InceptionSR: Recursive Symbolic Regression for Equation Synthesis

InceptionSR is a symbolic-regression method that freezes promising partial equations and reuses them as features in later search rounds. It is tested on planetary-system instability and a triangle-area formula.

Presented at the AAAI 2025 workshop on AI to Accelerate Science and Engineering by Edward Gu, Simon Alford, Omar Costilla-Reyes, Miles Cranmer, and Kevin Ellis, with authors from Cornell University, MIT, and the University of Cambridge, InceptionSR is a variant of symbolic regression, the task of finding mathematical equations that fit a dataset. Its base algorithm is the open-source PySR system, which evolves a population of candidate equations and keeps the best trade-offs between prediction error and equation size. InceptionSR is motivated by gradient boosting, in which later models correct earlier ones: it runs the base algorithm, freezes promising partial equations, and seeds the next round of search with those frozen expressions as new features, so later candidates can build on components found earlier. The authors also describe it as a simple form of library learning, a technique from program synthesis in which useful building blocks are discovered and reused.

The main test is a real scientific problem: predicting how long a system of several planets stays stable, an n-body problem that has resisted analytical treatment and has previously been addressed with neural networks that offer little scientific insight. Compared at equal run time and across repeated runs, the authors report that every InceptionSR variant modestly but consistently reduced training-set error relative to PySR, and by more than simply letting PySR run for longer. They describe larger equations on the frontier as building on a simpler relationship found first, and suggest the method may yield more interpretable equations.

A second test used a synthetic dataset of triangles to see whether the method could recover Heron’s formula for the area of a triangle. In this test, adding the library-learning tool Stitch between rounds identified the key subexpression, the semiperimeter, and let later rounds reuse it, which PySR alone, even recursively, did not find.

InceptionSR was developed during Dr. Costilla Reyes’s work as a research scientist at MIT CSAIL on the Understanding the World Through Code expedition, alongside LaSR, a symbolic-regression method that uses a learned concept library, and it shares the goal of producing closed-form equations that scientists can read and test. The authors plan to extend the method to standard benchmarks and other scientific domains.