This paper presents a structure-preserving Bayesian approach for learning nonseparable Hamiltonian systems using stochastic dynamic models allowing for statistically-dependent, vector-valued additive and multiplicative measurement noise. The approach is comprised of three main facets. First, we derive a Gaussian filter for a statistically-dependent, vector-valued, additive and multiplicative noise model that is needed to evaluate the likelihood within the Bayesian posterior. Second, we develop a novel algorithm for cost-effective application of Bayesian system identification to high-dimensional systems. Third, we demonstrate how structure-preserving methods can be incorporated into the proposed framework, using nonseparable Hamiltonians as an illustrative system class. We compare the Bayesian method to a state-of-the-art machine learning method on a canonical nonseparable Hamiltonian model and a chaotic double pendulum model with small, noisy training datasets. The results show that using the Bayesian posterior as a training objective can yield upwards of 724 times improvement in Hamiltonian mean squared error using training data with up to 10% multiplicative noise compared to a standard training objective. Lastly, we demonstrate the utility of the novel algorithm for parameter estimation of a 64-dimensional model of the spatially-discretized nonlinear Schr\"odinger equation with data corrupted by up to 20% multiplicative noise.

翻译：本文提出一种保持结构的贝叶斯方法，用于利用允许统计依赖的向量值加性和乘性测量噪声的随机动态模型，学习不可分离哈密顿系统。该方法包含三个主要方面。首先，我们针对统计依赖的向量值加性和乘性噪声模型推导出高斯滤波器，该滤波器用于评估贝叶斯后验中的似然函数。其次，我们开发了一种新颖算法，用于将贝叶斯系统辨识经济高效地应用于高维系统。第三，我们以不可分离哈密顿系统为例，展示了如何将保持结构的方法纳入所提出的框架中。我们将贝叶斯方法与一种先进的机器学习方法，在一个典型不可分离哈密顿模型和一个存在小规模含噪训练数据集的混沌双摆模型上进行了比较。结果表明，与标准训练目标相比，使用贝叶斯后验作为训练目标，在训练数据含高达10%乘性噪声的情况下，可使哈密顿均方误差获得高达724倍的改善。最后，我们展示了该新颖算法在空间离散化非线性薛定谔方程的64维模型参数估计中的实用性，其中所用数据受高达20%乘性噪声污染。