Dimension Reduction for Nonlinear Stochastic Systems

预测计算模型不仅在工程和科学中发挥着核心作用，而且在整个社会中也发挥着核心作用。这个项目解决了几乎所有计算模型共同的基本问题。如果一个模型过于简单，底层现象就不能被忠实地描述；如果一个模型过于复杂，它的预测能力将是最小的。在某个地方，有一个幸福的中间；这个项目是关于如何在模型复杂性方面找到正确的平衡。一个计算模型应该多复杂才有用？长期以来，数学界和该领域的科学家都在担心最小复杂度的下限。模型必须忠实于潜在的生物学、化学或物理学，但一个模型应该包含多少科学知识呢？量子和相对论效应在许多问题上可以被安全地忽略，但在描述复杂的化学反应或生理模型时，哪些应该包括，哪些可以忽略？该项目旨在为手头的任务找到足够的最低复杂性。然而，复杂系统的模型所呈现的特征使得用量化的不确定性进行预测的任务具有挑战性。这些特征包括高维不确定输入参数，时间和/或空间相关的兴趣量，固有的随机性，以及模拟复杂模型的计算费用。该项目的研究将通过发展数学技术和计算方法来解决复杂系统模型中的这些挑战，从而在不确定性下的建模方面取得关键进展。总体目标是开发方法，使领域科学家能够简化他们的模型，以便以合理的成本促进前向不确定性量化和参数估计。学生将在这个项目的跨学科方面接受培训和指导。此外，他的项目还包括开发新的课程材料，以反映和解决当今科学计算中的挑战。本项目的研究为不确定条件下的计算建模提供了数学理论和算法，包括：(1)随机隔室模型的多尺度灵敏度分析，(2)基于导数和方差的时变随机系统灵敏度分析方法，(3)隔室模型的模型复杂度降低，以及(4)面向目标的快速参数估计的多层降维。这些算法将应用于生物化学和生理学的模型。本项目将采用基于现有小鼠数据的降维、模型复杂度降低和参数估计等方法，对一系列复杂的神经血管耦合模型进行分析。模型复杂性降低框架可以在广泛的模型中使用，建模者可以从删除某些模型组件中获益。总的来说，通过在广泛的模型类别中实现降维，该项目在建模、不确定性下的预测和参数估计过程中架起了桥梁：将发现具有最基本组成部分的模型，并且计算预算可以专注于量化那些对模型输出最重要的模型参数中的不确定性。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。

Predictive computational models play a central role not only in engineering and the sciences, but also in society at large. This project addresses fundamental issues common to virtually all computational models. If a model is too simple, the underlying phenomenon will not be faithfully described; if a model is too complex, its predictive power will be minimal. There is, somewhere, a happy middle; this project is about how to find the right balance in terms of model complexity. How complex should a computational model be to be useful? Both the mathematical community and domain scientists have long been worried about the lower bound – the minimum complexity. Models have be faithful to the underlying biology, chemistry or physics but how much of the science should one include? Quantum and relativistic effects can safely be ignored on many problems but what should be included and what can be left out to describe complex chemical reactions or physiological models? This project is about finding the lowest complexity sufficient for the tasks at hand. Models of complex systems, however, present features that render the task of making prediction with quantified uncertainties challenging. Such features include high-dimensional uncertain input parameters, time and/or space dependent quantities of interest, inherent stochasticity, as well as computational expense of simulating complex models. Research in this project will bring about key advances in modeling under uncertainty by developing mathematical techniques and computational methods to address such challenges in models of complex systems. The overarching goal is to develop methods that allow domain scientists to simplify their models so as to facilitate forward uncertainty quantification and parameter estimation at reasonable costs. Students will be trained and mentored in the interdisciplinary aspects of this project. In addition, his project involves the development of new course material that reflects and addresses challenges in present-day scientific computing.The research in this project makes important contributions to computational modeling under uncertainty by developing mathematical theory and algorithms for (i) multiscale sensitivity analysis of stochastic compartment models, (ii) derivative- and variance-based sensitivity analysis methods for time-dependent stochastic systems, (iii) model complexity reduction in compartment models, and (iv) goal oriented multilevel dimension reduction for fast parameter estimation. The algorithms will be applied to models from biochemistry and physiology. A family of complex models of neurovascular coupling will be analyzed with the methods to be developed during the project, including dimension reduction, model complexity reduction and parameter estimation based on existing murine data. The model complexity reduction framework can be used in a broad range of models, where modelers can benefit from removing certain model components. Overall, by enabling dimension reduction in broad classes of models, the project bridges gaps in the processes of modeling, prediction under uncertainty, and parameter estimation: models with most essential components will be discovered and computational budget can be focused on quantifying uncertainty in only those model parameters that are most important to model output.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.