We consider the approximation of the smallest eigenvalue of a large parameter-dependent Hermitian matrix over a continuum compact domain. Our approach is based on approximating the smallest eigenvalue by the one obtained by projecting the large matrix onto a suitable small subspace, a practice widely employed in the literature. The projection subspaces are constructed iteratively (to reduce the error of the approximation where it is large) with the addition of the eigenvectors of the parameter-dependent matrix at the parameter values where a surrogate error is maximal. The surrogate error is the gap between the approximation and a lower bound for the smallest eigenvalue proposed in [Sirkovic and Kressner, SIAM J. Matrix Anal. Appl., 37(2), 2016]. Unlike the classical approaches, such as the successive constraint method, that maximize such surrogate errors over a discrete and finite set, we maximize the surrogate error over the continuum of all permissible parameter values globally. We put particular attention to the lower bound, which enables us to formally prove the global convergence of our framework both in finite-dimensional and infinite-dimensional settings. In the second part, we focus on the approximation of the smallest singular value of a large parameter-dependent matrix, in case it is non-Hermitian, and propose another subspace framework to construct a small parameter-dependent non-Hermitian matrix whose smallest singular value approximates the original large-scale smallest singular value. We perform numerical experiments on synthetic examples, as well as on real examples arising from parametric PDEs. The numerical experiments show that the proposed techniques are able to drastically reduce the size of the large parameter-dependent matrix, while ensuring an approximation error for the smallest eigenvalue/singular value below the prescribed tolerance.
{"title":"Uniform Approximation of Eigenproblems of a Large-Scale Parameter-Dependent Hermitian Matrix","authors":"Mattia Manucci, Emre Mengi, Nicola Guglielmi","doi":"arxiv-2409.05791","DOIUrl":"https://doi.org/arxiv-2409.05791","url":null,"abstract":"We consider the approximation of the smallest eigenvalue of a large\u0000parameter-dependent Hermitian matrix over a continuum compact domain. Our\u0000approach is based on approximating the smallest eigenvalue by the one obtained\u0000by projecting the large matrix onto a suitable small subspace, a practice\u0000widely employed in the literature. The projection subspaces are constructed\u0000iteratively (to reduce the error of the approximation where it is large) with\u0000the addition of the eigenvectors of the parameter-dependent matrix at the\u0000parameter values where a surrogate error is maximal. The surrogate error is the\u0000gap between the approximation and a lower bound for the smallest eigenvalue\u0000proposed in [Sirkovic and Kressner, SIAM J. Matrix Anal. Appl., 37(2), 2016].\u0000Unlike the classical approaches, such as the successive constraint method, that\u0000maximize such surrogate errors over a discrete and finite set, we maximize the\u0000surrogate error over the continuum of all permissible parameter values\u0000globally. We put particular attention to the lower bound, which enables us to\u0000formally prove the global convergence of our framework both in\u0000finite-dimensional and infinite-dimensional settings. In the second part, we\u0000focus on the approximation of the smallest singular value of a large\u0000parameter-dependent matrix, in case it is non-Hermitian, and propose another\u0000subspace framework to construct a small parameter-dependent non-Hermitian\u0000matrix whose smallest singular value approximates the original large-scale\u0000smallest singular value. We perform numerical experiments on synthetic\u0000examples, as well as on real examples arising from parametric PDEs. The\u0000numerical experiments show that the proposed techniques are able to drastically\u0000reduce the size of the large parameter-dependent matrix, while ensuring an\u0000approximation error for the smallest eigenvalue/singular value below the\u0000prescribed tolerance.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"26 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181978","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 introduce a two-level trust-region method (TLTR) for solving unconstrained nonlinear optimization problems. Our method uses a composite iteration step, which is based on two distinct search directions. The first search direction is obtained through minimization in the full/high-resolution space, ensuring global convergence to a critical point. The second search direction is obtained through minimization in the randomly generated subspace, which, in turn, allows for convergence acceleration. The efficiency of the proposed TLTR method is demonstrated through numerical experiments in the field of machine learning
{"title":"Two-level trust-region method with random subspaces","authors":"Andrea Angino, Alena Kopaničáková, Rolf Krause","doi":"arxiv-2409.05479","DOIUrl":"https://doi.org/arxiv-2409.05479","url":null,"abstract":"We introduce a two-level trust-region method (TLTR) for solving unconstrained\u0000nonlinear optimization problems. Our method uses a composite iteration step,\u0000which is based on two distinct search directions. The first search direction is\u0000obtained through minimization in the full/high-resolution space, ensuring\u0000global convergence to a critical point. The second search direction is obtained\u0000through minimization in the randomly generated subspace, which, in turn, allows\u0000for convergence acceleration. The efficiency of the proposed TLTR method is\u0000demonstrated through numerical experiments in the field of machine learning","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"50 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181983","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}
Aurelien Junior Noupelah, Jean Daniel Mukam, Antoine Tambue
The aim of this work is to provide the strong convergence results of numerical approximations of a general second order non-autonomous semilinear stochastic partial differential equation (SPDE) driven simultaneously by an additive fractional Brownian motion (fBm) with Hurst parameter H in (1/2,1) and a Poisson random measure, more realistic in modelling real world phenomena. Approximations in space are performed by the standard finite element method and in time by the stochastic Magnus-type integrator or the linear semi-implicit Euler method. We investigate the mean-square errors estimates of our fully discrete schemes and the results show how the convergence orders depend on the regularity of the initial data and the driven processes. To the best of our knowledge, these two schemes are the first numerical methods to approximate the non-autonomous semilinear stochastic partial differential equation (SPDE) driven simultaneously by an additive fractional Brownian motion with Hurst parameter H and a Poisson random measure.
