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Bayesian system ID: optimal management of parameter, model, and measurement uncertainty
Nonlinear Dynamics ( IF 5.2 ) Pub Date : 2020-09-07 , DOI: 10.1007/s11071-020-05925-8
Nicholas Galioto , Alex Arkady Gorodetsky

System identification of dynamical systems is often posed as a least squares minimization problem. The aim of these optimization problems is typically to learn either propagators or the underlying vector fields from trajectories of data. In this paper, we study a first principles derivation of appropriate objective formulations for system identification based on probabilistic principles. We compare the resulting inference objective to those used by emerging data-driven methods based on dynamic mode decomposition (DMD) and system identification of nonlinear dynamics (SINDy). We show that these and related least squares formulations are specific cases of a more general objective function. We also show that the more general objective function yields more robust and reliable recovery in the presence of sparse data and noisy measurements. We attribute this success to an explicit accounting of imperfect model forms, parameter uncertainty, and measurement uncertainty. We study the computational complexity of an approximate marginal Markov Chain Monte Carlo method to solve the resulting inference problem and numerically compare our results on a number of canonical systems: linear pendulum, nonlinear pendulum, the Van der Pol oscillator, the Lorenz system, and a reaction–diffusion system. The results of these comparisons show that in cases where DMD and SINDy excel, the Bayesian approach performs equally well, and in cases where DMD and SINDy fail to produce reasonable results, the Bayesian approach remains robust and can still deliver reliable results.



中文翻译:

贝叶斯系统ID:参数,模型和测量不确定度的最佳管理

动态系统的系统识别通常被提出为最小二乘最小化问题。这些优化问题的目的通常是从数据轨迹中学习传播子或基础向量场。在本文中,我们研究了基于概率原理的系统识别合适的目标公式的第一原理推导。我们将得出的推理目标与基于动态模式分解(DMD)和非线性动力学的系统识别(SINDy)的新兴数据驱动方法所使用的推理目标进行比较。我们显示这些和相关的最小二乘公式是更通用的目标函数的特定情况。我们还表明,在存在稀疏数据和嘈杂测量的情况下,更通用的目标函数可产生更强大和可靠的恢复。我们将这一成功归因于对不完美模型形式,参数不确定性和测量不确定性的明确考虑。我们研究了近似边际马尔可夫链蒙特卡罗方法的计算复杂性,以解决由此产生的推理问题,并在许多规范系统上进行了数值比较,这些规范系统包括:线性摆,非线性摆,范德波尔振子,洛伦兹系统和反应扩散系统。这些比较的结果表明,在DMD和SINDy优异的情况下,贝叶斯方法的效果同样好;在DMD和SINDy无法产生合理结果的情况下,贝叶斯方法仍然很健壮,仍然可以提供可靠的结果。我们研究了近似边际马尔可夫链蒙特卡罗方法的计算复杂性,以解决由此产生的推理问题,并在许多规范系统上进行了数值比较,这些规范系统包括:线性摆,非线性摆,范德波尔振子,洛伦兹系统和反应扩散系统。这些比较的结果表明,在DMD和SINDy优异的情况下,贝叶斯方法的效果同样好;在DMD和SINDy无法产生合理结果的情况下,贝叶斯方法仍然很健壮,仍然可以提供可靠的结果。我们研究了近似边际马尔可夫链蒙特卡罗方法的计算复杂性,以解决由此产生的推理问题,并在许多规范系统上进行了数值比较,这些规范系统包括:线性摆,非线性摆,范德波尔振子,洛伦兹系统和反应扩散系统。这些比较的结果表明,在DMD和SINDy优异的情况下,贝叶斯方法的效果同样好;在DMD和SINDy无法产生合理结果的情况下,贝叶斯方法仍然很健壮,仍然可以提供可靠的结果。洛伦兹系统和反应扩散系统。这些比较的结果表明,在DMD和SINDy优异的情况下,贝叶斯方法的效果同样好;在DMD和SINDy无法产生合理结果的情况下,贝叶斯方法仍然很健壮,仍然可以提供可靠的结果。洛伦兹系统和反应扩散系统。这些比较的结果表明,在DMD和SINDy优异的情况下,贝叶斯方法的效果同样好;在DMD和SINDy无法产生合理结果的情况下,贝叶斯方法仍然很健壮,并且仍然可以提供可靠的结果。

更新日期:2020-09-08
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