Model Order Reduction and Physics-Based Machine Learning: Challenges, Progress, and Impact

Abstract

While machine learning algorithms have been around for a long time, many were developed under different names. Collectively, Projection-based Model Order Reduction (PMOR) methods constitute a case in point. They have been data-driven, physics-informed, machine learning methods since their inception, before these buzzwords became fashionable. For parametric linear problems, they are mature and have already made some impact in industry. For parametric nonlinear problems, their state of the art has been significantly advanced during the last decade on all of theoretical, algorithmic, and application fronts. Furthermore, many rational myth theories about PMOR have been debunked, particularly for highly nonlinear problems. Impressive results have been demonstrated for complex geometries and industrial-grade problems in many applications including nonlinear multi-scale solid mechanics, nonlinear structural dynamics with contact, convection-dominated turbulent flows, wave propagation in both the frequency and time domains, linear and nonlinear multi-disciplinary shape optimization, and uncertainty quantification. On the other hand, high-dimensional parameter spaces, topological changes, and the Kolmogorov n-width issue remain outstanding challenges; nevertheless, significant progress has also been made in these areas. First, this talk will rapidly overview the aforementioned state assessment of PMOR. Next, it will contrast the state of the art of this computational technology with alternative, popular, surrogate modeling techniques including those based on artificial neural networks, before presenting a recently developed, disruptive PMOR approach and reporting on its performance for real-world applications. Then, the talk will discuss the concept of mechanics-informed neural networks for data-driven constitutive modeling, where the informing process is not associated with any physics-based residual or differential equation, but with teaching a neural network mechanics concepts such as objectivity, isotropy (as needed), consistency, material stability, and dynamic stability. Finally, the talk will report on applications pertaining to the recent landing of Perseverance on Mars.