SIAM Journal on Applied Dynamical Systems, Volume 22, Issue 4, Page 3208-3232, December 2023. Abstract.The deep linear network (DLN) is a model for implicit regularization in gradient based optimization of overparametrized learning architectures. Training the DLN corresponds to a Riemannian gradient flow, where the Riemannian metric is defined by the architecture of the network and the loss function is defined by the learning task. We extend this geometric framework, obtaining explicit expressions for the volume form, including the case when the network has infinite depth. We investigate the link between the Riemannian geometry and the training asymptotics for matrix completion with rigorous analysis and numerics. We propose that under small initialization, implicit regularization is a result of bias towards high state space volume.
应用动力系统学报，第22卷，第4期，第3208-3232页，2023年12月。摘要。深度线性网络(deep linear network, DLN)是一种基于梯度优化的隐式正则化模型。训练DLN对应于黎曼梯度流，其中黎曼度量由网络的体系结构定义，损失函数由学习任务定义。我们扩展了这个几何框架，得到了体积形式的显式表达式，包括网络具有无限深度的情况。我们用严格的分析和数值研究了黎曼几何和矩阵补全的训练渐近之间的联系。我们提出在小初始化下，隐式正则化是偏向于高状态空间体积的结果。
{"title":"Deep Linear Networks for Matrix Completion—an Infinite Depth Limit","authors":"Nadav Cohen, Govind Menon, Zsolt Veraszto","doi":"10.1137/22m1530653","DOIUrl":"https://doi.org/10.1137/22m1530653","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 22, Issue 4, Page 3208-3232, December 2023. <br/> Abstract.The deep linear network (DLN) is a model for implicit regularization in gradient based optimization of overparametrized learning architectures. Training the DLN corresponds to a Riemannian gradient flow, where the Riemannian metric is defined by the architecture of the network and the loss function is defined by the learning task. We extend this geometric framework, obtaining explicit expressions for the volume form, including the case when the network has infinite depth. We investigate the link between the Riemannian geometry and the training asymptotics for matrix completion with rigorous analysis and numerics. We propose that under small initialization, implicit regularization is a result of bias towards high state space volume.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"43 ","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138505528","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
SIAM Journal on Applied Dynamical Systems, Volume 22, Issue 4, Page 3165-3207, December 2023. Abstract. In an epidemic network, lags due to travel time between populations, latent period, and recovery period can significantly change the epidemic behavior and result in successive echoing waves of the spread between various population clusters. Moreover, external shocks to a given population can propagate to other populations within the network, potentially snowballing into waves of resurgent epidemics. The main objective of this study is to investigate the effect of time delay and small shocks/uncertainties on the linear susceptible-infectious-susceptible (SIS) dynamics of epidemic networks. In this regard, the asymptotic stability of this class of networks is first studied, and then its performance loss due to small shocks/uncertainties is evaluated based on the notion of the [math] norm. It is shown that network performance loss is correlated with the structure of the underlying graph, intrinsic time delays, epidemic characteristics, and external shocks. This performance measure is then used to develop an optimal traffic restriction algorithm for network performance enhancement, resulting in reduced infection in the metapopulation. A novel epidemic-based centrality index is also defined to evaluate the impact of every subpopulation on network performance, and its asymptotic behavior is investigated. It is shown that for specific choices of parameters, the output of the epidemic-based centrality index converges to the results obtained by local or eigenvector centralities. Moreover, given that epidemic-based centrality depends on the epidemic properties of the disease, it may yield distinct node rankings as the disease characteristics slowly change over time or as different types of infections spread. This node interlacing phenomenon is not observed in other centralities that rely solely on network structure. This unique characteristic of epidemic-based centrality enables it to adjust to various epidemic features. The derived centrality index is then adopted to improve the network robustness against external shocks on the epidemic network. The numerical results, along with the theoretical expectations, highlight the role of time delay as well as small shocks in investigating the most effective methods of epidemic containment.
