Rohit Tripathy

Rohit Tripathy, Ilias Bilionis

A problem of considerable importance within the field of uncertainty quantification (UQ) is the development of efficient methods for the construction of accurate surrogate models. Such efforts are particularly important to applications constrained by high-dimensional uncertain parameter spaces. The difficulty of accurate surrogate modeling in such systems, is further compounded by data scarcity brought about by the large cost of forward model evaluations. Traditional response surface techniques, such as Gaussian process regression (or Kriging) and polynomial chaos are difficult to scale to high dimensions. To make surrogate modeling tractable in expensive high-dimensional systems, one must resort to dimensionality reduction of the stochastic parameter space. A recent dimensionality reduction technique that has shown great promise is the method of `active subspaces'. The classical formulation of active subspaces, unfortunately, requires gradient information from the forward model - often impossible to obtain. In this work, we present a simple, scalable method for recovering active subspaces in high-dimensional stochastic systems, without gradient-information that relies on a reparameterization of the orthogonal active subspace projection matrix, and couple this formulation with deep neural networks. We demonstrate our approach on synthetic and real world datasets and show favorable predictive comparison to classical active subspaces.

Rohit Tripathy, Ilias Bilionis

State-of-the-art computer codes for simulating real physical systems are often characterized by a vast number of input parameters. Performing uncertainty quantification (UQ) tasks with Monte Carlo (MC) methods is almost always infeasible because of the need to perform hundreds of thousands or even millions of forward model evaluations in order to obtain convergent statistics. One, thus, tries to construct a cheap-to-evaluate surrogate model to replace the forward model solver. For systems with large numbers of input parameters, one has to deal with the curse of dimensionality - the exponential increase in the volume of the input space, as the number of parameters increases linearly. In this work, we demonstrate the use of deep neural networks (DNN) to construct surrogate models for numerical simulators. We parameterize the structure of the DNN in a manner that lends the DNN surrogate the interpretation of recovering a low dimensional nonlinear manifold. The model response is a parameterized nonlinear function of the low dimensional projections of the input. We think of this low dimensional manifold as a nonlinear generalization of the notion of the active subspace. Our approach is demonstrated with a problem on uncertainty propagation in a stochastic elliptic partial differential equation (SPDE) with uncertain diffusion coefficient. We deviate from traditional formulations of the SPDE problem by not imposing a specific covariance structure on the random diffusion coefficient. Instead, we attempt to solve a more challenging problem of learning a map between an arbitrary snapshot of the diffusion field and the response.