(246d) Data-Driven Dynamic Latent Variable Analysis of Chaotic and Oscillatory Chemical Systems

Complex, highly nonlinear chemical and biological systems often exhibit chaotic and oscillatory behaviors. These systems include the well-studied Lorenz attractors and Lorenz-like chaotic systems, the Belousov–Zhabotinsky chemical reaction systems (Zhang et al., 1993), biological systems that exhibit Turing patterns (Esptein and Pojman, 1998), atmospheric and environmental systems, and life-like chemical materials that naturally oscillate at steady state (Isokova et al., 2019).

Traditionally these systems have been studied from the point of nonlinear differential equations. For example, low dimensional models for the Lorenz system can describe the convective motion of a fluid heated from below. Under certain conditions, a laser can be described by the same equations, although the variables have different physical meaning. However, it is generally not possible to find analytical solutions of these complex nonlinear systems. Approximation techniques and qualitative analysis have been utilized to understand various behaviors of such systems. Another approach is to resort to numerical solutions first and then try to interpret the system behaviors from simulated data trajectories. This approach, however, leads to another daunting task when the dimensions of the simulated systems are high.

In recent years, time series analysis and statistical modeling of data from systems have gained popularity in understanding the dynamics and characteristics of these systems (Kantz and Schreiber, 2004). The dataset of Lorenz-like chaos in an NH3 laser created at the PTB Braunschweig in Germany has been made popular through competitions organized by the Santa Fe Institute. Recently, many experimental datasets are produced from chemical systems that exhibit oscillatory and chaotic dynamics (Isokova et al., 2019).

Linear and nonlinear time series analysis of observational data can reveal from data about the nature of the system without a priori insight into the system that produced it. Typical time series models include vector autoregressive (VAR) models, state-space reconstruction with the method of delay (MoD) for coordinate embedding, dynamic mode decomposition based on the Koopman operator (Brunton et al., 2016), neural networks, etc. (Zhang, 1993, Bradley and Kantz, 2015). By way of lifting, the Koopman operator allows one to transform the study of nonlinear system properties into a linear dynamic problem of the observables (Kamb et al., 2020). However, several major challenges remain, which include, i) the curse of dimensionality, where the number of data samples are often not much greater or even less than the dimension of the variables; ii) the large number of observational variables are highly collinear, making the modeling and inference procedures unreliable and even ill-conditioned; and iii) complex neural network solutions are often difficult to interpret.

In this paper, we propose to apply dynamic latent variable analysis (Dong and Qin, 2018, Dong et al. 2020) to chaotic and nonlinear time series to reveal structures, patterns, and possibly invariants exhibited in these complex systems. The dynamic latent analytics are good at extracting low dimensional dynamics from high dimensional data that are both cross- and auto-correlated. These methods have demonstrated superior efficacy in capturing the most predictable factors by projecting high dimensional data to a few dimensions. Owing to the dimension reduction as well as time series modeling in the low dimensional latent space, the methods are powerful in extracting low dimensional features and patterns for visualization and interpretation.

We use three datasets from chemical and physical nonlinear systems to illustrate the power of such methods to analyze observational data series, which are, i) a 3D Lorenz attractor measured in six very noisy variables, ii) the real dataset of Lorenz-like chaos in an NH3 laser (Kantz and Schreiber, 2004) embedded in low dimensions, and iii) data from the Pd‑catalyzed oscillatory carbonylation reactions in ethanol (Isokova et al., 2019). Comparisons to other methods such as principal component analysis (PCA) and singular spectral analysis are given. An example of the reconstructed Lorenz trajectories from the six-dimensional noisy data with DiCCA is shown in the attached figure. It is clearly shown that the dynamic latent variable analysis reconstructs the signals nearly perfectly. PCA was applied to the same data and failed to recover the features buried in the noise.

References

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