应用动力系统学报，vol . 22, Issue 4, Page 3165-3207, December 2023。摘要。在疫情网络中，由于种群间的传播时间、潜伏期和恢复期的滞后会显著改变疫情行为，导致不同种群间传播的连续回声波。此外，对某一特定人群的外部冲击可能会传播到网络内的其他人群，从而可能滚雪球般地形成一波又一波的流行病。本研究的主要目的是研究时间延迟和小冲击/不确定性对流行病网络线性易感-感染-易感(SIS)动力学的影响。在这方面，首先研究了这类网络的渐近稳定性，然后基于[数学]范数的概念评估了其由于小冲击/不确定性而造成的性能损失。结果表明，网络性能损失与底层图的结构、固有时滞、流行特征和外部冲击有关。然后使用该性能度量来开发网络性能增强的最优流量限制算法，从而减少元种群中的感染。定义了一种新的基于流行度的中心性指数来评价每个子种群对网络性能的影响，并研究了其渐近行为。结果表明，对于特定参数的选择，基于流行病的中心性指数的输出收敛于局部中心性或特征向量中心性得到的结果。此外，鉴于基于流行病的中心性取决于疾病的流行病特性，随着疾病特征随时间缓慢变化或不同类型的感染传播，它可能产生不同的节点排名。这种节点交错现象在其他仅依赖于网络结构的中心性中没有观察到。这种基于流行病的中心性的独特特征使其能够适应各种流行病特征。然后采用导出的中心性指数来提高网络对疫情网络外部冲击的鲁棒性。数值结果以及理论期望突出了时间延迟和小冲击在研究最有效的流行病控制方法中的作用。
{"title":"Centrality-Based Traffic Restriction in Delayed Epidemic Networks","authors":"Atefe Darabi, Milad Siami","doi":"10.1137/22m1507760","DOIUrl":"https://doi.org/10.1137/22m1507760","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 22, Issue 4, Page 3165-3207, December 2023. <br/> Abstract. In an epidemic network, lags due to travel time between populations, latent period, and recovery period can significantly change the epidemic behavior and result in successive echoing waves of the spread between various population clusters. Moreover, external shocks to a given population can propagate to other populations within the network, potentially snowballing into waves of resurgent epidemics. The main objective of this study is to investigate the effect of time delay and small shocks/uncertainties on the linear susceptible-infectious-susceptible (SIS) dynamics of epidemic networks. In this regard, the asymptotic stability of this class of networks is first studied, and then its performance loss due to small shocks/uncertainties is evaluated based on the notion of the [math] norm. It is shown that network performance loss is correlated with the structure of the underlying graph, intrinsic time delays, epidemic characteristics, and external shocks. This performance measure is then used to develop an optimal traffic restriction algorithm for network performance enhancement, resulting in reduced infection in the metapopulation. A novel epidemic-based centrality index is also defined to evaluate the impact of every subpopulation on network performance, and its asymptotic behavior is investigated. It is shown that for specific choices of parameters, the output of the epidemic-based centrality index converges to the results obtained by local or eigenvector centralities. Moreover, given that epidemic-based centrality depends on the epidemic properties of the disease, it may yield distinct node rankings as the disease characteristics slowly change over time or as different types of infections spread. This node interlacing phenomenon is not observed in other centralities that rely solely on network structure. This unique characteristic of epidemic-based centrality enables it to adjust to various epidemic features. The derived centrality index is then adopted to improve the network robustness against external shocks on the epidemic network. The numerical results, along with the theoretical expectations, highlight the role of time delay as well as small shocks in investigating the most effective methods of epidemic containment.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"50 ","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-11-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138505525","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Tea Vojkovic, David Quero, Charles Poussot-Vassal, Pierre Vuillemin
SIAM Journal on Applied Dynamical Systems, Volume 22, Issue 4, Page 3130-3164, December 2023. Abstract.In this work, we present a novel data-driven method for identifying parametric MIMO generalized state-space or descriptor systems of low order that accurately capture the frequency and time domain behavior of large-scale linear dynamical systems. The low-order parametric descriptor systems are identified from transfer matrix samples by means of two-variable Lagrange rational matrix interpolation. This is done within the Loewner framework by deploying the new matrix-valued barycentric formula given in both right and left polynomial matrix fraction forms, which enables the construction of minimal parametric descriptor systems with rectangular transfer matrices. The developed method allows the reduction of order and parameter dependence complexity of the constructed system. Stability of the system is preserved by the postprocessing technique based on flipping signs of unstable poles. The developed methodology is illustrated with a few academic examples and applied to low-order parametric state-space identification of an aerodynamic system.
