Model-Embedded Gaussian Process Regression for Parameter Estimation in Dynamical System

Ying Zhou, Jinglai Li, Xiang Zhou, Hongqiao Wang
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Abstract

Identifying dynamical system (DS) is a vital task in science and engineering. Traditional methods require numerous calls to the DS solver, rendering likelihood-based or least-squares inference frameworks impractical. For efficient parameter inference, two state-of-the-art techniques are the kernel method for modeling and the "one-step framework" for jointly inferring unknown parameters and hyperparameters. The kernel method is a quick and straightforward technique, but it cannot estimate solutions and their derivatives, which must strictly adhere to physical laws. We propose a model-embedded "one-step" Bayesian framework for joint inference of unknown parameters and hyperparameters by maximizing the marginal likelihood. This approach models the solution and its derivatives using Gaussian process regression (GPR), taking into account smoothness and continuity properties, and treats differential equations as constraints that can be naturally integrated into the Bayesian framework in the linear case. Additionally, we prove the convergence of the model-embedded Gaussian process regression (ME-GPR) for theoretical development. Motivated by Taylor expansion, we introduce a piecewise first-order linearization strategy to handle nonlinear dynamic systems. We derive estimates and confidence intervals, demonstrating that they exhibit low bias and good coverage properties for both simulated models and real data.
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用于动态系统参数估计的模型嵌入式高斯过程回归
识别动力学系统(DS)是科学与工程领域的一项重要任务。传统方法需要多次调用动力学系统求解器,这使得基于似然法或最小二乘法的推理框架变得不切实际。为实现高效的参数推断,目前有两种最先进的技术,一种是用于建模的核方法,另一种是用于联合推断未知参数和超参数的 "一步法框架"。核方法是一种快速、直接的技术,但它无法估计解及其衍生物,因为它们必须严格遵守物理规律。我们提出了一种嵌入模型的 "一步式 "贝叶斯框架,通过最大化边际似然来联合推断未知参数和超参数。这种方法使用高斯过程回归(GPR)对解及其导数进行建模,同时考虑到平滑性和连续性特性,并将微分方程视为约束条件,在线性情况下可自然集成到贝叶斯框架中。此外,我们还证明了模型嵌入式高斯过程回归(ME-GPR)的收敛性,以促进理论发展。受泰勒展开的启发,我们引入了逐一一阶线性化策略来处理非线性动态系统。我们推导出了估计值和置信区间,证明它们对模拟模型和真实数据都具有低偏差和良好的覆盖特性。
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