Collaborative Research: SI2-SSI: Integrating Data with Complex Predictive Models under Uncertainty: An Extensible Software Framework for Large-Scale Bayesian Inversion

科学家经常使用数学模型来预测自然系统和工程系统的行为。因此，这些模型是科学和工程进步的基础，因此与NSF的科学任务相关。大多数现实物理系统的模型使用涉及许多变量的复杂公式(例如，偏微分方程式)。当使用这样的模型来预测系统的未来行为时，科学家必须提供所有变量的初始值。这可能很困难，因为输入值可能无法直接测量。因此，科学家常常必须使用“逆”运算来计算系统模型变量的初始输入值，这是基于对现实世界的外部观察。换句话说，科学家试图从输出的真实观测数据中推断出物理过程的计算机模型的输入。有许多逆计算的例子，从通过CAT扫描计算器官的重要尺寸，通过测量其在不同位置的体积和频率来重建声源，通过测量其重力场来计算地球的密度，或者计算大气的初始条件(温度、压力等)。来自卫星和气象站在一段时间间隔内的观测。逆问题在所有科学和工程领域(甚至更远的领域)中无处不在。逆问题有许多解决方案，即使数据与观测数据相匹配的解决方案。然而，在确定的解决方案中存在差异。也就是说，反问题的解是不确定的。贝叶斯推理为表征这种不确定性提供了一个系统的数学框架。然而，大规模复杂模型反问题的贝叶斯解需要巨大的计算能力。直到最近，才开始出现在计算上容易掌握的算法。然而，由于这些算法的复杂性和对正演模型更深层次信息的需要，这些算法仍然超出了解决逆问题的主流科学家的能力范围。该项目旨在开发、分发和支持开源软件，该软件编码用于解决大规模复杂贝叶斯逆问题的最先进算法，并且健壮、可伸缩、灵活、模块化、可广泛访问且易于使用。该项目在很大程度上建立在团队一直在开发的两个互补的开源软件库之上：麻省理工学院的MUQ和加州大学奥斯汀分校/加州大学默塞德分校的hIPPYlib。MUQ提供了一系列强大的贝叶斯反演模型和算法，但预计正演模型将配备梯度/海森，以允许大规模求解。HIPPYlib在一个可以自动生成所需导数的环境中实现了强大的大规模渐变/基于海森的逆求解器，但它缺乏完整的贝叶斯功能。通过集成这两个互补库，该项目将产生一个强大、可扩展和高效的软件框架，实现每个库的好处，以解决广泛的科学和工程学科中复杂的大规模贝叶斯逆问题。由此产生的软件将以开放源码许可证的形式分发，它将为快速开发配备梯度/海森信息的逆模型提供环境；为评估和比较算法提供基准问题；为培训和测试目的提供辅导问题。

Scientists often use mathematical models to predict the behavior of natural and engineered systems. These models are therefore fundamental to scientific and engineering progress and hence relevant to NSF's science mission. Most models of realistic physical systems use complex formulae (such as, partial differential equations) involving many variables. When using such a model for predicting the future behavior of a system, a scientist has to provide initial values for all the variables. This can be difficult because input values may not be directly measureable. Thus, scientists often must use "inverse" computations to calculate the initial input values of the variables of a system model based on external observations of the real world. In other words, scientists seek to infer inputs to a computer model of a physical process from real observational data of the outputs. There are many examples of inverse computations, ranging from computing the important dimensions of an organ from its CAT scan, reconstructing the source of a sound by measuring its volume and frequency at various places, calculating the density of the Earth from measurements of its gravity field, or calculating the initial condition of the atmosphere (temperature, pressure, etc.) from satellite and weather station observations over a time interval. Inverse problems are ubiquitous across all of science and engineering (and beyond). Many solutions exist for inverse problems, i.e. solutions that fit the data to the observations. However, there are variations in the solutions identified. That is, the solutions of an inverse problem are subject to uncertainty. Bayesian inferencing provides a systematic mathematical framework for characterizing this uncertainty. However, the Bayesian solution of inverse problems for large-scale complex models require enormous computational power. Only recently have algorithms begun to emerge that are computationally tractable. However, these algorithms have remained out of the reach of the mainstream of scientists who solve inverse problems, due to their complexity and the need for deeper information from the forward model. This project aims to develop, distribute, and support open-source software that encodes state-of-the-art algorithms for the solution of large-scale complex Bayesian inverse problems and is robust, scalable, flexible, modular, widely accessible, and easy to use.The project builds heavily on two complementary open-source software libraries the team has been developing: MUQ at MIT, and hIPPYlib at UT-Austin/UC-Merced. MUQ provides a spectrum of powerful Bayesian inversion models and algorithms, but expects forward models to come equipped with gradients/Hessians to permit large-scale solution. hIPPYlib implements powerful large-scale gradient/Hessian-based inverse solvers in an environment that can automatically generate needed derivatives, but it lacks full Bayesian capabilities. By integrating these two complementary libraries, the project will result in a robust, scalable, and efficient software framework that realizes the benefits of each to tackle complex large-scale Bayesian inverse problems across a broad spectrum of scientific and engineering disciplines. The resulting software, that will be distributed under an open-source license, will provide an environment for rapid development of inverse models equipped with gradient/Hessian information; benchmark problems for evaluation and comparison of algorithms; and tutorial problems for training and testing purposes.