Efficient reliability analysis with multifidelity Gaussian processes and normalizing flows

Estimating probability of failure in aerospace systems is a critical requirement for flight certification and qualification. Furthermore, constraining design optimization with failure probabilities have immense potential toward building safety into the design of complex engi- neered systems. Failure probability estimation (FPE) involves resolving tails of probability distribution and Monte Carlo (MC) sampling methods are intractable when expensive high-fidelity simulations have to be queried, and can lead to large variance in the estimators. Despite the limitations, MC methods are attractive given the simplicity of the approach and its implementation. Computational efficiency of MC methods can be improved by developing surrogate models of the underlying function and variance reduction can be achieved via a variety of methods such as importance sampling (IS). Specifically, the surrogate model offers a cheap means to construct a biasing distribution for IS which can re-weight the MC estimator for better sample efficiency and reduced variance. In this work, we seek to further improve this (surrogate + IS) paradigm by contributing on two fronts: more efficient surrogate models with multifidelity Gaussian processes and estimating IS biasing distributions with normalizing flows. The use of multifidelity Gaussian process models is to efficiently fuse models at multiple fidelity and thereby offering a cheap surrogate model that emulates the original model at all fidelities. Furthermore, this lends itself very well towards adaptive surrogate model development via the Bayesian decision theoretic framework and experiment design. The normalizing flows then offers a more flexible way to estimating the biasing density for IS. We demonstrate our method on the FPE on synthetic test functions as well as the reliability analysis of a gas turbine engine blade. Our results show that using normalizing flows results in biasing densities that are more realistic and closer to the optimal biasing densities (in the KL divergence sense) and predicts failure probabilities more accurately with smaller variance.