Model inference for Ordinary Differential Equations by parametric polynomial kernel regression
The Alan Turing Institute（艾伦·图灵研究所）；Mathematics Institute, University of Warwick（华威大学数学研究所）
David K. E. Green, Filip Rindler
备注：
23 pages, 7 figures. Submission to 3rd International Conference on Uncertainty Quantification in Computational Sciences and Engineering (UNCECOMP), Crete, Greece, 24-26 June 2019
Model inference for dynamical systems aims to estimate the future behaviour of a system from observations. Purely model-free statistical methods, such as Artificial Neural Networks, tend to perform poorly for such tasks. They are therefore not well suited to many questions from applications, for example in Bayesian filtering and reliability estimation.
This work introduces a parametric polynomial kernel method that can be used for inferring the future behaviour of Ordinary Differential Equation models, including chaotic dynamical systems, from observations. Using numerical integration techniques, parametric representations of Ordinary Differential Equations can be learnt using Backpropagation and Stochastic Gradient Descent. The polynomial technique presented here is based on a nonparametric method, kernel ridge regression. However, the time complexity of nonparametric kernel ridge regression scales cubically with the number of training data points. Our parametric polynomial method avoids this manifestation of the curse of dimensionality, which becomes particularly relevant when working with large time series data sets.
Two numerical demonstrations are presented. First, a simple regression test case is used to illustrate the method and to compare the performance with standard Artificial Neural Network techniques. Second, a more substantial test case is the inference of a chaotic spatio-temporal dynamical system, the Lorenz--Emanuel system, from observations. Our method was able to successfully track the future behaviour of the system over time periods much larger than the training data sampling rate. Finally, some limitations of the method are presented, as well as proposed directions for future work to mitigate these limitations.
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École Polytechnique（巴黎综合理工学院）；Institut Polytechnique de Paris（巴黎理工学院）；Lagrange Mathematics and Computing Research Center（拉格朗日数学与计算研究中心）；Télécom Paris（巴黎电信学院）；CREST（国家科学研究中心）；ENSAI（法国国家统计与信息分析学院）；UCLouvain（鲁汶大学）
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University of Bristol（布里斯托大学）；University of Tartu（塔尔图大学）
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Univ. of Alberta（阿尔伯塔大学）；Amii（阿尔伯塔机器智能研究院）；Vector Institute（向量研究所）；Univ. of Toronto（多伦多大学）
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MTA SZTAKI: The Institute for Computer Science and Control, Hungarian Academy of Sciences（匈牙利科学院MTA SZTAKI计算机科学与控制研究所）
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Accounting for shared covariates in semi-parametric Bayesian additive regression trees
School of Mathematical Sciences, Lancaster University, United Kingdom.；Hamilton Institute, Maynooth University, Co. Kildare, Ireland.；Department of Mathematics \& Statistics, Maynooth University, Co. Kildare, Ireland.；Insight Centre for Data Analytics, Maynooth University, Co. Kildare, Ireland.；Department of Mathematics \& Statistics, Lancaster University, United Kingdom.
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University of California San Diego（加州大学圣地亚哥分校）；Alexa AI（Alexa人工智能部门）
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Bosch Center for Artificial Intelligence（博世人工智能中心）；University of British Columbia（不列颠哥伦比亚大学）
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Department of Mathematics University of California San Diego（加利福尼亚大学圣地亚哥分校数学系）
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e:fs TechHub GmbH（e:fs科技中心有限公司）；Ruhr West University of Applied Sciences（鲁尔西区应用科学大学）
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University of Alberta（阿尔伯塔大学）；Harvard University（哈佛大学）
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Mila - Quebec AI Institute（米拉-魁北克人工智能研究所）；University of Toronto Institute for Aerospace Studies（多伦多大学航空航天研究所）；University of Toronto（多伦多大学）
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