Online learning in bifurcating dynamic systems via SINDy and Kalman filtering

Abstract

We propose the use of the Extended Kalman Filter (EKF) for online data assimilation and update of a dynamic model, preliminary identified through the Sparse Identification of Nonlinear Dynamics (SINDy). This data-driven technique may avoid biases due to incorrect modelling assumptions and exploits SINDy to approximate the system dynamics leveraging a predefined library of functions, where active terms are selected and weighted by a sparse set of coefficients. This results in a physically-sound and interpretable dynamic model allowing to reduce epistemic uncertainty often affecting machine learning approaches. Treating the SINDy model coefficients as random variables, we propose to update them while acquiring (possibly noisy) system measurements, thus enabling the online identification of time-varying systems. These changes can stem from, e.g., varying operational conditions or unforeseen events. The EKF performs model adaptation through joint state-parameters estimation, with the Jacobian matrices required to computed the model sensitivity inexpensively evaluated from the SINDy model formulation. The effectiveness of this approach is demonstrated through three case studies: (i) a Lokta-Volterra model in which all parameters simultaneously evolve during the observation period; (ii) a Selkov model where the system undergoes a bifurcation not seen during the SINDy training; (iii) a MEMS arch exhibiting a 1:2 internal resonance. The ability of EKF of recovering inactivated functional terms from the SINDy library, or discarding unnecessary contribution, is also highlighted. Based on the presented applications, this method shows strong promise for handling time-varying nonlinear dynamic systems possibly experiencing bifurcating behaviours.

Figures (8)

• Figure 1: Online model adaptation combining SINDy and EKF. In the offline stage, a SINDy model is trained. In the online stage, this model is used in the predictor phase of the EKF to advance the system state over time. The EKF corrector phase then updates both the states and parameters of the model by assimilating measurements acquired on the physical system. The procedure can potentially recover inactivated functional terms from the SINDy library or, conversely, suppress unnecessary contributions.
• Figure 2: Lokta-Volterra model. In the top figure (a), the estimated states are plotted (red dotted line) against the assimilated noisy signals (black solid lines). Additionally, noise-free signals are reported in pink solid lines. In the bottom figure (b), the time-evolution of the estimated SINDy-model parameter are plotted (red dotted line) against the target time evolution (black solid line). The red shaded area represents 95% confidence interval of the estimates, determined using the posterior covariance.
• Figure 3: Selkov model. In the top figure (a), the estimated states are plotted (red dotted line) against the assimilated noisy signals (gray solid lines) using a 3D state-space-time representation. Additionally, noise-free signals are reported in black solid lines. In the bottom plot (b), the projection of this 3D representation onto the 2D planes is reported. Colouring depending on the estimate $\hat{\rho}$ of the parameter ruling the two types of dynamics of the system (stable focus and stable limit cycles). The blue colour (further marked with a star in the $(t,x_1)$ and $(t,x_2)$ planes) indicates when the system undergoes a Hopf bifurcation.
• Figure 4: Selkov model. The time-evolution of the estimated SINDy-model parameter are plotted (red dotted line) against the target time evolution (black solid line). The top figure reports the evolution of the parameters entering $\dot{x}_1=f_1(x_1,x_2)$; the bottom figure the evolution of the parameters entering $\dot{x}_2=f_2(x_1,x_2)$. The red shaded area represents $95\%$ confidence interval of the estimates, determined using the posterior covariance.
• Figure 5: MEMS arch geometry. Dimensions in $\mu \text{m}$.