DMS/NIGMS 1: Design and Analysis of Machine Learning Approaches for Long Timescale Prediction from Short Trajectory Data

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Events like the changes in molecular interactions underlying functions in our bodies, or the extremely intense hurricanes that cause devastating damage to our coastal cities, are difficult to study computationally because they occur very infrequently on feasible simulation timescales. Remarkably, it is theoretically possible to bypass this issue by appealing to certain high-dimensional equations characterizing long-timescale statistics in terms of short-timescale properties. Leveraging these equations and new tools in statistical and machine learning to solve them, this project introduces algorithms to learn long-time statistics using a data set consisting of many short simulations. The development of these algorithms will be informed by mathematical analysis aimed at fully characterizing their potential utility. The research will complement and support significant and diverse applications of critical societal interest. Examples include studies of protein assemblies relevant to treating diabetes and wound healing and changes in atmospheric conditions that lead to polar vortices and tropical cyclones. Because the methods developed in this project avoid the need to make simplifying assumptions when formulating the models, they promise to reveal the underlying physical mechanisms of these and other processes in unprecedented detail. The project will provide opportunities to graduate students to be involved in the research.This project concerns the development and analysis of a family of algorithms that assemble forecasts of events occurring over extremely long times using only a data set consisting of short trajectories. In this approach, forecasts (conditional expectations of future behavior) are cast as solutions to equations involving the operator determining the statistics of the underlying dynamical system. Building on significant and promising preliminary efforts employing a basis expansion approximation of the target predictive functions, this project will develop more expressive approximations that remain robust and reliable. A first aim will explore kernelized extensions of the basis expansion approach, which will allow careful control of the degree of approximation flexibility. A second aim will explore variational representations, allowing the introduction of neural network approximations and their extreme expressive power. Introducing this higher level of approximation flexibility while maintaining reliability and reproducibility will be a significant challenge. A third but parallel thrust will provide a careful and complete mathematical analysis of the basis expansion and kernel-based methods with an emphasis on building a theory that informs even the more complicated neural network-based approaches. All computational approaches will be extensively validated on a benchmark protein folding/unfolding data set. Their development will be accompanied by careful mathematical error analysis aimed at understanding the effect of various design choices such as the measure with respect to which the data is sampled and the length of the short simulations in the data set.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.