{"title":"Low-Order Parametric State-Space Modeling of MIMO Systems in the Loewner Framework","authors":"Tea Vojkovic, David Quero, Charles Poussot-Vassal, Pierre Vuillemin","doi":"10.1137/22m1509898","DOIUrl":"https://doi.org/10.1137/22m1509898","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 22, Issue 4, Page 3130-3164, December 2023. <br/> Abstract.In this work, we present a novel data-driven method for identifying parametric MIMO generalized state-space or descriptor systems of low order that accurately capture the frequency and time domain behavior of large-scale linear dynamical systems. The low-order parametric descriptor systems are identified from transfer matrix samples by means of two-variable Lagrange rational matrix interpolation. This is done within the Loewner framework by deploying the new matrix-valued barycentric formula given in both right and left polynomial matrix fraction forms, which enables the construction of minimal parametric descriptor systems with rectangular transfer matrices. The developed method allows the reduction of order and parameter dependence complexity of the constructed system. Stability of the system is preserved by the postprocessing technique based on flipping signs of unstable poles. The developed methodology is illustrated with a few academic examples and applied to low-order parametric state-space identification of an aerodynamic system.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"37 ","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138505529","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Olivier Hénot, Jean-Philippe Lessard, Jason D. Mireles James
SIAM Journal on Applied Dynamical Systems, Volume 22, Issue 4, Page 3093-3129, December 2023. Abstract. We present a computational method for studying transverse homoclinic orbits for periodic solutions of delay differential equations, a phenomenon that we refer to as the Poincaré scenario. The strategy is geometric in nature and consists of viewing the connection as the zero of a nonlinear map, such that the invertibility of its Fréchet derivative implies the transversality of the intersection. The map is defined by a projected boundary value problem (BVP), with boundary conditions in the (finite dimensional) unstable and (infinite dimensional) stable manifolds of the periodic orbit. The parameterization method is used to compute the unstable manifold, and the BVP is solved using a discrete time dynamical system approach (defined via the method of steps) and Chebyshev series expansions. We illustrate this technique by computing transverse homoclinic orbits in the cubic Ikeda and Mackey–Glass systems.
{"title":"Numerical Computation of Transverse Homoclinic Orbits for Periodic Solutions of Delay Differential Equations","authors":"Olivier Hénot, Jean-Philippe Lessard, Jason D. Mireles James","doi":"10.1137/23m1562858","DOIUrl":"https://doi.org/10.1137/23m1562858","url":null,"abstract":"SIAM Journal on Applied Dynamical Systems, Volume 22, Issue 4, Page 3093-3129, December 2023. <br/> Abstract. We present a computational method for studying transverse homoclinic orbits for periodic solutions of delay differential equations, a phenomenon that we refer to as the Poincaré scenario. The strategy is geometric in nature and consists of viewing the connection as the zero of a nonlinear map, such that the invertibility of its Fréchet derivative implies the transversality of the intersection. The map is defined by a projected boundary value problem (BVP), with boundary conditions in the (finite dimensional) unstable and (infinite dimensional) stable manifolds of the periodic orbit. The parameterization method is used to compute the unstable manifold, and the BVP is solved using a discrete time dynamical system approach (defined via the method of steps) and Chebyshev series expansions. We illustrate this technique by computing transverse homoclinic orbits in the cubic Ikeda and Mackey–Glass systems.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"56 ","pages":""},"PeriodicalIF":2.1,"publicationDate":"2023-11-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138505522","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The spectrum of the Koopman operator has been shown to encode many important statistical and geometric properties of a dynamical system. In this work, we consider induced linear operators acting on the space of sections of the state space’s tangent, cotangent, and tensor bundles. We first demonstrate how these operators are natural generalizations of Koopman operators acting on functions. We then draw connections between the various operators’ spectra and characterize the algebraic and differential topological properties of their spectra. We describe the discrete spectrum of these operators for linear dynamical systems and derive spectral expansions for linear vector fields. We define the notion of an “eigendistribution,” provide conditions for an eigendistribution to be integrable, and demonstrate how to recover the foliations arising from their integral manifolds. Last, we demonstrate that the characteristic Lyapunov exponents of a uniformly hyperbolic dynamical system are in the spectrum of the induced operators on sections of the tangent or cotangent bundle. We conclude with an application to differential geometry where the well-known fact that the flows of commuting vector fields commute is generalized, and we recover the original statement as a particular case of our result. We also apply our results to recover the Lyapunov exponents and the stable/unstable foliations of Arnold’s cat map via the spectrum of the induced operator on sections of the tangent bundle.
