Expressive Symbolic Regression for Interpretable Models of Discrete-Time Dynamical Systems

TL;DR

This work makes a modification to the model pipeline to optimize the regression, then characterize the behavior of the adjusted model in identifying several classical chaotic maps, and shows that the modified SymANNTEx model properly identifies single-state maps and achieves moderate success in approximating a dual-state attractor.

Abstract

Interpretable mathematical expressions defining discrete-time dynamical systems (iterated maps) can model many phenomena of scientific interest, enabling a deeper understanding of system behaviors. Since formulating governing expressions from first principles can be difficult, it is of particular interest to identify expressions for iterated maps given only their data streams. In this work, we consider a modified Symbolic Artificial Neural Network-Trained Expressions (SymANNTEx) architecture for this task, an architecture more expressive than others in the literature. We make a modification to the model pipeline to optimize the regression, then characterize the behavior of the adjusted model in identifying several classical chaotic maps. With the goal of parsimony, sparsity-inducing weight regularization and information theory-informed simplification are implemented. We show that our modified SymANNTEx model properly identifies single-state maps and achieves moderate success in approximating a dual-state attractor. These performances offer significant promise for data-driven scientific discovery and interpretation.

Figures (4)

• Figure 1: Logistic Map experiment visualizations of (a) state space data and best identified models, (b) trajectories of true and best identified models for 30 iterations, and (c) all RRMSEs at AIC and OLS stages. Note differing vertical axes scales in (c).
• Figure 2: Gaussian Map experiment visualizations of (a) state space data and best identified models and (b) trajectories of true and best identified models for 30 iterations.
• Figure 3: Tinkerbell Map experiment visualizations of (a) three- dimensional state spaces for true and identified $x_{t+1}$ over $x_t$ and $y_t$, (b) similar state spaces for $y_{t+1}$, and (c) true and identified trajectories for 10 iterations. Training data is omitted from the state space plots for clarity.
• Figure 4: Larger-domain Gaussian Map experiment visualizations of (a) state space data and best identified models and (b) trajectories of true and best identified models.