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

• URL: http://arxiv.org/abs/2401.12476v3
• Date: Sat, 20 Jul 2024 13:04:54 GMT
• Title: Bayesian identification of nonseparable Hamiltonians with multiplicative noise using deep learning and reduced-order modeling
• Authors: Nicholas Galioto, Harsh Sharma, Boris Kramer, Alex Arkady Gorodetsky,
• Abstract summary: This paper presents a structure-preserving Bayesian approach for learning nonseparable Hamiltonian systems. We develop a novel algorithm for cost-effective application of Bayesian system identification to high-dimensional systems. We show that using the Bayesian posterior as a training objective can yield upwards of 724 times improvement in Hamiltonian mean squared error.
• Score: 0.5999777817331317
• License: http://creativecommons.org/licenses/by/4.0/
• 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 Schr\"odinger equation with data corrupted by up to 20% multiplicative noise.

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