Time-series machine-learning error models for approximate solutions to parameterized dynamical systems

Abstract

This work proposes a machine-learning framework for modeling the error incurred by approximate solutions to parameterized dynamical systems. In particular, we extend the machine-learning error models (MLEM) framework proposed in Ref. 15 to dynamical systems. The proposed Time-Series Machine-Learning Error Modeling (T-MLEM) method constructs a regression model that maps features–which comprise error indicators that are derived from standard a posteriori error-quantification techniques–to a random variable for the approximate-solution error at each time instance. The proposed framework considers a wide range of candidate features, regression methods, and additive noise models. We consider primarily recursive regression techniques developed for time-series modeling, including both classical time-series models (e.g., autoregressive models) and recurrent neural networks (RNNs), but also analyze standard non-recursive regression techniques (e.g., feed-forward neural networks) for comparative purposes. Numerical experiments conducted on multiple benchmark problems illustrate that the long short-term memory (LSTM) neural network, which is a type of RNN, outperforms other methods and yields substantial improvements in error predictions over traditional approaches.

Eric Parish

Senior member of technical staff

Eric is a research staff member at Sandia National Laboratories. Previously, Eric was a John von Neumann postdoctoral fellow at Sandia, and before that he earned his Ph.D. from the University of Michigan in Aerospace Engineering. Eric’s research focuses on the development of engineering technologies that enable rapid simulation of complex multiscale and multiphysics systems through computational engineering, applied math, and machine learning. He is particulaly interested in reduced-order modeling, numerical methods for PDEs, scientific machine learning, and computational fluid dynamics.