Bayesian identification of nonseparable Hamiltonians with multiplicative
  noise using deep learning and reduced-order modeling
Nicholas Galioto Harsh Sharma Boris Kramer Alex Arkady Gorodetsky
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.