Symbolic-Diffusion: Deep Learning Based Symbolic Regression with D3PM Discrete Token Diffusion

Symbolic regression refers to the task of finding a closed-form mathematical expression to fit a set of data points. Genetic programming based techniques are the most common algorithms used to tackle this problem, but recently, neural-network based approaches have gained popularity. Most of the leading neural-network based models used for symbolic regression utilize transformer-based autoregressive models to generate an equation conditioned on encoded input points. However, autoregressive generation is limited to generating tokens left-to-right, and future generated tokens are conditioned only on previously generated tokens. Motivated by the desire to generate all tokens simultaneously to produce improved closed-form equations, we propose Symbolic Diffusion, a D3PM based discrete state-space diffusion model which simultaneously generates all tokens of the equation at once using discrete token diffusion. Using the bivariate dataset developed for SymbolicGPT, we compared our diffusion-based generation approach to an autoregressive model based on SymbolicGPT, using equivalent encoder and transformer architectures. We demonstrate that our novel approach of using diffusion-based generation for symbolic regression can offer comparable and, by some metrics, improved performance over autoregressive generation in models using similar underlying architectures, opening new research opportunities in neural-network based symbolic regression.