Highly accurate simulations of complex phenomena governed by partial differential equations (PDEs) typically require intrusive methods and entail expensive computational costs, which might become prohibitive when approximating steady-state solutions of PDEs for multiple combinations of control parameters and initial conditions. Therefore, constructing efficient reduced order models (ROMs) that enable accurate but fast predictions, while retaining the dynamical characteristics of the physical phenomenon as parameters vary, is of paramount importance. In this work, a data-driven, non-intrusive framework which combines ROM construction with reduced dynamics identification, is presented. Starting from a limited amount of full order solutions, the proposed approach leverages autoencoder neural networks with parametric sparse identification of nonlinear dynamics (SINDy) to construct a low-dimensional dynamical model. This model can be queried to efficiently compute full-time solutions at new parameter instances, as well as directly fed to continuation algorithms. These aim at tracking the evolution of periodic steady-state responses as functions of system parameters, avoiding the computation of the transient phase, and allowing to detect instabilities and bifurcations. Featuring an explicit and parametrized modeling of the reduced dynamics, the proposed data-driven framework presents remarkable capabilities to generalize with respect to both time and parameters. Applications to structural mechanics and fluid dynamics problems illustrate the effectiveness and accuracy of the proposed method.

翻译：偏微分方程控制的复杂现象的高精度模拟通常需要侵入式方法，并带来高昂的计算成本，当需要针对多组控制参数和初始条件逼近偏微分方程的稳态解时，这种成本可能变得难以承受。因此，构建能够实现精确且快速预测、同时在参数变化时保留物理现象动力学特征的高效降阶模型至关重要。本文提出了一种数据驱动的非侵入式框架，该框架将降阶模型构建与降阶动力学识别相结合。该框架从有限的全阶解出发，利用自编码器神经网络与带参数稀疏非线性动力学识别方法，构建低维动力学模型。该模型可用于高效计算新参数实例下的全时域解，也可直接输入延续算法。这些算法旨在追踪周期稳态响应随系统参数变化的演化，避免瞬态阶段的计算，并能够检测不稳定性和分岔。通过显式且参数化的降阶动力学建模，所提出的数据驱动框架在时间和参数泛化方面展现出卓越能力。在结构力学和流体动力学问题中的应用验证了所提方法的有效性和准确性。