Symbolic regression

Symbolic regression searches for a mathematical expression — an actual equation involving variables, operators, and constants — that best describes a dataset. Unlike neural networks, which produce black-box numerical functions, symbolic regression outputs a human-readable formula that can be examined, critiqued, and integrated into theoretical frameworks.

A classic application: given measurements of planetary positions over time, symbolic regression can recover Kepler’s third law from the data without being told what form the relationship should take.

Why it matters for physics and astronomy:

Physics is fundamentally about finding governing equations. Symbolic regression aligns with that goal in a way that standard ML doesn’t: if the correct underlying law is expressible as a compact equation (and physical laws often are), symbolic regression can find it. Researchers have used it to rediscover known relationships and to discover new empirical relationships in areas where the underlying theory is incomplete.

Key tools:

• PySR (Miles Cranmer, Princeton): the most widely used open-source implementation; runs a multi-population evolutionary search and scales to large datasets
• Eureqa (earlier commercial tool, now discontinued): introduced many researchers to the approach
• AI Feynman: a deep learning approach to symbolic regression that incorporates dimensional analysis

Limitations:

• Scales poorly with the number of variables — search space grows exponentially
• Can find equations that fit training data but don’t generalize (overfitting to spurious correlations in noisy data)
• The “correct” symbolic form isn’t unique — mathematically equivalent expressions can look very different
• Works best when the true relationship is relatively simple; highly complex underlying physics may not yield a clean symbolic form

Related guide: Physics & Astronomy