{"title":"Spectral Properties of Pullback Operators on Vector Bundles of a Dynamical System","authors":"Allan M. Avila, Igor Mezić","doi":"10.1137/22m1492064","DOIUrl":"https://doi.org/10.1137/22m1492064","url":null,"abstract":"The spectrum of the Koopman operator has been shown to encode many important statistical and geometric properties of a dynamical system. In this work, we consider induced linear operators acting on the space of sections of the state space’s tangent, cotangent, and tensor bundles. We first demonstrate how these operators are natural generalizations of Koopman operators acting on functions. We then draw connections between the various operators’ spectra and characterize the algebraic and differential topological properties of their spectra. We describe the discrete spectrum of these operators for linear dynamical systems and derive spectral expansions for linear vector fields. We define the notion of an “eigendistribution,” provide conditions for an eigendistribution to be integrable, and demonstrate how to recover the foliations arising from their integral manifolds. Last, we demonstrate that the characteristic Lyapunov exponents of a uniformly hyperbolic dynamical system are in the spectrum of the induced operators on sections of the tangent or cotangent bundle. We conclude with an application to differential geometry where the well-known fact that the flows of commuting vector fields commute is generalized, and we recover the original statement as a particular case of our result. We also apply our results to recover the Lyapunov exponents and the stable/unstable foliations of Arnold’s cat map via the spectrum of the induced operator on sections of the tangent bundle.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":" 24","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135286406","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Developing a model to capture all aspects of a complex dynamical system is an immense task, and each model will have deficiencies in some areas, such as global climate models having difficulty in capturing tropical intraseasonal variability such as the Madden–Julian oscillation. Besides complex models, it is possible to create simplified, low-dimensional models to capture specific phenomena while ignoring many aspects of the full system. Here, we propose a strategy to allow complex models to communicate with simplified models throughout a simulation. The communication allows one to leverage the strengths of each model, without needing to change their dynamics, to mitigate model error. Furthermore, to ensure ease of implementation in complex systems, the strategy is based on common data assimilation techniques that are normally used to combine models and real-world data. This strategy is investigated here in a test case that is nonlinear, non-Gaussian, and high-dimensional (approximately degrees of freedom), and the multiple models have different state spaces. In particular, it is an idealized tropical climate model in three spatial dimensions. The multimodel communication strategy is seen to mitigate model error and reproduce statistical features akin to those of the truth model when the communication is sufficiently frequent. In these tests, the low-dimensional model contributes only two degrees of freedom, which suggests that, in some systems, large amounts of model error can possibly be reduced by focusing on a small set of model components.
