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 assess the method's performance based on the forecasting accuracy of a model estimated from single-trajectory data. 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：本文提出一种结构保持的贝叶斯方法，用于学习具有统计相关的向量值加性与乘性测量噪声的随机动力学模型所描述的非可分离哈密顿系统。该方法包含三个主要方面。首先，我们推导了适用于统计相关的向量值加性与乘性噪声模型的高斯滤波器，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，可将贝叶斯系统辨识经济高效地应用于高维系统。第三，我们以非可分离哈密顿系统为例，展示了如何将结构保持方法融入所提出的框架中。我们基于从单轨迹数据估计的模型的预测精度来评估该方法的性能。我们在一个经典的非可分离哈密顿模型和一个混沌双摆模型上，使用小型、含噪声的训练数据集，将贝叶斯方法与一种先进的机器学习方法进行了比较。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，在训练数据包含高达10%乘性噪声的情况下，哈密顿量的均方误差最高可改善724倍。最后，我们通过一个空间离散化的非线性薛定谔方程的64维模型，展示了该新颖算法在参数估计中的效用，所用数据受到高达20%乘性噪声的污染。