这项工作的目的是提供一般二阶非自治半线性随机偏微分方程（SPDE）的数值近似的强收敛性结果，该近似同时由Hurst参数H （1/2,1）和泊松随机度量的加分布朗运动（fBm）驱动，在模拟现实世界的现象时更为现实。空间逼近采用标准有限元法，时间逼近采用随机马格努斯型积分法或线性隐式欧拉法。我们研究了完全离散方案的均方误差估计，结果表明收敛阶数如何依赖于初始数据和驱动过程的规则性。据我们所知，这两种方案是第一种近似非自治半线性随机偏微分方程（SPDE）的数值方法，该方程同时由具有 Hurst 参数 H 的加性分数布朗运动和泊松随机度量驱动。
{"title":"Strong convergence of some Magnus-type schemes for the finite element discretization of non-autonomous parabolic SPDEs driven by additive fractional Brownian motion and Poisson random measure","authors":"Aurelien Junior Noupelah, Jean Daniel Mukam, Antoine Tambue","doi":"arxiv-2409.06045","DOIUrl":"https://doi.org/arxiv-2409.06045","url":null,"abstract":"The aim of this work is to provide the strong convergence results of\u0000numerical approximations of a general second order non-autonomous semilinear\u0000stochastic partial differential equation (SPDE) driven simultaneously by an\u0000additive fractional Brownian motion (fBm) with Hurst parameter H in (1/2,1)\u0000and a Poisson random measure, more realistic in modelling real world phenomena. Approximations in space are performed by the standard finite element method\u0000and in time by the stochastic Magnus-type integrator or the linear\u0000semi-implicit Euler method. We investigate the mean-square errors estimates of\u0000our fully discrete schemes and the results show how the convergence orders\u0000depend on the regularity of the initial data and the driven processes. To the\u0000best of our knowledge, these two schemes are the first numerical methods to\u0000approximate the non-autonomous semilinear stochastic partial differential\u0000equation (SPDE) driven simultaneously by an additive fractional Brownian motion\u0000with Hurst parameter H and a Poisson random measure.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"55 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181970","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 implicit Euler method integrates systems of ordinary differential equations $$frac{d x}{d t}=G(t,x(t))$$ with differentiable right-hand side $G : R times R^n rightarrow R^n$ from an initial state $x=x(0) in R^n$ to a target time $t in R$ as $x(t)=E(t,m,x)$ using an equidistant discretization of the time interval $[0,t]$ yielding $m>0$ time steps. We aim to compute the product of its inverse Jacobian $$ (E')^{-1} equiv left (frac{d E}{d x}right )^{-1} in R^{n times n} $$ with a given vector efficiently. We show that the differential inverse $(E')^{-1} cdot v$ can be evaluated for given $v in R^n$ with a computational cost of $mathcal{O}(m cdot n^2)$ as opposed to the standard $mathcal{O}(m cdot n^3)$ or, naively, even $mathcal{O}(m cdot n^4).$ The theoretical results are supported by actual run times. A reference implementation is provided.
