Bayesian identification of nonseparable Hamiltonians with multiplicative noise using deep learning and reduced-order modeling

TL;DR

The paper tackles the problem of identifying nonseparable Hamiltonians from noisy data in the presence of multiplicative noise by formulating a structure-preserving Bayesian approach. It combines a Gaussian filter tailored to multiplicative noise with physics-informed parameterization to preserve symplectic structure, and scales to high-dimensional systems via reduced-order modeling and cotangent-lift projections. The authors demonstrate data-efficient learning on Tao’s example, a chaotic double pendulum, and a high-dimensional NLSE discretization, achieving substantial improvements in Hamiltonian accuracy and invariant preservation over standard training objectives. This work enables reliable, physically consistent identification of complex Hamiltonian dynamics from limited, noisy observations, with practical impact for physics-informed forecasting and control of high-dimensional systems.

Abstract

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 Schrödinger equation with data corrupted by up to 20% multiplicative noise.

Figures (15)

• Figure 1: Schematic showing how Hamiltonian structure is preserved when evaluating the likelihood.
• Figure 2: Schematic showing how Hamiltonian structure is preserved in reduced dimensions when evaluating the likelihood. In this setting, the parameter vector is partitioned into quadratic $\theta_{\text{quad}}$ and nonlinear components $\theta_{\text{nl}}$, where $\theta_{\text{quad}}$ is determined by $\theta_{\text{nl}}$ through the H-OpInf algorithm.
• Figure 3: Tao's example: $\log_{10}$ MSE \ref{['eq:mse']} of models trained using $-\log\pi(\boldsymbol{\uptheta}|\mathcal{Y}_{N})$ and the $L_1$ norm \ref{['eq:objective']} as objective functions. The label $\pi(\boldsymbol{\uptheta}|\mathcal{Y}_{N})$ represents the MAP estimate, and 'Difference' represents the $\log_{10}$ MSE of the MAP estimate minus the $\log_{10}$ MSE of the $L_1$ estimate. The median MSEs of the MAP estimates are lower than those of the $L_1$ estimates on all datasets, and the minimum MSEs of the MAP estimates are lower than those of the $L_1$ estimates on all datasets with noise.
• Figure 4: Tao's example: Estimated trajectories from the median MSE models. The MAP estimate closely matches the truth when the data are noiseless and degrades gracefully as noise increases. The $L_1$ estimates reflect that the optimizer gets caught in local minima when the training dataset is small.
• Figure 5: Tao's example: Estimated trajectories from the minimum MSE models. The MAP estimates match the truth even on noisy datasets, whereas the $L_1$ estimates only match the truth on the noiseless datasets.