Differentiable programming for flows with discontinuities

可微性是研究许多实际相关现象的数值程序的一个非常理想的甚至是本质的性质。激励性的例子包括航天器的设计，例如火箭、航天飞机或再入飞行器，或在飞行中调整飞行构型，要求通过周围气流将结构上的载荷降至最低。这些复杂领域的决策通常依赖于通过数值模拟结果获得的感兴趣数量的敏感性。因此，导数是科学计算中各种最先进方法的关键组成部分，从基本参数敏感度分析、误差和不确定性量化、通过偏微分方程约束下的非线性优化，到数据驱动和混合模拟方法，以及添加了机器学习等人工智能元素的方法。作为计算机程序的模型的可微性及其数值实现的实际差异性都是算法差分法(AD)的主题。AD编译器和/或运行时库使得能够(半自动)区分可区分的程序。后者通常通过数十万行源代码实现高度复杂的数值算法。它们通常在大规模并行的高性能计算机上运行。需要解决数值软件技术中的挑战，特别关注可区分的程序分析和特定领域的程序转换。因此，大多数AD的实际应用都需要计算机科学家、应用数学家和工程师的共同努力。这一提议汇集了所有三个领域的研究人员，旨在为出现激波的极端流动状态下的高度敏感流动的可差分编程取得实质性进展。这一建议旨在开发一种针对具有间断的流型的灵敏度演算，该流型可以修正为差分规划。为了实现这一目标，必须将这三个领域的专门知识结合起来。我们的结果将作为一种可扩展的AD软件解决方案发布，用于一系列以激波结构为特征的无粘流动模拟。将开发相应的程序转换技术，包括适当的派生代码设计模式，这将促使计算机科学在该项目中发挥突出的协调作用。

Differentiability is a highly desirable or even essential property of numerical programs for studying many practically relevant phenomena. Motivating examples include the design of spacecraft, e.g., rockets, shuttles, or re-entry vehicles, or in in--flight adaption of flight configurations, requiring the minimization of loads on the structure by the surrounding flow. Decision making in those complex fields typically relies on sensitivities of quantities of interest obtained through results of numerical simulations. Consequently, derivatives are crucial ingredients of a wide range of state of the art methods in scientific computing ranging from basic parameter sensitivity analysis, error and uncertainty quantification via nonlinear optimization under constraints given by partial differential equations to data-driven and hybrid simulation methods augmented with elements of artificial intelligence such as machine learning. Both differentiability of the models and the actual differentiation of their numerical implementations as computer programs are the subject of algorithmic differentiation (AD). AD compilers and/or run time libraries enable the (semi-)automatic differentiation of differentiable programs. The latter typically implement highly sophisticated numerical algorithms by means of hundreds of thousands of lines of source code. They typically run on massively parallel high-performance computers. Challenges in numerical software technology with a particular focus on differentiable program analysis and domain-specific program transformation need to be addressed. Hence, most real-world applications of AD require collaborative efforts by computer scientists, applied mathematicians, and engineers. This proposal brings together researchers from all three domains aiming to obtain substantial progress in differentiable programming for highly sensitive flows in extreme flow regimes where shocks appear. This proposal aims to develop a sensitivity calculus for flow regimes with discontinuities that are amendable to differential programming. To achieve this objective the expertise from all three domains has to be combined. Our results will be published as an extensible AD software solution for a range of inviscid flow simulations featuring shock structures. Corresponding program transformation techniques will be developed including suitable derivative code design patterns, which motivates the prominent, coordinating role of computer science in this project.