some convergence estimates for two-grid methods with approximate coarse\u0000solvers, including both linear and general cases. A key feature of our analysis\u0000is that it does not require any additional assumptions on the system matrix,\u0000especially on its null space.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"45 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259142","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper is concerned with the inverse elastic scattering problem to determine the shape and location of an elastic cavity. By establishing a one-to-one correspondence between the Herglotz wave function and its kernel, we introduce the far-field operator which is crucial in the factorization method. We present a theoretical factorization of the far-field operator and rigorously prove the properties of its associated operators involved in the factorization. Unlike the Dirichlet problem where the boundary integral operator of the single-layer potential involved in the factorization of the far-field operator is weakly singular, the boundary integral operator of the conormal derivative of the double-layer potential involved in the factorization of the far-field operator with Neumann boundary conditions is hypersingular, which forces us to prove that this operator is isomorphic using Fredholm's theorem. Meanwhile, we present theoretical analyses of the factorization method for various illumination and measurement cases, including compression-wave illumination and compression-wave measurement, shear-wave illumination and shear-wave measurement, and full-wave illumination and full-wave measurement. In addition, we also consider the limited aperture problem and provide a rigorous theoretical analysis of the factorization method in this case. Numerous numerical experiments are carried out to demonstrate the effectiveness of the proposed method, and to analyze the influence of various factors, such as polarization direction, frequency, wavenumber, and multi-scale scatterers on the reconstructed results.
{"title":"Factorization method for inverse elastic cavity scattering","authors":"Shuxin Li, Junliang Lv, Yi Wang","doi":"arxiv-2409.09434","DOIUrl":"https://doi.org/arxiv-2409.09434","url":null,"abstract":"This paper is concerned with the inverse elastic scattering problem to\u0000determine the shape and location of an elastic cavity. By establishing a\u0000one-to-one correspondence between the Herglotz wave function and its kernel, we\u0000introduce the far-field operator which is crucial in the factorization method.\u0000We present a theoretical factorization of the far-field operator and rigorously\u0000prove the properties of its associated operators involved in the factorization.\u0000Unlike the Dirichlet problem where the boundary integral operator of the\u0000single-layer potential involved in the factorization of the far-field operator\u0000is weakly singular, the boundary integral operator of the conormal derivative\u0000of the double-layer potential involved in the factorization of the far-field\u0000operator with Neumann boundary conditions is hypersingular, which forces us to\u0000prove that this operator is isomorphic using Fredholm's theorem. Meanwhile, we\u0000present theoretical analyses of the factorization method for various\u0000illumination and measurement cases, including compression-wave illumination and\u0000compression-wave measurement, shear-wave illumination and shear-wave\u0000measurement, and full-wave illumination and full-wave measurement. In addition,\u0000we also consider the limited aperture problem and provide a rigorous\u0000theoretical analysis of the factorization method in this case. Numerous\u0000numerical experiments are carried out to demonstrate the effectiveness of the\u0000proposed method, and to analyze the influence of various factors, such as\u0000polarization direction, frequency, wavenumber, and multi-scale scatterers on\u0000the reconstructed results.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"204 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259108","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the last decade, tensors have shown their potential as valuable tools for various tasks in numerical linear algebra. While most of the research has been focusing on how to compress a given tensor in order to maintain information as well as reducing the storage demand for its allocation, the solution of linear tensor equations is a less explored venue. Even if many of the routines available in the literature are based on alternating minimization schemes (ALS), we pursue a different path and utilize Krylov methods instead. The use of Krylov methods in the tensor realm is not new. However, these routines often turn out to be rather expensive in terms of computational cost and ALS procedures are preferred in practice. We enhance Krylov methods for linear tensor equations with a panel of diverse randomization-based strategies which remarkably increase the efficiency of these solvers making them competitive with state-of-the-art ALS schemes. The up-to-date randomized approaches we employ range from sketched Krylov methods with incomplete orthogonalization and structured sketching transformations to streaming algorithms for tensor rounding. The promising performance of our new solver for linear tensor equations is demonstrated by many numerical results.
