  We develop a novel computational method for evaluating the extreme excursion
probabilities arising from random initialization of nonlinear dynamical
systems. The method uses excursion probability theory to formulate a sequence
of Bayesian inverse problems that, when solved, yields the biasing
distribution. Solving multiple Bayesian inverse problems can be expensive; more
so in higher dimensions. To alleviate the computational cost, we build
machine-learning-based surrogates to solve the Bayesian inverse problems that
give rise to the biasing distribution. This biasing distribution can then be
used in an importance sampling procedure to estimate the extreme excursion
probabilities.
