# Constructing Parsimonious Analytic Models for Dynamic Systems via Symbolic Regression

## TL;DR

This paper introduces a symbolic regression approach to construct simple, accurate analytic models for dynamic systems, improving over neural networks and linear regression, and enabling effective reinforcement learning control with limited data.

## Contribution

The paper presents a novel application of symbolic regression with genetic programming algorithms to build parsimonious models for complex dynamic systems, including real-world experiments.

## Key findings

- SR outperforms neural networks and linear regression in modeling accuracy.
- The method successfully models high-dimensional systems like robots.
- A real pendulum experiment demonstrates effective RL control with minimal data.

## Abstract

Developing mathematical models of dynamic systems is central to many disciplines of engineering and science. Models facilitate simulations, analysis of the system's behavior, decision making and design of automatic control algorithms. Even inherently model-free control techniques such as reinforcement learning (RL) have been shown to benefit from the use of models, typically learned online. Any model construction method must address the tradeoff between the accuracy of the model and its complexity, which is difficult to strike. In this paper, we propose to employ symbolic regression (SR) to construct parsimonious process models described by analytic equations. We have equipped our method with two different state-of-the-art SR algorithms which automatically search for equations that fit the measured data: Single Node Genetic Programming (SNGP) and Multi-Gene Genetic Programming (MGGP). In addition to the standard problem formulation in the state-space domain, we show how the method can also be applied to input-output models of the NARX (nonlinear autoregressive with exogenous input) type. We present the approach on three simulated examples with up to 14-dimensional state space: an inverted pendulum, a mobile robot, and a bipedal walking robot. A comparison with deep neural networks and local linear regression shows that SR in most cases outperforms these commonly used alternative methods. We demonstrate on a real pendulum system that the analytic model found enables a RL controller to successfully perform the swing-up task, based on a model constructed from only 100 data samples.