# Neuro-Symbolic ODE Discovery with Latent Grammar Flow

arXiv: https://arxiv.org/abs/2604.16232 · HTML: https://mlsanity.com/paper/2604.16232/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 25 of its 43 references that are in the map.
- First posted on arXiv 17 Apr 2026.

## Builds on (most cited first, 25 of 25)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Neural Discrete Representation Learning](https://mlsanity.com/paper/1711.00937/) (2017)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)](https://mlsanity.com/paper/1511.07289/) (2015)
- [AlphaEvolve: A coding agent for scientific and algorithmic discovery](https://mlsanity.com/paper/2506.13131/) (2025)
- [Finite Scalar Quantization: VQ-VAE Made Simple](https://mlsanity.com/paper/2309.15505/) (2023)
- [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)
- [Generative Flows on Discrete State-Spaces: Enabling Multimodal Flows with Applications to Protein Co-Design](https://mlsanity.com/paper/2402.04997/) (2024)
- [Grammar Variational Autoencoder](https://mlsanity.com/paper/1703.01925/) (2017)
- [Deep Learning for Symbolic Mathematics](https://mlsanity.com/paper/1912.01412/) (2019)
- [Learning Equations for Extrapolation and Control](https://mlsanity.com/paper/1806.07259/) (2018)
- [LLM-SR: Scientific Equation Discovery via Programming with Large Language Models](https://mlsanity.com/paper/2404.18400/) (2024)
- [Unlocking Guidance for Discrete State-Space Diffusion and Flow Models](https://mlsanity.com/paper/2406.01572/) (2024)
- [SINDy-PI: A Robust Algorithm for Parallel Implicit Sparse Identification of Nonlinear Dynamics](https://mlsanity.com/paper/2004.02322/) (2020)
- [ODEFormer: Symbolic Regression of Dynamical Systems with Transformers](https://mlsanity.com/paper/2310.05573/) (2023)
- [Deep Symbolic Regression for Recurrent Sequences](https://mlsanity.com/paper/2201.04600/) (2022)
- [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)
- [Predicting Ordinary Differential Equations with Transformers](https://mlsanity.com/paper/2307.12617/) (2023)
- [GENSR: Symbolic Regression Based in Equation Generative Space](https://mlsanity.com/paper/2602.20557/) (2026)
- [D-CIPHER: Discovery of Closed-form Partial Differential Equations](https://mlsanity.com/paper/2206.10586/) (2022)
- [Grammar-based Ordinary Differential Equation Discovery](https://mlsanity.com/paper/2504.02630/) (2025)
- [A Reinforcement Learning Approach to Domain-Knowledge Inclusion Using Grammar Guided Symbolic Regression](https://mlsanity.com/paper/2202.04367/) (2022)
- [A Neural Operator-Based Approach to Symbolic Discovery of PDEs](https://mlsanity.com/paper/2501.08086/) (2025)

## Built on by (most cited first, 2 of 2)

- [Verifier-Guided Model Discovery for Physical Dynamical Systems with Pretrained Symbolic Transformers](https://mlsanity.com/paper/2608.02662/) (2026)
- [Learning the Geometry of Admissible Hypotheses through Inductive Bias in Training Distributions](https://mlsanity.com/paper/2608.31028/) (2026)

## Abstract (arXiv)

Understanding natural and engineered systems often relies on symbolic formulations, such as differential equations, which provide interpretability and transferability beyond black-box models. We introduce Latent Grammar Flow (LGF), a neuro-symbolic generative framework for discovering ordinary differential equations from data. LGF embeds equations as grammar-based representations into a discrete latent space and forces semantically similar equations to be positioned closer together with a behavioural loss. Then, a discrete flow model guides the sampling process to recursively generate candidate equations that best fit the observed data. Domain knowledge and constraints, such as stability, can be either embedded into the rules or used as conditional predictors.