📄 AI Feynman: a Physics-Inspired Method for Symbolic Regression

Summary

Symbolic regression means recovering an exact closed-form formula from a table of input–output values. It is likely NP-hard in general because the space of candidate expressions grows exponentially. Udrescu and Tegmark argue that formulas from physics are rarely generic. They usually have known units, low-order polynomial parts, compositional structure, smoothness, symmetries and separability. Their algorithm, AI Feynman, exploits these properties through a set of modules applied recursively:

• dimensional analysis
• polynomial fitting
• brute-force search over expressions
• a neural network fitted to the data, which serves as a probe for translational symmetry and additive or multiplicative separability

When such structure is found, the problem is split into smaller sub-problems with fewer variables, and each is solved by a fresh run of the same procedure. On 100 equations taken from the Feynman Lectures on Physics, the method recovers every one, while the strongest earlier tool (Eureqa) solves 71. On a harder set of physics formulas, the success rate rises from 15% to 90%. The work shows that combining neural fitting with physics-style simplification can make automated equation discovery practical.

Key points

• Benchmark results: all 100 Feynman Lectures equations were solved, against 71 for Eureqa, a commercial genetic-algorithm tool. On a more difficult physics test set, the success rate went from 15% to 90%.
• Dimensional analysis: each variable's units are encoded as a vector of 5 integer exponents (meter, second, kilogram, kelvin, volt). Solving a linear system then yields dimensionless variables, which reduces how many inputs remain.
• Polynomial and brute-force modules:
  • Polynomials of degree 0 to 4 are fitted by linear least squares.
  • The brute-force search enumerates syntactically valid expressions in reverse Polish notation, from shortest to longest.
• Worked example: for Newton's gravitation law with 9 inputs, the method reduces the problem to 6 dimensionless variables. It then detects two translational symmetries and a multiplicative factorization, and finishes with polynomial fits.
• Code: the implementation is openly available at github.com/SJ001/AI-Feynman.