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

像我们身体中潜在功能的分子相互作用的变化，或者对沿海城市造成毁灭性破坏的极端强烈的飓风这样的事件，很难通过计算来研究，因为它们在可行的模拟时间尺度上很少发生。值得注意的是，理论上可以通过利用某些高维方程来用短时间尺度性质来描述长时间尺度统计来绕过这个问题。利用这些方程以及统计学和机器学习中的新工具来解决它们，该项目引入了使用由许多短模拟组成的数据集来学习长期统计数据的算法。这些算法的发展将通过旨在充分表征其潜在效用的数学分析来了解。这项研究将补充和支持重要的社会利益的重要和多样化的应用。例如，研究与治疗糖尿病和伤口愈合有关的蛋白质组合，以及导致极地涡旋和热带气旋的大气条件变化。由于该项目开发的方法在制定模型时避免了简化假设的需要，因此它们有望以前所未有的细节揭示这些过程和其他过程的潜在物理机制。该项目将为研究生提供参与研究的机会。该项目涉及一系列算法的开发和分析，这些算法仅使用由短轨迹组成的数据集，对极长时间内发生的事件进行预测。在这种方法中，预测（对未来行为的条件期望）被转换为方程的解，这些方程涉及决定潜在动力系统统计量的算子。在采用目标预测函数的基扩展近似的重要和有希望的初步努力的基础上，该项目将开发更具表达性的近似，保持鲁棒性和可靠性。第一个目标将探索基展开方法的核化扩展，这将允许仔细控制近似灵活性的程度。第二个目标将探索变分表示，允许引入神经网络近似及其极端的表达能力。在保持可靠性和可重复性的同时引入这种更高水平的近似灵活性将是一个重大挑战。第三个平行的重点是对基扩展和基于核的方法进行仔细而完整的数学分析，重点是建立一个理论，为更复杂的基于神经网络的方法提供指导。所有计算方法将在基准蛋白质折叠/展开数据集上进行广泛验证。它们的发展将伴随着仔细的数学误差分析，旨在理解各种设计选择的影响，例如数据采样的度量和数据集中短模拟的长度。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.