# Active Symbolic Discovery of Ordinary Differential Equations via Phase Portrait Sketching

arXiv: https://arxiv.org/abs/2409.01416 · HTML: https://mlsanity.com/paper/2409.01416/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 65 references that are in the map.
- First posted on arXiv 2 Sep 2024.

## Builds on (most cited first, 28 of 28)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [Deep Batch Active Learning by Diverse, Uncertain Gradient Lower Bounds](https://mlsanity.com/paper/1906.03671/) (2019)
- [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)
- [Ensemble-SINDy: Robust sparse model discovery in the low-data, high-noise limit, with active learning and control](https://mlsanity.com/paper/2111.10992/) (2021)
- [Discovering physical concepts with neural networks](https://mlsanity.com/paper/1807.10300/) (2018)
- [Active Learning for Nonlinear System Identification with Guarantees](https://mlsanity.com/paper/2006.10277/) (2020)
- [Transformer-based Planning for Symbolic Regression](https://mlsanity.com/paper/2303.06833/) (2023)
- [AI Poincar'e: Machine Learning Conservation Laws from Trajectories](https://mlsanity.com/paper/2011.04698/) (2020)
- [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)
- [Learning unknown ODE models with Gaussian processes](https://mlsanity.com/paper/1803.04303/) (2018)
- [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)
- [Active Learning for Identification of Linear Dynamical Systems](https://mlsanity.com/paper/2002.00495/) (2020)
- [A computational framework for physics-informed symbolic regression with straightforward integration of domain knowledge](https://mlsanity.com/paper/2209.06257/) (2022)
- [Actively Learning Gaussian Process Dynamics](https://mlsanity.com/paper/1911.09946/) (2019)
- [Predicting Ordinary Differential Equations with Transformers](https://mlsanity.com/paper/2307.12617/) (2023)
- [Neural Program Synthesis with Priority Queue Training](https://mlsanity.com/paper/1801.03526/) (2018)
- [Taylor Genetic Programming for Symbolic Regression](https://mlsanity.com/paper/2205.09751/) (2022)
- [Active Learning Improves Performance on Symbolic RegressionTasks in StackGP](https://mlsanity.com/paper/2202.04708/) (2022)
- [Vertical Symbolic Regression via Deep Policy Gradient](https://mlsanity.com/paper/2402.00254/) (2024)
- [Active Learning in Symbolic Regression with Physical Constraints](https://mlsanity.com/paper/2305.10379/) (2023)
- [Active Learning in Genetic Programming: Guiding Efficient Data Collection for Symbolic Regression](https://mlsanity.com/paper/2308.00672/) (2023)
- [Online Symbolic Regression with Informative Query](https://mlsanity.com/paper/2302.10539/) (2023)

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

- [LLM-AutoSciLab: Closed-Loop Scientific Discovery via Active Experimentation with LLMs](https://mlsanity.com/paper/2605.24043/) (2026)
- [LLM-ACES: Closed-Loop Discovery of Dynamical Systems with LLM-Guided Adaptive Search](https://mlsanity.com/paper/2606.25039/) (2026)
- [ALDER: Discovering the Laws of a World by Acting in It](https://mlsanity.com/paper/2609.33728/) (2026)

## Abstract (arXiv)

The symbolic discovery of Ordinary Differential Equations (ODEs) from trajectory data plays a pivotal role in AI-driven scientific discovery. Existing symbolic methods predominantly rely on fixed, pre-collected training datasets, which often result in suboptimal performance, as demonstrated in our case study in Figure 1. Drawing inspiration from active learning, we investigate strategies to query informative trajectory data that can enhance the evaluation of predicted ODEs. However, the butterfly effect in dynamical systems reveals that small variations in initial conditions can lead to drastically different trajectories, necessitating the storage of vast quantities of trajectory data using conventional active learning. To address this, we introduce Active Symbolic Discovery of Ordinary Differential Equations via Phase Portrait Sketching (APPS). Instead of directly selecting individual initial conditions, our APPS first identifies an informative region within the phase space and then samples a batch of initial conditions from this region. Compared to traditional active learning methods, APPS mitigates the gap of maintaining a large amount of data. Extensive experiments demonstrate that APPS consistently discovers more accurate ODE expressions than baseline methods using passively collected datasets.