# Learning Symbolic Expressions: Mixed-Integer Formulations, Cuts, and Heuristics

arXiv: https://arxiv.org/abs/2102.08351 · HTML: https://mlsanity.com/paper/2102.08351/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 23 references that are in the map.
- First posted on arXiv 16 Feb 2021.

## Builds on (most cited first, 5 of 5)

- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [Discovering Symbolic Models from Deep Learning with Inductive Biases](https://mlsanity.com/paper/2006.11287/) (2020)
- [AI Feynman 2.0: Pareto-optimal symbolic regression exploiting graph modularity](https://mlsanity.com/paper/2006.10782/) (2020)
- [Globally Optimal Symbolic Regression](https://mlsanity.com/paper/1710.10720/) (2017)
- [Symbolic Regression using Mixed-Integer Nonlinear Optimization](https://mlsanity.com/paper/2006.06813/) (2020)

## Built on by (most cited first, 3 of 3)

- [Taming Binarized Neural Networks and Mixed-Integer Programs](https://mlsanity.com/paper/2310.04469/) (2023)
- [Symbolic Density Estimation for Discrete Distributions](https://mlsanity.com/paper/2605.21813/) (2026)
- [Decision-Making under Combinatorial Risk](https://mlsanity.com/paper/2606.10092/) (2026)

## Abstract (arXiv)

In this paper we consider the problem of learning a regression function without assuming its functional form. This problem is referred to as symbolic regression. An expression tree is typically used to represent a solution function, which is determined by assigning operators and operands to the nodes. The symbolic regression problem can be formulated as a nonconvex mixed-integer nonlinear program (MINLP), where binary variables are used to assign operators and nonlinear expressions are used to propagate data values through nonlinear operators such as square, square root, and exponential. We extend this formulation by adding new cuts that improve the solution of this challenging MINLP. We also propose a heuristic that iteratively builds an expression tree by solving a restricted MINLP. We perform computational experiments and compare our approach with a mixed-integer program-based method and a neural-network-based method from the literature.