# Learning Equations for Extrapolation and Control

arXiv: https://arxiv.org/abs/1806.07259 · HTML: https://mlsanity.com/paper/1806.07259/

## Key facts

- Cited by 156 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Jun 2018.

## Built on by (most cited first, 30 of 156)

- [End-to-end symbolic regression with transformers](https://mlsanity.com/paper/2204.10532/) (2022)
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- [Rediscovering orbital mechanics with machine learning](https://mlsanity.com/paper/2202.02306/) (2022)
- [Symbolic Physics Learner: Discovering governing equations via Monte Carlo tree search](https://mlsanity.com/paper/2205.13134/) (2022)
- [SymFormer: End-to-end symbolic regression using transformer-based architecture](https://mlsanity.com/paper/2205.15764/) (2022)
- [Deep Symbolic Regression for Recurrent Sequences](https://mlsanity.com/paper/2201.04600/) (2022)
- [Interpretable Machine Learning for Science with PySR and SymbolicRegression.jl](https://mlsanity.com/paper/2305.01582/) (2023)
- [Physics-Guided Deep Learning for Dynamical Systems: A Survey](https://mlsanity.com/paper/2107.01272/) (2021)
- [Pruner-Zero: Evolving Symbolic Pruning Metric from scratch for Large Language Models](https://mlsanity.com/paper/2406.02924/) (2024)
- [Generating Synergistic Formulaic Alpha Collections via Reinforcement Learning](https://mlsanity.com/paper/2306.12964/) (2023)
- [Discovering Causal Relations and Equations from Data](https://mlsanity.com/paper/2305.13341/) (2023)
- [gLaSDI: Parametric Physics-informed Greedy Latent Space Dynamics Identification](https://mlsanity.com/paper/2204.12005/) (2022)
- [Shape-constrained Symbolic Regression -- Improving Extrapolation with Prior Knowledge](https://mlsanity.com/paper/2103.15624/) (2021)
- [Structure-preserving Sparse Identification of Nonlinear Dynamics for Data-driven Modeling](https://mlsanity.com/paper/2109.05364/) (2021)
- [AlphaForge: A Framework to Mine and Dynamically Combine Formulaic Alpha Factors](https://mlsanity.com/paper/2406.18394/) (2024)
- [Predicting Ordinary Differential Equations with Transformers](https://mlsanity.com/paper/2307.12617/) (2023)
- [Discovering Nonlinear PDEs from Scarce Data with Physics-encoded Learning](https://mlsanity.com/paper/2201.12354/) (2022)
- [Deep Learning and Symbolic Regression for Discovering Parametric Equations](https://mlsanity.com/paper/2207.00529/) (2022)
- [DrSR: LLM based Scientific Equation Discovery with Dual Reasoning from Data and Experience](https://mlsanity.com/paper/2506.04282/) (2025)
- [Taylor Genetic Programming for Symbolic Regression](https://mlsanity.com/paper/2205.09751/) (2022)
- [Automated Scientific Discovery: From Equation Discovery to Autonomous Discovery Systems](https://mlsanity.com/paper/2305.02251/) (2023)
- [Physics-informed Spline Learning for Nonlinear Dynamics Discovery](https://mlsanity.com/paper/2105.02368/) (2021)
- [A Comprehensive Survey on Self-Interpretable Neural Networks](https://mlsanity.com/paper/2501.15638/) (2025)
- [End-to-End Neuro-Symbolic Reinforcement Learning with Textual Explanations](https://mlsanity.com/paper/2403.12451/) (2024)
- [Call for Action: towards the next generation of symbolic regression benchmark](https://mlsanity.com/paper/2505.03977/) (2025)
- [A Neural-Guided Dynamic Symbolic Network for Exploring Mathematical Expressions from Data](https://mlsanity.com/paper/2309.13705/) (2023)
- [Think like a Scientist: Physics-guided LLM Agent for Equation Discovery](https://mlsanity.com/paper/2602.12259/) (2026)

## Abstract (arXiv)

We present an approach to identify concise equations from data using a shallow neural network approach. In contrast to ordinary black-box regression, this approach allows understanding functional relations and generalizing them from observed data to unseen parts of the parameter space. We show how to extend the class of learnable equations for a recently proposed equation learning network to include divisions, and we improve the learning and model selection strategy to be useful for challenging real-world data. For systems governed by analytical expressions, our method can in many cases identify the true underlying equation and extrapolate to unseen domains. We demonstrate its effectiveness by experiments on a cart-pendulum system, where only 2 random rollouts are required to learn the forward dynamics and successfully achieve the swing-up task.