在过去十年中，张量已经显示出其作为数值线性代数中各种任务的重要工具的潜力。虽然大部分研究都集中在如何压缩给定张量以保持信息以及减少其分配的存储需求上，但线性张量方程的求解是一个探索较少的领域。尽管文献中的许多例程都是基于交替最小化方案（ALS），但我们却另辟蹊径，采用了克雷洛夫方法。在张量领域使用克雷洛夫方法并不新鲜。然而，这些例程的计算成本往往相当昂贵，因此 ALS 程序在实践中更受青睐。我们通过一系列基于随机化的策略来增强线性张量方程的 Krylov 方法，这些策略显著提高了求解器的效率，使其与最先进的 ALS 方案相媲美。我们采用的最新随机化方法包括具有不完全正交化和结构化草图变换的草图 Krylov 方法，以及用于张量包围的流算法。许多数值结果表明，我们的新求解器对线性张弦具有良好的性能。
{"title":"Randomized sketched TT-GMRES for linear systems with tensor structure","authors":"Alberto Bucci, Davide Palitta, Leonardo Robol","doi":"arxiv-2409.09471","DOIUrl":"https://doi.org/arxiv-2409.09471","url":null,"abstract":"In the last decade, tensors have shown their potential as valuable tools for\u0000various tasks in numerical linear algebra. While most of the research has been\u0000focusing on how to compress a given tensor in order to maintain information as\u0000well as reducing the storage demand for its allocation, the solution of linear\u0000tensor equations is a less explored venue. Even if many of the routines\u0000available in the literature are based on alternating minimization schemes\u0000(ALS), we pursue a different path and utilize Krylov methods instead. The use\u0000of Krylov methods in the tensor realm is not new. However, these routines often\u0000turn out to be rather expensive in terms of computational cost and ALS\u0000procedures are preferred in practice. We enhance Krylov methods for linear\u0000tensor equations with a panel of diverse randomization-based strategies which\u0000remarkably increase the efficiency of these solvers making them competitive\u0000with state-of-the-art ALS schemes. The up-to-date randomized approaches we\u0000employ range from sketched Krylov methods with incomplete orthogonalization and\u0000structured sketching transformations to streaming algorithms for tensor\u0000rounding. The promising performance of our new solver for linear tensor\u0000equations is demonstrated by many numerical results.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259106","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Although Lattice Boltzmann Method (LBM) is relatively straightforward, it demands a well-crafted framework to handle the complex partial differential equations involved in multiphase flow simulations. This document presents some potential strategies for developing an Eulerian-Eulerian LBM solver tailored for multiphase systems. The paper first states what are the starting equations governing a multiphase flow in classical CFD. Secondly, it derives a pseudo-compressible (targeting the incompressible limit) system of equations for deriving the Eulerian-Eulerian LBM framework to simulate multiphase flows. Finally, a dispersed phase volume fraction equation is provided to balance the degree of freedom less due to the pressure gradient coupling. The effectiveness of these approaches can only be confirmed through rigorous numerical experimentation.
{"title":"Lattice Boltzmann framework for multiphase flows by Eulerian-Eulerian Navier-Stokes equations","authors":"Matteo Maria Piredda, Pietro Asinari","doi":"arxiv-2409.10399","DOIUrl":"https://doi.org/arxiv-2409.10399","url":null,"abstract":"Although Lattice Boltzmann Method (LBM) is relatively straightforward, it\u0000demands a well-crafted framework to handle the complex partial differential\u0000equations involved in multiphase flow simulations. This document presents some\u0000potential strategies for developing an Eulerian-Eulerian LBM solver tailored\u0000for multiphase systems. The paper first states what are the starting equations\u0000governing a multiphase flow in classical CFD. Secondly, it derives a\u0000pseudo-compressible (targeting the incompressible limit) system of equations\u0000for deriving the Eulerian-Eulerian LBM framework to simulate multiphase flows.\u0000Finally, a dispersed phase volume fraction equation is provided to balance the\u0000degree of freedom less due to the pressure gradient coupling. The effectiveness\u0000of these approaches can only be confirmed through rigorous numerical\u0000experimentation.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"17 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259094","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Deep operator networks (DeepONet) and neural operators have gained significant attention for their ability to map infinite-dimensional function spaces and perform zero-shot super-resolution. However, these models often require large datasets for effective training. While physics-informed operators offer a data-agnostic learning approach, they introduce additional training complexities and convergence issues, especially in highly nonlinear systems. To overcome these challenges, we introduce Finite Basis Physics-Informed HyperDeepONet (FB-HyDON), an advanced operator architecture featuring intrinsic domain decomposition. By leveraging hypernetworks and finite basis functions, FB-HyDON effectively mitigates the training limitations associated with existing physics-informed operator learning methods. We validated our approach on the high-frequency harmonic oscillator, Burgers' equation at different viscosity levels, and Allen-Cahn equation demonstrating substantial improvements over other operator learning models.
