Differentiable programming for flows with discontinuities

• Applicants: Professor Dr. Michael Herty; Professor Dr. Uwe Naumann; Professor Dr.-Ing. Wolfgang Schröder
• Subject Area: Mathematics ; Software Engineering and Programming Languages ; Fluid Mechanics
• Term: since 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 513718742

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.

• DFG Programme: Research Grants