{"title":"Mitigating Model Error via a Multimodel Method and Application to Tropical Intraseasonal Oscillations","authors":"Jason L. Torchinsky, Samuel Stechmann","doi":"10.1137/22m152551x","DOIUrl":"https://doi.org/10.1137/22m152551x","url":null,"abstract":"Developing a model to capture all aspects of a complex dynamical system is an immense task, and each model will have deficiencies in some areas, such as global climate models having difficulty in capturing tropical intraseasonal variability such as the Madden–Julian oscillation. Besides complex models, it is possible to create simplified, low-dimensional models to capture specific phenomena while ignoring many aspects of the full system. Here, we propose a strategy to allow complex models to communicate with simplified models throughout a simulation. The communication allows one to leverage the strengths of each model, without needing to change their dynamics, to mitigate model error. Furthermore, to ensure ease of implementation in complex systems, the strategy is based on common data assimilation techniques that are normally used to combine models and real-world data. This strategy is investigated here in a test case that is nonlinear, non-Gaussian, and high-dimensional (approximately degrees of freedom), and the multiple models have different state spaces. In particular, it is an idealized tropical climate model in three spatial dimensions. The multimodel communication strategy is seen to mitigate model error and reproduce statistical features akin to those of the truth model when the communication is sufficiently frequent. In these tests, the low-dimensional model contributes only two degrees of freedom, which suggests that, in some systems, large amounts of model error can possibly be reduced by focusing on a small set of model components.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"230 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135775132","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Cris R. Hasan, Ruaidhrí Mac Cárthaigh, Sebastian Wieczorek
{"title":"Rate-Induced Tipping in Heterogeneous Reaction-Diffusion Systems: An Invariant Manifold Framework and Geographically Shifting Ecosystems","authors":"Cris R. Hasan, Ruaidhrí Mac Cárthaigh, Sebastian Wieczorek","doi":"10.1137/22m1536625","DOIUrl":"https://doi.org/10.1137/22m1536625","url":null,"abstract":"","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"40 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136068099","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Joshua W. Moore, Trevor C. Dale, Thomas E. Woolley
Recent advances in high-resolution experimental methods have highlighted the significance of cell signal pathway crosstalk and localized signaling activity in the development and disease of numerous biological systems. The investigation of multiple signal pathways often introduces different methods of cell-cell communication, i.e., contact-based or diffusive signaling, which generates both a spatial and a temporal dependence on cell behaviors. Motivated by cellular mechanisms that control cell-fate decisions in developing bilayer tissues, we use dynamical systems coupled with multilayer graphs to analyze the role of signaling polarity and pathway crosstalk in fine-grain pattern formation of protein activity. Specifically, we study how multilayer graph edge structures and weights influence the layerwise (laminar) patterning of cells in bilayer structures, which are commonly found in glandular tissues. We present sufficient conditions for the existence, uniqueness, and instability of homogeneous cell states in the large-scale spatially discrete dynamical system. Using methods of pattern templating by graph partitioning to generate quotient systems, in combination with concepts from monotone dynamical systems, we exploit the extensive dimensionality reduction to provide existence conditions for the polarity required to induce fine-grain laminar patterns with multiple spatially dependent intracellular components. We then explore the spectral links between the quotient and large-scale dynamical systems to extend the laminar patterning criteria from existence to convergence for sufficiently large amounts of cellular polarity in the large-scale dynamical system, independent of spatial dimension and number of cells in the tissue.
