High-dimensional/high-fidelity nonlinear dynamical systems appear naturally when the goal is to accurately model real-world phenomena. Many physical properties are thereby encoded in the internal differential structure of these resulting large-scale nonlinear systems. The high dimensionality of the dynamics causes computational bottlenecks, especially when these large-scale systems need to be simulated for a variety of situations such as different forcing terms. This motivates model reduction where the goal is to replace the full-order dynamics with accurate reduced-order surrogates. Interpolation-based model reduction has been proven to be an effective tool for the construction of cheap-to-evaluate surrogate models that preserve the internal structure in the case of weak nonlinearities. In this paper, we consider the construction of multivariate interpolants in frequency domain for structured quadratic-bilinear systems. We propose definitions for structured variants of the symmetric subsystem and generalized transfer functions of quadratic-bilinear systems and provide conditions for structure-preserving interpolation by projection. The theoretical results are illustrated using two numerical examples including the simulation of molecular dynamics in crystal structures.

当目标是准确地模拟现实世界的现象时，高维/高保真非线性动力系统就会自然出现。许多物理特性因此被编码在这些由此产生的大规模非线性系统的内部微分结构中。动力学的高维性会导致计算瓶颈，特别是当这些大型系统需要针对各种情况（例如不同的强迫项）进行模拟时。这促进了模型简化，其目标是用精确的降阶代理代替全阶动态。基于插值的模型简化已被证明是构建评估成本低廉的替代模型的有效工具，该模型在弱非线性的情况下保留了内部结构。在本文中，我们考虑在频域中构造结构化二次双线性系统的多元插值。我们提出了对称子系统的结构化变体和二次双线性系统的广义传递函数的定义，并为通过投影进行结构保持插值提供了条件。使用两个数值例子说明了理论结果，包括晶体结构中分子动力学的模拟。