{"title":"FB-HyDON: Parameter-Efficient Physics-Informed Operator Learning of Complex PDEs via Hypernetwork and Finite Basis Domain Decomposition","authors":"Milad Ramezankhani, Rishi Yash Parekh, Anirudh Deodhar, Dagnachew Birru","doi":"arxiv-2409.09207","DOIUrl":"https://doi.org/arxiv-2409.09207","url":null,"abstract":"Deep operator networks (DeepONet) and neural operators have gained\u0000significant attention for their ability to map infinite-dimensional function\u0000spaces and perform zero-shot super-resolution. However, these models often\u0000require large datasets for effective training. While physics-informed operators\u0000offer a data-agnostic learning approach, they introduce additional training\u0000complexities and convergence issues, especially in highly nonlinear systems. To\u0000overcome these challenges, we introduce Finite Basis Physics-Informed\u0000HyperDeepONet (FB-HyDON), an advanced operator architecture featuring intrinsic\u0000domain decomposition. By leveraging hypernetworks and finite basis functions,\u0000FB-HyDON effectively mitigates the training limitations associated with\u0000existing physics-informed operator learning methods. We validated our approach\u0000on the high-frequency harmonic oscillator, Burgers' equation at different\u0000viscosity levels, and Allen-Cahn equation demonstrating substantial\u0000improvements over other operator learning models.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"9 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259157","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Eddel Elí Ojeda Avilés, Jae-Hun Jung, Daniel Olmos Liceaga
Reaction-Diffusion systems arise in diverse areas of science and engineering. Due to the peculiar characteristics of such equations, analytic solutions are usually not available and numerical methods are the main tools for approximating the solutions. In the last decade, artificial neural networks have become an active area of development for solving partial differential equations. However, several challenges remain unresolved with these methods when applied to reaction-diffusion equations. In this work, we focus on two main problems. The implementation of homogeneous Neumann boundary conditions and long-time integrations. For the homogeneous Neumann boundary conditions, we explore four different neural network methods based on the PINN approach. For the long time integration in Reaction-Diffusion systems, we propose a domain splitting method in time and provide detailed comparisons between different implementations of no-flux boundary conditions. We show that the domain splitting method is crucial in the neural network approach, for long time integration in Reaction-Diffusion systems. We demonstrate numerically that domain splitting is essential for avoiding local minima, and the use of different boundary conditions further enhances the splitting technique by improving numerical approximations. To validate the proposed methods, we provide numerical examples for the Diffusion, the Bistable and the Barkley equations and provide a detailed discussion and comparisons of the proposed methods.
{"title":"Neural network Approximations for Reaction-Diffusion Equations -- Homogeneous Neumann Boundary Conditions and Long-time Integrations","authors":"Eddel Elí Ojeda Avilés, Jae-Hun Jung, Daniel Olmos Liceaga","doi":"arxiv-2409.08941","DOIUrl":"https://doi.org/arxiv-2409.08941","url":null,"abstract":"Reaction-Diffusion systems arise in diverse areas of science and engineering.\u0000Due to the peculiar characteristics of such equations, analytic solutions are\u0000usually not available and numerical methods are the main tools for\u0000approximating the solutions. In the last decade, artificial neural networks\u0000have become an active area of development for solving partial differential\u0000equations. However, several challenges remain unresolved with these methods\u0000when applied to reaction-diffusion equations. In this work, we focus on two\u0000main problems. The implementation of homogeneous Neumann boundary conditions\u0000and long-time integrations. For the homogeneous Neumann boundary conditions, we\u0000explore four different neural network methods based on the PINN approach. For\u0000the long time integration in Reaction-Diffusion systems, we propose a domain\u0000splitting method in time and provide detailed comparisons between different\u0000implementations of no-flux boundary conditions. We show that the domain\u0000splitting method is crucial in the neural network approach, for long time\u0000integration in Reaction-Diffusion systems. We demonstrate numerically that\u0000domain splitting is essential for avoiding local minima, and the use of\u0000different boundary conditions further enhances the splitting technique by\u0000improving numerical approximations. To validate the proposed methods, we\u0000provide numerical examples for the Diffusion, the Bistable and the Barkley\u0000equations and provide a detailed discussion and comparisons of the proposed\u0000methods.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"49 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142259158","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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