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.

翻译：高维/高保真非线性动力系统在需要精确建模真实世界现象时自然出现。许多物理属性由此被编码在这些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，尤其是当这些大规模系统需要针对不同外力项等多种情景进行模拟时。这引出了模型降阶的需求，其目标是用精确的降阶替代模型替代全阶动力学。基于插值的模型降阶已被证明是构建廉价评估替代模型的有效工具，该模型在弱非线性情况下能保持内部结构。本文考虑在频域内为结构化二次-双线性系统构建多元插值函数。我们提出了二次-双线性系统对称子系统和广义传递函数的结构化变体定义，并给出了通过投影实现保结构插值的条件。通过两个数值算例（包括晶体结构中的分子动力学模拟）对理论结果进行了验证。