# Population Dynamics in Genetic Programming for Dynamic Symbolic Regression
## 1 Introduction
In industrial use cases and production systems, the application of several models is sometimes limited due to their black-box nature. The aforementioned deep NNs are often undesired because experts do not want to rely on predictions where they do not understand the underlying prediction processes, and thus the validation and interoperability of a prediction model is difficult. On the other hand, white-box models, in the form of symbolic models, are easily readable by a human and can therefore be validated by an expert, which greatly improves the acceptance of a machine learning model. Having a symbolic model also allows for easy implementation in any system or programming language because it is, in essence, only a mathematical expression. Also, symbolic models allow for many mathematical transformations such as simplifications, or even the definition of a derivative, which can also be very useful in certain scenarios. Symbolic regression (SR) via genetic programming (GP) emerged as a widespread approach for generating symbolic models. The underlying algorithm, a genetic algorithm (GA), evolves a population of symbolic models represented as expression trees [4]. While various forms of SR exist, this paper predominantly focuses on GP due to the suitability of population-based evolutionary algorithms in handling dynamic changes in datasets, aligning with their inspiration drawn from natural evolution and the adaptation of individuals to changing environments [6]. This paper centers its attention on two critical aspects: first, the consideration of a dynamic environment where training data evolve over time, and second, the exploration of symbolic models through GP as a promising means to address the evolving underlying optimization problem. In this context, the following research questions are addressed in this paper:  
* How can changing training data be modeled within the context of a dynamic optimization problem? * Can a dynamic symbolic regression problem be effectively solved using genetic programming? * Which variants of genetic programming prove effective for dynamic symbolic regression problems? * How do the population dynamics of GP evolve in the context of dynamic symbolic regression problems? The term \"dynamic\" in the context of GP holds various meanings in different scientific papers. Noteworthy examples include the work of [PERSON] et al. [7] on the dynamic Santa Fe Ant Trail problem and that of [PERSON] et al. [8], who applied GP in a dynamic real-world pricing environment. In these scenarios, the environment in which a GP solution operates is dynamic, but the solution itself remains unchanged. The aim is to find a single generalizing solution that performs well across various states of a dynamic system. This differs from the dynamicity explored in this paper, where different solutions emerge over time, tailored to short periods and becoming obsolete with system changes. Another aspect of dynamicity involves identifying dynamic systems as demonstrated by [PERSON] et al. [9]. Here, GP is used to derive differential equations that best capture system measurements. Yet another dynamic facet includes GP's self-adaptation, such as dynamically adjusting the population size during a run. However, these applications, while noteworthy, do not form the primary focus of this paper. The exploration of GP for dynamic symbolic regression is still in its early stages, with limited attention. For instance, [PERSON] et al. [7] provides only a brief exploration of symbolic regression, while [PERSON] et al. [10] concentrates on accelerating evolution in dynamic environments. This paper seeks to extend the scope of dynamic symbolic regression with GP, both in terms of the benchmark instances employed and the depth of analysis conducted, providing valuable insights into the inner workings of GP in dynamic contexts. This paper is structured as follows.