  We introduce neural information field filter, a Bayesian state and parameter
estimation method for high-dimensional nonlinear dynamical systems given large
measurement datasets. Solving such a problem using traditional methods, such as
Kalman and particle filters, is computationally expensive. Information field
theory is a Bayesian approach that can efficiently reconstruct dynamical model
state paths and calibrate model parameters from noisy measurement data. To
apply the method, we parameterize the time evolution state path using the span
of a finite linear basis. The existing method has to reparameterize the state
path by initial states to satisfy the initial condition. Designing an
expressive yet simple linear basis before knowing the true state path is
crucial for inference accuracy but challenging. Moreover, reparameterizing the
state path using the initial state is easy to perform for a linear basis, but
is nontrivial for more complex and expressive function parameterizations, such
as neural networks. The objective of this paper is to simplify and enrich the
class of state path parameterizations using neural networks for the information
field theory approach. To this end, we propose a generalized physics-informed
conditional prior using an auxiliary initial state. We show the existing
reparameterization is a special case. We parameterize the state path using a
residual neural network that consists of a linear basis function and a Fourier
encoding fully connected neural network residual function. The residual
function aims to correct the error of the linear basis function. To sample from
the intractable posterior distribution, we develop an optimization algorithm,
nested stochastic variational inference, and a sampling algorithm, nested
preconditioned stochastic gradient Langevin dynamics. A series of numerical and
experimental examples verify and validate the proposed method.
