Symbolic Regression: Evolving Physical Laws from Data

Physical Law to Discover

True law: —

Genetic Programming

Population size 30

Mutation rate 25%

Data noise 8%

Best Individual

Generation

Best RMSE

—

Tree size

—

ŷ = —

How it works

Symbolic regression searches the space of mathematical expressions themselves — not just their coefficients — for a formula that fits a dataset. This simulator runs it as genetic programming (GP): each candidate is a small expression tree built from x, constants, and the operators + − × ÷ √ sin (·)².

Fitness(tree) = RMSE( tree(xᵢ), yᵢ )  over all data points i
Selection:  tournament of 3, elitism keeps the top 2 unchanged
Crossover:  swap a random subtree between two parents
Mutation:   replace a random subtree, or nudge a constant

Each generation's best-fit curve is drawn at the front of the 3D tunnel; every prior generation's curve is pushed one step further back and dimmed, so the whole run reads as a corridor of curves converging toward the data as evolution refines the population — exactly how tools like Eureqa or PySR discover closed-form physical laws (Kepler's third law itself, T² ∝ r³, was found this way from planetary data centuries before calculus existed to derive it analytically).

Population size — more candidates explored per generation, at the cost of more evaluations.
Mutation rate — probability a child is produced by mutation instead of crossover; too low stalls in a local optimum, too high turns search into random guessing.
Data noise — measurement noise added to the synthetic dataset; realistic experimental data is never perfectly clean.

Measured data points Current best fit Earlier generations

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