{"title":"Differential Inversion of the Implicit Euler Method: Symbolic Analysis","authors":"Uwe Naumann","doi":"arxiv-2409.05445","DOIUrl":"https://doi.org/arxiv-2409.05445","url":null,"abstract":"The implicit Euler method integrates systems of ordinary differential\u0000equations $$frac{d x}{d t}=G(t,x(t))$$ with differentiable right-hand side $G\u0000: R times R^n rightarrow R^n$ from an initial state $x=x(0) in R^n$ to a\u0000target time $t in R$ as $x(t)=E(t,m,x)$ using an equidistant discretization of\u0000the time interval $[0,t]$ yielding $m>0$ time steps. We aim to compute the\u0000product of its inverse Jacobian $$ (E')^{-1} equiv left (frac{d E}{d x}right )^{-1} in R^{n times n} $$ with a given vector efficiently. We show that the differential inverse\u0000$(E')^{-1} cdot v$ can be evaluated for given $v in R^n$ with a computational\u0000cost of $mathcal{O}(m cdot n^2)$ as opposed to the standard $mathcal{O}(m\u0000cdot n^3)$ or, naively, even $mathcal{O}(m cdot n^4).$ The theoretical\u0000results are supported by actual run times. A reference implementation is\u0000provided.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"32 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181984","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}
A recently developed coupling strategy for two nonconservative hyperbolic systems is employed to investigate a collapsing vapor bubble embedded in a liquid near a solid. For this purpose, an elastic solid modeled by a linear system of conservation laws is coupled to the two-phase Baer-Nunziato-type model for isothermal fluids, a nonlinear hyperbolic system with non-conservative products. For the coupling of the two systems the Jin-Xin relaxation concept is employed and embedded in a second order finite volume scheme. For a proof of concept simulations in one space dimension are performed.
{"title":"A relaxation approach to the coupling of a two-phase fluid with a linear-elastic solid","authors":"Niklas Kolbe, Siegfried Müller","doi":"arxiv-2409.05473","DOIUrl":"https://doi.org/arxiv-2409.05473","url":null,"abstract":"A recently developed coupling strategy for two nonconservative hyperbolic\u0000systems is employed to investigate a collapsing vapor bubble embedded in a\u0000liquid near a solid. For this purpose, an elastic solid modeled by a linear\u0000system of conservation laws is coupled to the two-phase Baer-Nunziato-type\u0000model for isothermal fluids, a nonlinear hyperbolic system with\u0000non-conservative products. For the coupling of the two systems the Jin-Xin\u0000relaxation concept is employed and embedded in a second order finite volume\u0000scheme. For a proof of concept simulations in one space dimension are\u0000performed.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"31 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142227626","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 combination of machine learning and physical laws has shown immense potential for solving scientific problems driven by partial differential equations (PDEs) with the promise of fast inference, zero-shot generalisation, and the ability to discover new physics. Examples include the use of fundamental physical laws as inductive bias to machine learning algorithms, also referred to as physics-driven machine learning, and the application of machine learning to represent features not represented in the differential equations such as closures for unresolved spatiotemporal scales. However, the simulation of complex physical systems by coupling advanced numerics for PDEs with state-of-the-art machine learning demands the composition of specialist PDE solving frameworks with industry-standard machine learning tools. Hand-rolling either the PDE solver or the neural net will not cut it. In this work, we introduce a generic differentiable programming abstraction that provides scientists and engineers with a highly productive way of specifying end-to-end differentiable models coupling machine learning and PDE-based components, while relying on code generation for high performance. Our interface automates the coupling of arbitrary PDE-based systems and machine learning models and unlocks new applications that could not hitherto be tackled, while only requiring trivial changes to existing code. Our framework has been adopted in the Firedrake finite-element library and supports the PyTorch and JAX ecosystems, as well as downstream libraries.
