In this talk, we outline a computational framework for engineering design of multiscale and multiphysics systems. The framework revolves around (1) reducing errors in numerical simulations via multiscale modeling, (2) leveraging data to construct approximate models suitable for many-query and time-critical problems, and (3) quantifying the error incurred by these approximate models via machine learning.
In the first part of the talk, we focus on reducing errors in numerical simulations of transient partial differential equations (PDEs). We outline a novel closure modeling and stabilization framework, termed MZ—VMS, that combines the Mori—Zwanzig formalism of statistical mechanics with the variational multiscale method. We show that unresolved effects in numerical simulations yield a non-local residual-based memory term, the inclusion of which yields substantially improved numerical simulations. Next, we discuss how we can integrate data into the MZ—VMS framework to develop accurate projection-based reduced-order models for time-critical and many-query applications. The final part of the talk focuses on quantifying the error incurred by these approximate models via a posteriori machine learning error models. We outline the Time-Series Machine-Learning Error Modeling (T-MLEM) approach. T-MLEM constructs a regression model that maps features—which comprise residual-based error indicators—to a random variable for the solution error at each time instance. We consider recursive regression techniques from deep learning, including recurrent and long short-term memory neural networks. Together, these technologies provide the tools for a design framework capable of many-query and time-critical predictions with statistically certified error bounds.
