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的科学使命相关。大多数现实物理系统的模型使用复杂的公式（如偏微分方程），涉及许多变量。当使用这样的模型来预测系统的未来行为时，科学家必须为所有变量提供初始值。这可能很困难，因为输入值可能无法直接测量。因此，科学家经常必须使用“逆”计算来计算基于真实世界的外部观察的系统模型变量的初始输入值。换句话说，科学家们试图从实际观测数据的输出中推断出物理过程的计算机模型的输入。反计算的例子有很多，从通过CAT扫描计算器官的重要尺寸，通过测量不同地方的音量和频率来重建声源，通过测量地球的重力场计算地球的密度，或者通过卫星和气象站在一段时间间隔内的观测计算大气的初始状态（温度、压力等）。逆问题在所有科学和工程（以及其他领域）中无处不在。反问题存在许多解，即解使数据与观测值拟合。然而，在确定的解决方案中存在差异。也就是说，反问题的解受制于不确定性。贝叶斯推理为描述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可计算处理的算法。然而，由于这些算法的复杂性和需要从正演模型中获得更深入的信息，这些算法仍然超出了解决逆问题的主流科学家的范围。该项目旨在开发、分发和支持开源软件，该软件编码解决大规模复杂贝叶斯反问题的最先进算法，并且具有鲁棒性、可扩展性、灵活性、模块化、广泛可访问性和易于使用性。该项目在很大程度上建立在两个互补的开源软件库上：麻省理工学院的MUQ和UT-Austin/UC-Merced的hIPPYlib。MUQ提供了一系列强大的贝叶斯反演模型和算法，但希望正演模型配备梯度/Hessians以允许大规模解决。hipylib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梯度/ hessian反求解器，但它缺乏完整的Bayesian功能。通过集成这两个互补的库，该项目将产生一个健壮的、可伸缩的、高效的软件框架，它实现了每个库在处理跨越广泛的科学和工程学科的复杂的大规模贝叶斯逆问题方面的优势。最终的软件将在开源许可下发布，将为配备梯度/黑森信息的逆模型的快速开发提供环境；评估和比较算法的基准问题；以及用于培训和测试目的的教程问题。

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.