# Grammar-based Ordinary Differential Equation Discovery

arXiv: https://arxiv.org/abs/2504.02630 · HTML: https://mlsanity.com/paper/2504.02630/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 27 of its 64 references that are in the map.
- First posted on arXiv 3 Apr 2025.

## Builds on (most cited first, 27 of 27)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Learning Phrase Representations using RNN Encoder-Decoder for Statistical Machine Translation](https://mlsanity.com/paper/1406.1078/) (2014)
- [Neural Machine Translation of Rare Words with Subword Units](https://mlsanity.com/paper/1508.07909/) (2015)
- [Fast and Accurate Deep Network Learning by Exponential Linear Units (ELUs)](https://mlsanity.com/paper/1511.07289/) (2015)
- [Sigmoid-Weighted Linear Units for Neural Network Function Approximation in Reinforcement Learning](https://mlsanity.com/paper/1702.03118/) (2017)
- [The CMA Evolution Strategy: A Tutorial](https://mlsanity.com/paper/1604.00772/) (2016)
- [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)
- [Grammar Variational Autoencoder](https://mlsanity.com/paper/1703.01925/) (2017)
- [Deep Learning for Symbolic Mathematics](https://mlsanity.com/paper/1912.01412/) (2019)
- [End-to-end symbolic regression with transformers](https://mlsanity.com/paper/2204.10532/) (2022)
- [SINDy-PI: A Robust Algorithm for Parallel Implicit Sparse Identification of Nonlinear Dynamics](https://mlsanity.com/paper/2004.02322/) (2020)
- [Symbolic Regression via Neural-Guided Genetic Programming Population Seeding](https://mlsanity.com/paper/2111.00053/) (2021)
- [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)
- [SNIP: Bridging Mathematical Symbolic and Numeric Realms with Unified Pre-training](https://mlsanity.com/paper/2310.02227/) (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)
- [Nonlinear state-space identification using deep encoder networks](https://mlsanity.com/paper/2012.07697/) (2020)
- [Predicting Ordinary Differential Equations with Transformers](https://mlsanity.com/paper/2307.12617/) (2023)
- [The Foundations of Tokenization: Statistical and Computational Concerns](https://mlsanity.com/paper/2407.11606/) (2024)
- [Physics-guided Deep Markov Models for Learning Nonlinear Dynamical Systems with Uncertainty](https://mlsanity.com/paper/2110.08607/) (2021)
- [Symbolic Recovery of Differential Equations: The Identifiability Problem](https://mlsanity.com/paper/2210.08342/) (2022)
- [Baseline Results for Selected Nonlinear System Identification Benchmarks](https://mlsanity.com/paper/2405.10779/) (2024)
- [Recursive Tree Grammar Autoencoders](https://mlsanity.com/paper/2012.02097/) (2020)
- [A Survey on Solving and Discovering Differential Equations Using Deep Neural Networks](https://mlsanity.com/paper/2304.13807/) (2023)
- [Neuro-Symbolic AI for Analytical Solutions of Differential Equations](https://mlsanity.com/paper/2502.01476/) (2025)

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

- [Structured Kolmogorov-Arnold Neural ODEs for Interpretable Learning and Symbolic Discovery of Nonlinear Dynamics](https://mlsanity.com/paper/2506.18339/) (2025)
- [Learning to Disentangle Latent Reasoning Rules with Language VAEs: A Systematic Study](https://mlsanity.com/paper/2506.19418/) (2025)
- [Data-driven Discovery of Digital Twins in Biomedical Research](https://mlsanity.com/paper/2508.21484/) (2025)
- [Neuro-Symbolic ODE Discovery with Latent Grammar Flow](https://mlsanity.com/paper/2604.16232/) (2026)
- [Learning the Geometry of Admissible Hypotheses through Inductive Bias in Training Distributions](https://mlsanity.com/paper/2608.31028/) (2026)
- [Exhaustive Canonical Skeletons Generation: A Formal Dataset for Decomposable Symbolic Regression](https://mlsanity.com/paper/2610.20108/) (2026)

## Abstract (arXiv)

The understanding and modeling of complex physical phenomena through dynamical systems has historically driven scientific progress, as it provides the tools for predicting the behavior of different systems under diverse conditions through time. The discovery of dynamical systems has been indispensable in engineering, as it allows for the analysis and prediction of complex behaviors for computational modeling, diagnostics, prognostics, and control of engineered systems. Joining recent efforts that harness the power of symbolic regression in this domain, we propose a novel framework for the end-to-end discovery of ordinary differential equations (ODEs), termed Grammar-based ODE Discovery Engine (GODE). The proposed methodology combines formal grammars with dimensionality reduction and stochastic search for efficiently navigating high-dimensional combinatorial spaces. Grammars allow us to seed domain knowledge and structure for both constraining, as well as, exploring the space of candidate expressions. GODE proves to be more sample- and parameter-efficient than state-of-the-art transformer-based models and to discover more accurate and parsimonious ODE expressions than both genetic programming- and other grammar-based methods for more complex inference tasks, such as the discovery of structural dynamics. Thus, we introduce a tool that could play a catalytic role in dynamics discovery tasks, including modeling, system identification, and monitoring tasks.