{"title":"Differentiable programming across the PDE and Machine Learning barrier","authors":"Nacime Bouziani, David A. Ham, Ado Farsi","doi":"arxiv-2409.06085","DOIUrl":"https://doi.org/arxiv-2409.06085","url":null,"abstract":"The combination of machine learning and physical laws has shown immense\u0000potential for solving scientific problems driven by partial differential\u0000equations (PDEs) with the promise of fast inference, zero-shot generalisation,\u0000and the ability to discover new physics. Examples include the use of\u0000fundamental physical laws as inductive bias to machine learning algorithms,\u0000also referred to as physics-driven machine learning, and the application of\u0000machine learning to represent features not represented in the differential\u0000equations such as closures for unresolved spatiotemporal scales. However, the\u0000simulation of complex physical systems by coupling advanced numerics for PDEs\u0000with state-of-the-art machine learning demands the composition of specialist\u0000PDE solving frameworks with industry-standard machine learning tools.\u0000Hand-rolling either the PDE solver or the neural net will not cut it. In this\u0000work, we introduce a generic differentiable programming abstraction that\u0000provides scientists and engineers with a highly productive way of specifying\u0000end-to-end differentiable models coupling machine learning and PDE-based\u0000components, while relying on code generation for high performance. Our\u0000interface automates the coupling of arbitrary PDE-based systems and machine\u0000learning models and unlocks new applications that could not hitherto be\u0000tackled, while only requiring trivial changes to existing code. Our framework\u0000has been adopted in the Firedrake finite-element library and supports the\u0000PyTorch and JAX ecosystems, as well as downstream libraries.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181973","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}
L. Chacon, Jason Hamilton, Natalia Krasheninnikova
We propose a second-order temporally implicit, fourth-order-accurate spatial discretization scheme for the strongly anisotropic heat transport equation characteristic of hot, fusion-grade plasmas. Following [Du Toit et al., Comp. Phys. Comm., 228 (2018)], the scheme transforms mixed-derivative diffusion fluxes (which are responsible for the lack of a discrete maximum principle) into nonlinear advective fluxes, amenable to nonlinear-solver-friendly monotonicity-preserving limiters. The scheme enables accurate multi-dimensional heat transport simulations with up to seven orders of magnitude of heat-transport-coefficient anisotropies with low cross-field numerical error pollution and excellent algorithmic performance, with the number of linear iterations scaling very weakly with grid resolution and grid anisotropy, and scaling with the square-root of the implicit timestep. We propose a multigrid preconditioning strategy based on a second-order-accurate approximation that renders the scheme efficient and scalable under grid refinement. Several numerical tests are presented that display the expected spatial convergence rates and strong algorithmic performance, including fully nonlinear magnetohydrodynamics simulations of kink instabilities in a Bennett pinch in 2D helical geometry and of ITER in 3D toroidal geometry.
{"title":"A robust fourth-order finite-difference discretization for the strongly anisotropic transport equation in magnetized plasmas","authors":"L. Chacon, Jason Hamilton, Natalia Krasheninnikova","doi":"arxiv-2409.06070","DOIUrl":"https://doi.org/arxiv-2409.06070","url":null,"abstract":"We propose a second-order temporally implicit, fourth-order-accurate spatial\u0000discretization scheme for the strongly anisotropic heat transport equation\u0000characteristic of hot, fusion-grade plasmas. Following [Du Toit et al., Comp.\u0000Phys. Comm., 228 (2018)], the scheme transforms mixed-derivative diffusion\u0000fluxes (which are responsible for the lack of a discrete maximum principle)\u0000into nonlinear advective fluxes, amenable to nonlinear-solver-friendly\u0000monotonicity-preserving limiters. The scheme enables accurate multi-dimensional\u0000heat transport simulations with up to seven orders of magnitude of\u0000heat-transport-coefficient anisotropies with low cross-field numerical error\u0000pollution and excellent algorithmic performance, with the number of linear\u0000iterations scaling very weakly with grid resolution and grid anisotropy, and\u0000scaling with the square-root of the implicit timestep. We propose a multigrid\u0000preconditioning strategy based on a second-order-accurate approximation that\u0000renders the scheme efficient and scalable under grid refinement. Several\u0000numerical tests are presented that display the expected spatial convergence\u0000rates and strong algorithmic performance, including fully nonlinear\u0000magnetohydrodynamics simulations of kink instabilities in a Bennett pinch in 2D\u0000helical geometry and of ITER in 3D toroidal geometry.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"98 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181974","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}
Neural network approaches have been demonstrated to work quite well to solve partial differential equations in practice. In this context approaches like physics-informed neural networks and the Deep Ritz method have become popular. In this paper, we propose a similar approach to solve an infinite-dimensional total variation minimization problem using neural networks. We illustrate that the resulting neural network problem does not have a solution in general. To circumvent this theoretic issue, we consider an auxiliary neural network problem, which indeed has a solution, and show that it converges in the sense of $Gamma$-convergence to the original problem. For computing a numerical solution we further propose a discrete version of the auxiliary neural network problem and again show its $Gamma$-convergence to the original infinite-dimensional problem. In particular, the $Gamma$-convergence proof suggests a particular discretization of the total variation. Moreover, we connect the discrete neural network problem to a finite difference discretization of the infinite-dimensional total variation minimization problem. Numerical experiments are presented supporting our theoretical findings.