{"title":"Modeling Polarity-Driven Laminar Patterns in Bilayer Tissues with Mixed Signaling Mechanisms","authors":"Joshua W. Moore, Trevor C. Dale, Thomas E. Woolley","doi":"10.1137/22m1522565","DOIUrl":"https://doi.org/10.1137/22m1522565","url":null,"abstract":"Recent advances in high-resolution experimental methods have highlighted the significance of cell signal pathway crosstalk and localized signaling activity in the development and disease of numerous biological systems. The investigation of multiple signal pathways often introduces different methods of cell-cell communication, i.e., contact-based or diffusive signaling, which generates both a spatial and a temporal dependence on cell behaviors. Motivated by cellular mechanisms that control cell-fate decisions in developing bilayer tissues, we use dynamical systems coupled with multilayer graphs to analyze the role of signaling polarity and pathway crosstalk in fine-grain pattern formation of protein activity. Specifically, we study how multilayer graph edge structures and weights influence the layerwise (laminar) patterning of cells in bilayer structures, which are commonly found in glandular tissues. We present sufficient conditions for the existence, uniqueness, and instability of homogeneous cell states in the large-scale spatially discrete dynamical system. Using methods of pattern templating by graph partitioning to generate quotient systems, in combination with concepts from monotone dynamical systems, we exploit the extensive dimensionality reduction to provide existence conditions for the polarity required to induce fine-grain laminar patterns with multiple spatially dependent intracellular components. We then explore the spectral links between the quotient and large-scale dynamical systems to extend the laminar patterning criteria from existence to convergence for sufficiently large amounts of cellular polarity in the large-scale dynamical system, independent of spatial dimension and number of cells in the tissue.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"3 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136115189","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Aaron Kirtland, Jonah Botvinick-Greenhouse, Marianne DeBrito, Megan Osborne, Casey Johnson, Robert S. Martin, Samuel J. Araki, Daniel Q. Eckhardt
To address noise inherent in electronic data acquisition systems and real-world sources, Araki et al. [Phys. D, 417 (2021), 132819] demonstrated a grid-based nonlinear technique to remove noise from a chaotic signal, leveraging a clean high-fidelity signal from the same dynamical system and ensemble averaging in multidimensional phase space. This method achieved denoising of a time series data with 100% added noise but suffered in regions of low data density. To improve this grid-based method, here an unstructured mesh based on triangulations and Voronoi diagrams is used to accomplish the same task. The unstructured mesh more uniformly distributes data samples over mesh cells to improve the accuracy of the reconstructed signal. By empirically balancing bias and variance errors in selecting the number of unstructured cells as a function of the number of available samples, the method achieves asymptotic statistical convergence with known test data and reduces synthetic noise on experimental signals from Hall effect thrusters with greater success than the original grid-based strategy.
{"title":"An Unstructured Mesh Approach to Nonlinear Noise Reduction for Coupled Systems","authors":"Aaron Kirtland, Jonah Botvinick-Greenhouse, Marianne DeBrito, Megan Osborne, Casey Johnson, Robert S. Martin, Samuel J. Araki, Daniel Q. Eckhardt","doi":"10.1137/22m152092x","DOIUrl":"https://doi.org/10.1137/22m152092x","url":null,"abstract":"To address noise inherent in electronic data acquisition systems and real-world sources, Araki et al. [Phys. D, 417 (2021), 132819] demonstrated a grid-based nonlinear technique to remove noise from a chaotic signal, leveraging a clean high-fidelity signal from the same dynamical system and ensemble averaging in multidimensional phase space. This method achieved denoising of a time series data with 100% added noise but suffered in regions of low data density. To improve this grid-based method, here an unstructured mesh based on triangulations and Voronoi diagrams is used to accomplish the same task. The unstructured mesh more uniformly distributes data samples over mesh cells to improve the accuracy of the reconstructed signal. By empirically balancing bias and variance errors in selecting the number of unstructured cells as a function of the number of available samples, the method achieves asymptotic statistical convergence with known test data and reduces synthetic noise on experimental signals from Hall effect thrusters with greater success than the original grid-based strategy.","PeriodicalId":49534,"journal":{"name":"SIAM Journal on Applied Dynamical Systems","volume":"54 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135919878","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Yen Ting Lin, Yifeng Tian, Danny Perez, Daniel Livescu
We propose to adopt statistical regression as the projection operator to enable data-driven learning of the operators in the Mori–Zwanzig formalism. We present a principled method to extract the Markov and memory operators for any regression models. We show that the choice of linear regression results in a recently proposed data-driven learning algorithm based on Mori’s projection operator, which is a higher-order approximate Koopman learning method. We show that more expressive nonlinear regression models naturally fill in the gap between the highly idealized and computationally efficient Mori’s projection operator and the most optimal yet computationally infeasible Zwanzig’s projection operator. We performed numerical experiments and extracted the operators for an array of regression-based projections, including linear, polynomial, spline, and neural network–based regressions, showing a progressive improvement as the complexity of the regression model increased. Our proposition provides a general framework to extract memory-dependent corrections and can be readily applied to an array of data-driven learning methods for stationary dynamical systems in the literature.
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