Reduced order modeling of parametrized systems through autoencoders and SINDy approach: continuation of periodic solutions

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. (c) 2023 Elsevier B.V. All rights reserved.

Automated summary

To accurately simulate complex phenomena governed by partial differential equations (PDEs) with reduced computational costs, a data-driven, non-intrusive framework is proposed in this work. This framework combines reduced order model (ROM) construction with reduced dynamics identification using autoencoder neural networks and parametric sparse identification of nonlinear dynamics (SINDy). It allows for efficient computation of full-time solutions at new parameter instances and tracking the evolution of periodic steady-state responses. The proposed method shows remarkable capabilities in generalizing with respect to both time and parameters, as demonstrated in applications to structural mechanics and fluid dynamics problems.