Amortized Equation Discovery in Hybrid Dynamical Systems

TL;DR

The paper tackles learning parsimonious, closed-form equations for hybrid dynamical systems with unknown mode switches. It introduces AMORE, an end-to-end amortized equation discovery framework that jointly infers discrete modes and learns mode-specific dynamical equations, plus AMORE-MIO for multi-object interactions via latent edge variables. The approach uses mode and count latent variables, a generative model, and sparsity regularization to jointly discover dynamics and switching behavior, demonstrated across ten datasets with superior segmentation and forecasting. The results show robust performance on both single- and multi-object systems and across hybrid and non-hybrid dynamics, indicating practical impact for interpretable modeling and forecasting in complex systems.

Abstract

Hybrid dynamical systems are prevalent in science and engineering to express complex systems with continuous and discrete states. To learn the laws of systems, all previous methods for equation discovery in hybrid systems follow a two-stage paradigm, i.e. they first group time series into small cluster fragments and then discover equations in each fragment separately through methods in non-hybrid systems. Although effective, these methods do not fully take advantage of the commonalities in the shared dynamics of multiple fragments that are driven by the same equations. Besides, the two-stage paradigm breaks the interdependence between categorizing and representing dynamics that jointly form hybrid systems. In this paper, we reformulate the problem and propose an end-to-end learning framework, i.e. Amortized Equation Discovery (AMORE), to jointly categorize modes and discover equations characterizing the dynamics of each mode by all segments of the mode. Experiments on four hybrid and six non-hybrid systems show that our method outperforms previous methods on equation discovery, segmentation, and forecasting.

Figures (6)

• Figure 1: (a) Previous methods for equation discovery in hybrid dynamical systems typically follow a two-stage paradigm, i.e. they first group time series into small cluster fragments and then apply methods proposed in non-hybrid systems, e.g. SINDy brunton2016discovering to discover equations in each fragment separately. (b) Different from all previous methods, we reformulate the problem and propose a one-stage end-to-end learning framework, Amortized Equation Discovery (a.k.a. AMORE), to jointly categorize hybrid systems into discrete modes and discover equations characterizing motion dynamics of each mode based on all segments belonging to the mode.
• Figure 2: Generative model for amortized equation discovery. $p(c_{t}|c_{t-1},z_{t-1})$ and $p(z_t|z_{t-1}, c_{t}, \mathbf{y}_{t-1})$ are count and mode transition probabilities, respectively. $p(\mathbf{y}_t|\mathbf{y}_{t-1},z_t)$ denotes the observation transition probability where equations are discovered to characterize the dynamics of each mode.
• Figure 3: (a) Generative model of AMORE-MIO. $p(\mathbf{e}_{t}^{1:N^2}\!|\mathbf{e}_{t-1}^{1:N^2}\!\!,\mathbf{z}_{t-1}^{1:N},\mathbf{c}_{t-1}^{1:N},\mathbf{y}_{t-1}^{1:N})$, $p(\mathbf{c}_{t}^{1:N}\!|\mathbf{c}_{t-1}^{1:N},\mathbf{z}_{t-1}^{1:N})$, $p(\mathbf{z}_{t}^{1:N}|\mathbf{z}_{t-1}^{1:N},\mathbf{c}_{t}^{1:N}\!\!,\mathbf{y}_{t-1}^{1:N},\mathbf{e}_{t}^{1:N^2})$, and $p(\mathbf{y}_{t}^{1:N}|\mathbf{y}_{t-1}^{1:N},\mathbf{z}_{t}^{1:N})$ denotes the edge, count, mode, and observation transition probabilities, respectively. Equations are modeled at $p(\mathbf{y}_{t}^{1:N}|\mathbf{y}_{t-1}^{1:N},\mathbf{z}_{t}^{1:N})$ which characterize object-shared and mode-specific dynamics. (b) Inference model of AMORE-MIO. Left: posterior approximate inference of edge variables $\mathbf{e}_{t}^{1:N^2}$. Right: Exact inference of discrete mode and count variables $\mathbf{z}_t^{1:N}$ and $\mathbf{c}_t^{1:N}$ based on observations $\mathbf{y}_t^{1:N}$ and the approximate edge variables $\mathbf{e}_{t}^{1:N^2}$. Orange arrows denote the approximate inference flow.
• Figure 4: Qualitative time series segmentation results of AMORE compared to Hybrid-SINDy brunton2016discovering on the Mass-spring Hopper dataset. For Hybrid-SINDy, we aggregate the discovered equations with the same number of coefficients as one mode. We can see that with joint learning of modes and equations, AMORE can categorize the exact number of modes and achieve superior segmentation results with fewer switching errors.
• Figure 5: Discovered equations on the Salsa-dancing dataset. Locations $(x,y)$ of the hip joints are used as observations.