  Dynamical system state estimation and parameter calibration problems are
ubiquitous across science and engineering. Bayesian approaches to the problem
are the gold standard as they allow for the quantification of uncertainties and
enable the seamless fusion of different experimental modalities. When the
dynamics are discrete and stochastic, one may employ powerful techniques such
as Kalman, particle, or variational filters. Practitioners commonly apply these
methods to continuous-time, deterministic dynamical systems after discretizing
the dynamics and introducing fictitious transition probabilities. However,
approaches based on time-discretization suffer from the curse of dimensionality
since the number of random variables grows linearly with the number of
time-steps. Furthermore, the introduction of fictitious transition
probabilities is an unsatisfactory solution because it increases the number of
model parameters and may lead to inference bias. To address these drawbacks,
the objective of this paper is to develop a scalable Bayesian approach to state
and parameter estimation suitable for continuous-time, deterministic dynamical
systems. Our methodology builds upon information field theory. Specifically, we
construct a physics-informed prior probability measure on the function space of
system responses so that functions that satisfy the physics are more likely.
This prior allows us to quantify model form errors. We connect the system's
response to observations through a probabilistic model of the measurement
process. The joint posterior over the system responses and all parameters is
given by Bayes' rule. To approximate the intractable posterior, we develop a
stochastic variational inference algorithm. In summary, the developed
methodology offers a powerful framework for Bayesian estimation in dynamical
systems.
