Presentation given by Kaushik Bhattacharya on 2 June 2021 in the one world seminar on the mathematics of machine learning on the topic "Learning based multi-scale modeling".
Abstract: The behavior of materials involve physics at multiple length and time scales: electronic, atomistic, domains, defects etc. The engineering properties that we observe and exploit in application are a sum total of all these interactions. Multiscale modeling seeks to understand how the physics at the finer scales affect the coarser scales. This can be challenging for two reasons. First, it is computationally expensive due to the need to repeatedly solve the finer scale model. Second, it requires a priori (empirical) knowledge of the aspects of the finer-scale behavior that affect the coarser scale (order parameters, state variables, descriptors, etc.). This is especially challenging in situations where the behavior depends on time. We regard the solution of the finer-scale model as an input-output map (possibly between infinite dimensional spaces), and introduce a a general framework for the data-driven approximation of such maps. The proposed approach is motivated by the recent successes of neural networks and deep learning, in combination with ideas from model reduction. This combination results in a neural network approximation that is computationally inexpensive, independent of the need for a priori knowledge, and can be used directly in the coarser scale calculations. We demonstrate the ideas with examples drawn from first principles study of defects and crystal plasticity study of inelastic impact. The work draws from collaborations with the Caltech PDE-ML group and in particular Burigede Liu, Nikola Kovachki and Ying Shi Teh.
