# Vertical Symbolic Regression via Deep Policy Gradient

arXiv: https://arxiv.org/abs/2402.00254 · HTML: https://mlsanity.com/paper/2402.00254/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Placed in Interpretable and logic-guided RL, one of 15 directions in Deep reinforcement learning methods.
- Builds on 17 of its 57 references that are in the map.
- First posted on arXiv 1 Feb 2024.

## Placed in

- Deep reinforcement learning methods / [Interpretable and logic-guided RL](https://mlsanity.com/deeprl/interpretable-and-logic-guided-rl/)

## Builds on (most cited first, 17 of 17)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Discovering Symbolic Models from Deep Learning with Inductive Biases](https://mlsanity.com/paper/2006.11287/) (2020)
- [Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients](https://mlsanity.com/paper/1912.04871/) (2019)
- [End-to-end symbolic regression with transformers](https://mlsanity.com/paper/2204.10532/) (2022)
- [Neural Symbolic Regression that Scales](https://mlsanity.com/paper/2106.06427/) (2021)
- [Discovering physical concepts with neural networks](https://mlsanity.com/paper/1807.10300/) (2018)
- [AI Poincar'e: Machine Learning Conservation Laws from Trajectories](https://mlsanity.com/paper/2011.04698/) (2020)
- [Symbolic Regression is NP-hard](https://mlsanity.com/paper/2207.01018/) (2022)
- [Symbolic Physics Learner: Discovering governing equations via Monte Carlo tree search](https://mlsanity.com/paper/2205.13134/) (2022)
- [ODEFormer: Symbolic Regression of Dynamical Systems with Transformers](https://mlsanity.com/paper/2310.05573/) (2023)
- [Deep Generative Symbolic Regression with Monte-Carlo-Tree-Search](https://mlsanity.com/paper/2302.11223/) (2023)
- [Probabilistic Grammars for Equation Discovery](https://mlsanity.com/paper/2012.00428/) (2020)
- [A computational framework for physics-informed symbolic regression with straightforward integration of domain knowledge](https://mlsanity.com/paper/2209.06257/) (2022)
- [Neural Program Synthesis with Priority Queue Training](https://mlsanity.com/paper/1801.03526/) (2018)
- [Symbolic Regression via Control Variable Genetic Programming](https://mlsanity.com/paper/2306.08057/) (2023)
- [Active Learning Improves Performance on Symbolic RegressionTasks in StackGP](https://mlsanity.com/paper/2202.04708/) (2022)
- [Vertical Symbolic Regression](https://mlsanity.com/paper/2312.11955/) (2023)

## Built on by (most cited first, 6 of 6)

- [Knowledge Integration for Physics-informed Symbolic Regression Using Pre-trained Large Language Models](https://mlsanity.com/paper/2509.03036/) (2025)
- [Complexity-Aware Deep Symbolic Regression with Robust Risk-Seeking Policy Gradients](https://mlsanity.com/paper/2406.06751/) (2024)
- [Diffusion-Based Symbolic Regression](https://mlsanity.com/paper/2505.24776/) (2025)
- [Active Symbolic Discovery of Ordinary Differential Equations via Phase Portrait Sketching](https://mlsanity.com/paper/2409.01416/) (2024)
- [EGG-SR: Embedding Symbolic Equivalence into Symbolic Regression via Equality Graph](https://mlsanity.com/paper/2511.05849/) (2025)
- [Data Enrichment for Symbolic Regression Using Diffusion Models](https://mlsanity.com/paper/2606.00988/) (2026)

## Abstract (arXiv)

Vertical Symbolic Regression (VSR) recently has been proposed to expedite the discovery of symbolic equations with many independent variables from experimental data. VSR reduces the search spaces following the vertical discovery path by building from reduced-form equations involving a subset of independent variables to full-fledged ones. Proved successful by many symbolic regressors, deep neural networks are expected to further scale up VSR. Nevertheless, directly combining VSR with deep neural networks will result in difficulty in passing gradients and other engineering issues. We propose Vertical Symbolic Regression using Deep Policy Gradient (VSR-DPG) and demonstrate that VSR-DPG can recover ground-truth equations involving multiple input variables, significantly beyond both deep reinforcement learning-based approaches and previous VSR variants. Our VSR-DPG models symbolic regression as a sequential decision-making process, in which equations are built from repeated applications of grammar rules. The integrated deep model is trained to maximize a policy gradient objective. Experimental results demonstrate that our VSR-DPG significantly outperforms popular baselines in identifying both algebraic equations and ordinary differential equations on a series of benchmarks.