Symbolic regression is a machine learning technique that aims to discover the underlying mathematical formula that best describes a given dataset. Instead of assuming a predefined model, it searches for both the structure of an equation and its numerical parameters, allowing patterns within the data to reveal appropriate mathematical relationships. Its primary goal is to uncover interpretable, closed-form expressions that accurately represent observed phenomena.
How Symbolic Regression Differs from Traditional Regression
Traditional regression methods, such as linear or polynomial regression, require a user to specify the form of the equation in advance. For example, one might assume a linear relationship like y = ax + b, and the algorithm then calculates the optimal values for parameters like ‘a’ and ‘b’ to fit the data. The model’s structure is fixed before the learning process begins.
Symbolic regression operates differently by not imposing prior assumptions on the model’s structure. Instead, it explores a vast space of possible mathematical expressions to find the equation that best fits the data, discovering both parameters and the operators and variables that form the equation itself. This allows it to uncover complex, non-linear relationships that might be missed by conventional methods.
Traditional regression optimizes numerical parameters for a fixed equation, while symbolic regression optimizes both the equation’s structure and its parameters. This flexibility allows it to uncover intrinsic relationships within datasets without human bias or unknown gaps in domain knowledge. While this search space is significantly larger, it often leads to more insightful and interpretable models.
The Process of Discovering Equations
Symbolic regression employs evolutionary algorithms, most commonly genetic programming, to navigate the immense space of possible equations. Drawing inspiration from biological evolution, the algorithm begins by generating an initial “population” of random mathematical expressions.
Each equation in this initial population is then evaluated for its “fitness,” which measures how well it fits the given dataset. This fitness score quantifies the accuracy of the equation in predicting the observed output values. Equations that exhibit a better fit to the data are considered “fitter” and are more likely to contribute to subsequent generations.
The fittest equations are then selected to “reproduce,” much like natural selection in biology. New equations, or “offspring,” are created through genetic operations such as crossover and mutation. Crossover involves combining parts of two parent equations to form new ones, while mutation introduces small, random changes within an equation’s structure.
This cycle of evaluation, selection, and reproduction repeats for many generations. Over time, the equations in the population evolve, becoming progressively better at describing the data. The process continues until a satisfactory level of accuracy is achieved or a predefined number of generations has passed.
Core Components and Building Blocks
The mathematical expressions discovered by symbolic regression are constructed from a predefined set of fundamental elements. These elements are categorized into two main types: terminals and functions. The user typically specifies these building blocks, which in turn defines the scope of the algorithm’s search for equations.
Terminals represent the input variables from the dataset, such as ‘x’, ‘y’, or ‘time’, along with numerical constants. Terminals form the “leaf nodes” of the expression trees that represent the equations.
Functions, also known as operators, are the mathematical operations that combine terminals and other functions to build complex expressions. Common functions include basic arithmetic operations like addition (+), subtraction (-), multiplication (), and division (/). More advanced functions, such as sine (sin), cosine (cos), exponentiation (exp), or logarithms (log), can also be included.
Real-World Applications and Discoveries
Symbolic regression has found application in various scientific and engineering fields, demonstrating its ability to uncover interpretable relationships from data. It has the potential to independently rediscover fundamental scientific laws. For example, symbolic regression has been applied to astronomical data and successfully derived Kepler’s laws of planetary motion, which describe the orbits of planets around the sun.
In materials science, this technique can identify formulas that predict a material’s properties, such as strength or conductivity, based on its composition or processing conditions. This capability allows researchers to understand how changes in inputs affect material behavior, leading to the design of new materials with desired characteristics.
Beyond scientific discovery, symbolic regression is used in areas like finance for developing novel trading models or in system identification to model the dynamics of complex systems. It can uncover governing equations for processes like fluid flow or the spread of infectious diseases, providing transparent and actionable insights that traditional “black box” machine learning models might not offer.