Uncertainty Quantification of Locally Nonlinear Dynamical Systems Using Neural Networks

Abstract

Models are often given in terms of differential equations to represent physical systems. In the presence of uncertainty, accurate prediction of the behavior of these systems using the models requires understanding the effect of uncertainty in the response. In uncertainty quantification, statistics such as mean and variance of the response of these physical systems are sought. To estimate these statistics, sampling-based methods like Monte Carlo often require many evaluations of the models’ governing differential equations for multiple realizations of the uncertainty. However, for large complex engineering systems, these methods become computationally burdensome as the solution of the models’ governing differential equations for such systems is expensive. In structural engineering, an otherwise linear structure often contains spatially local nonlinearities with uncertainty present in them. A standard nonlinear solver for them with sampling-based methods for uncertainty quantification incurs significant computational cost for estimating the statistics of the response. To ease this computational burden of uncertainty quantification of large-scale locally nonlinear dynamical systems, a method is proposed herein that decomposes the response into two parts: response of a deterministic nominal linear system and a corrective term. This corrective term is the response from a pseudoforce that contains the nonlinearity and uncertainty information. In this paper, neural network, a recently popular tool for universal function approximation in the scientific machine-learning community due to the advancement of computational capability as well as the availability of open-source packages, is used to estimate the pseudoforce. Since only the nonlinear and uncertain pseudoforce is modeled using the neural networks, the same network can be used to predict a different response of the system, and no new network is required to train if the statistics of a different response are sought.