Bayesian updating for nonlinear dynamics problems using machine learning

The probabilistic parameter and state estimation of highly nonlinear dynamical system can be quite challenging due to presence of both nonlinearity and non-Gaussianity. In this paper we show a computationally efficient approach to the previously mentioned estimation. By setting the problem in a Bayesian framework, we show that the posterior mean can be directly estimated given measurement data if the optimal map representing the conditional expectation of the quantity to be estimated given the observation variable is known. As an example, we explore the simple nonlinear feedforward neural network as an approximation of conditional expectation. The approach is tested on a highly nonlinear Lorenz-63 system by employing Monte Carlo sampling.