在实践中，神经网络方法在求解偏微分方程时已被证明非常有效。在这种情况下，物理信息神经网络和 Deep Ritz 方法等方法开始流行起来。在本文中，我们提出了一种类似的方法，利用神经网络解决无限维总变异最小化问题。本文提出了利用神经网络求解无限维总变异最小化问题的类似方法，并说明了由此产生的神经网络问题在一般情况下没有解。为了避免这个理论问题，我们考虑了一个辅助神经网络问题，它确实有一个解，并证明它在$Gamma$-收敛的意义上收敛于原始问题。为了计算数值解，我们进一步提出了离散版的辅助神经网络问题，并再次证明了它与原始无限维问题的$Gamma$收敛性。特别是，$Gamma$-收敛证明提出了总变异的特定离散化。此外，我们还将离散神经网络问题与无限维总变化最小化问题的有限差分离散化联系起来。数值实验支持我们的理论发现。
{"title":"DeepTV: A neural network approach for total variation minimization","authors":"Andreas Langer, Sara Behnamian","doi":"arxiv-2409.05569","DOIUrl":"https://doi.org/arxiv-2409.05569","url":null,"abstract":"Neural network approaches have been demonstrated to work quite well to solve\u0000partial differential equations in practice. In this context approaches like\u0000physics-informed neural networks and the Deep Ritz method have become popular.\u0000In this paper, we propose a similar approach to solve an infinite-dimensional\u0000total variation minimization problem using neural networks. We illustrate that\u0000the resulting neural network problem does not have a solution in general. To\u0000circumvent this theoretic issue, we consider an auxiliary neural network\u0000problem, which indeed has a solution, and show that it converges in the sense\u0000of $Gamma$-convergence to the original problem. For computing a numerical\u0000solution we further propose a discrete version of the auxiliary neural network\u0000problem and again show its $Gamma$-convergence to the original\u0000infinite-dimensional problem. In particular, the $Gamma$-convergence proof\u0000suggests a particular discretization of the total variation. Moreover, we\u0000connect the discrete neural network problem to a finite difference\u0000discretization of the infinite-dimensional total variation minimization\u0000problem. Numerical experiments are presented supporting our theoretical\u0000findings.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"10 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181981","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}
Ronald Katende, Henry Kasumba, Godwin Kakuba, John M. Mango
This paper addresses critical challenges in machine learning, particularly the stability, consistency, and convergence of neural networks under non-IID data, distribution shifts, and high-dimensional settings. We provide new theoretical results on uniform stability for neural networks with dynamic learning rates in non-convex settings. Further, we establish consistency bounds for federated learning models in non-Euclidean spaces, accounting for distribution shifts and curvature effects. For Physics-Informed Neural Networks (PINNs), we derive stability, consistency, and convergence guarantees for solving Partial Differential Equations (PDEs) in noisy environments. These results fill significant gaps in understanding model behavior in complex, non-ideal conditions, paving the way for more robust and reliable machine learning applications.
{"title":"Some Results on Neural Network Stability, Consistency, and Convergence: Insights into Non-IID Data, High-Dimensional Settings, and Physics-Informed Neural Networks","authors":"Ronald Katende, Henry Kasumba, Godwin Kakuba, John M. Mango","doi":"arxiv-2409.05030","DOIUrl":"https://doi.org/arxiv-2409.05030","url":null,"abstract":"This paper addresses critical challenges in machine learning, particularly\u0000the stability, consistency, and convergence of neural networks under non-IID\u0000data, distribution shifts, and high-dimensional settings. We provide new\u0000theoretical results on uniform stability for neural networks with dynamic\u0000learning rates in non-convex settings. Further, we establish consistency bounds\u0000for federated learning models in non-Euclidean spaces, accounting for\u0000distribution shifts and curvature effects. For Physics-Informed Neural Networks\u0000(PINNs), we derive stability, consistency, and convergence guarantees for\u0000solving Partial Differential Equations (PDEs) in noisy environments. These\u0000results fill significant gaps in understanding model behavior in complex,\u0000non-ideal conditions, paving the way for more robust and reliable machine\u0000learning applications.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"15 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181994","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}
Ensuring data integrity is a critical requirement in complex systems, especially in financial platforms where vast amounts of data must be consistently accurate and reliable. This paper presents a robust approach using polynomial interpolation methods to maintain data integrity across multiple indicators and dimensions.
{"title":"Polynomial Methods for Ensuring Data Integrity in Financial Systems","authors":"Ignacio Brasca","doi":"arxiv-2409.07490","DOIUrl":"https://doi.org/arxiv-2409.07490","url":null,"abstract":"Ensuring data integrity is a critical requirement in complex systems,\u0000especially in financial platforms where vast amounts of data must be\u0000consistently accurate and reliable. This paper presents a robust approach using\u0000polynomial interpolation methods to maintain data integrity across multiple\u0000indicators and dimensions.","PeriodicalId":501162,"journal":{"name":"arXiv - MATH - Numerical Analysis","volume":"118 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-09-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142181929","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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