Parametric Nonlinear Volterra Series via Machine Learning: Transonic Aerodynamics

In aerospace and mechanical engineering, the design process for new products relies on hierarchies of mathematical models, the physical complexity of which may be imposed by computational costs or dictated by regulations. These models typically incorporate parameters to account for various operating conditions and configurations. In the framework of optimization, for example, hundreds of parameters (design variables) may be required to define the configuration of a system. Similarly, uncertainty propagation may necessitate defining a complex parameter space to account for variations in geometrical imperfections, material properties, or flow conditions. The design process, especially during optimization and uncertainty quantification, often involves numerous evaluations of the system's mathematical models across a wide range of points in the parameter space. Computational costs vary with the level of model fidelity: lower fidelity models are traditionally used for computationally intensive evaluations, while higher fidelity models - often involving nonlinear partial differential equations discretized on fine grids - are typically reserved for later stages of the design process.