H. B. Keller Colloquium Series
The nonparametric probabilistic method (NPM) for modeling and quantifying model-form uncertainty introduced by Soize and Farhat in  is a multi-faceted, data-enhanced, computational modeling method. It is grounded in projection-based model order reduction and therefore is data-driven and computationally tractable. It can be described as a physics-based machine learning method for extracting from data information that is not captured by a deterministic high-dimensional model (HDM) of dimension N; and infusing it into a counterpart stochastic projection-based reduced-order model (PROM). Starting from a deterministic HDM, NPM constructs in three steps a stochastic PROM. First, it builds and trains a deterministic PROM of dimension n<<N. Next, it substitutes the deterministic reduced-order basis underlying the PROM with a stochastic counterpart that it constructs on a subset of a compact Stiefel manifold using a number of hyperparameters that grows as of O(n^2). Then, it identifies these hyperparameters by formulating and solving a data-driven statistical inverse problem. While the potential of NPM was demonstrated for a number of applications, its practicality is contingent upon the effective solution of the optimization problem governing the identification of its hyperparameters. Solving this problem, which is non-convex and stochastic, is challenging for n>20. This issue is addressed here by a two-pronged approach. First, a network of autoencoders that provides a nonlinear approximation of the dependence of the randomized reduced-order basis on its hyperparameters is developed to reduce their number from O(n^2) to O(n). Second, the complex web of operations underlying the construction of the stochastic objective function is tracked and an adjoint framework is developed to enable the analytical evaluation of the sensitivities of the objective function with respect to the hyperparameters. The result is a dramatic improvement of the robustness and performance of NPM that makes it a viable candidate for constructing digital twins of the instance type. This potential is demonstrated for one application pertaining to the crash analysis of a car; and another application pertaining to the analysis of the nozzle of a supersonic jet engine.