Data-driven identification of nonlinear dynamical systems with LSTM autoencoders and Normalizing Flows

TL;DR

This work tackles nonlinear system identification by coupling a LSTM-Autoencoder to extract temporal features with Normalizing Flows to map these features to system parameters, extending to fluid-flow problems through CNN-based spatial feature extraction. The approach is demonstrated on Duffing and Lorenz dynamical systems as well as 2-D flows past a cylinder and a lid-driven cavity, achieving accurate parameter inference in many single-parameter cases and reasonable performance in multi-parameter and higher-$Re$ regimes. Key contributions include a dual-path deep learning framework that handles both time-series and spatial-temporal data, a composite loss incorporating negative log-likelihood and reconstruction terms, and detailed architectural choices that enable end-to-end training with shared weights. The results suggest practical potential for nonlinear system identification in control and prediction tasks, though performance declines with increasing parameter coupling and flow complexity, motivating future validation on experimental data and broader regime coverage.

Abstract

While linear systems have been useful in solving problems across different fields, the need for improved performance and efficiency has prompted them to operate in nonlinear modes. As a result, nonlinear models are now essential for the design and control of these systems. However, identifying a nonlinear system is more complicated than identifying a linear one. Therefore, modeling and identifying nonlinear systems are crucial for the design, manufacturing, and testing of complex systems. This study presents using advanced nonlinear methods based on deep learning for system identification. Two deep neural network models, LSTM autoencoder and Normalizing Flows, are explored for their potential to extract temporal features from time series data and relate them to system parameters, respectively. The presented framework offers a nonlinear approach to system identification, enabling it to handle complex systems. As case studies, we consider Duffing and Lorenz systems, as well as fluid flows such as flows over a cylinder and the 2-D lid-driven cavity problem. The results indicate that the presented framework is capable of capturing features and effectively relating them to system parameters, satisfying the identification requirements of nonlinear systems.

Figures (12)

• Figure 1: The concept of Normalizing Flows can be defined as the process of transforming a simple probability distribution function ($\textit{P}_0$) to the original, more complex distribution function ($P_k$) by utilizing bijectors.
• Figure 2: In the Normalizing Flows concept, the dependency between the two original coordinates of a 2-DOF Duffing system can be reduced by passing through Normalizing Flows layers. Here, $Z_0$ represents the decomposed modal coordinates, and $X$ refers to the original coordinates.
• Figure 3: Case studies: a: 2 DOF Duffing system b: Lorenz system c: Stream-wise velocity over a cylinder d: Transverse velocity over a cylinder e: Vorticity of a 2-D driven-lid cavityproblem
• Figure 4: The architecture of the framework presented for non-fluid case studies involves two main components. Firstly, the temporal features ($\varphi$) of the input data (i.e., trajectories: $X$) are extracted using LSTM-Encoder ($\vartheta$). Next, Normalizing Flows (NF) is utilized to establish a relationship between these temporal features and the system parameters ($Y$). In order to ensure that the temporal features of the input data represent the most critical information, a LSTM-Decoder ($\vartheta^{-1}$) is utilized to reconstruct the data.
• Figure 5: The architecture of the framework presented for fluid case studies involves three main components. Firstly, the spatial features ($\mathcal{M}^\prime$) of the flow field ($\mathcal{M}$) is extracted using a CNN-Encoder ($\tau$) .Secondly, the temporal features ($\varphi$) of the spatial features data are extracted using LSTM-Encoder. Next, Normalizing Flows (NF) is utilized to establish a relationship between these temporal features and the system parameters ($Y$). In order to ensure that the spatial and temporal features of the input data represent the most critical information, a CNN-Decoder ($\tau^{-1}$) and a LSTM-Decoder ($\vartheta^{-1}$) are used respectively to reconstruct the data.