# Learning First Integrals via Backward-Generated Data and Guided Reinforcement Learning

arXiv: https://arxiv.org/abs/2605.21160 · HTML: https://mlsanity.com/paper/2605.21160/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in RL theory and sample complexity, one of 15 directions in Deep reinforcement learning methods.
- Builds on 29 of its 50 references that are in the map.
- First posted on arXiv 20 May 2026.

## Placed in

- Deep reinforcement learning methods / [RL theory and sample complexity](https://mlsanity.com/deeprl/rl-theory-and-sample-complexity/)

## Builds on (most cited first, 29 of 29)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [Qwen2 Technical Report](https://mlsanity.com/paper/2407.10671/) (2024)
- [Solving Quantitative Reasoning Problems with Language Models](https://mlsanity.com/paper/2206.14858/) (2022)
- [MetaMath: Bootstrap Your Own Mathematical Questions for Large Language Models](https://mlsanity.com/paper/2309.12284/) (2023)
- [Hamiltonian Neural Networks](https://mlsanity.com/paper/1906.01563/) (2019)
- [WizardMath: Empowering Mathematical Reasoning for Large Language Models via Reinforced Evol-Instruct](https://mlsanity.com/paper/2308.09583/) (2023)
- [Sequence Level Training with Recurrent Neural Networks](https://mlsanity.com/paper/1511.06732/) (2015)
- [Textbooks Are All You Need](https://mlsanity.com/paper/2306.11644/) (2023)
- [AI Feynman: a Physics-Inspired Method for Symbolic Regression](https://mlsanity.com/paper/1905.11481/) (2019)
- [Datasets: A Community Library for Natural Language Processing](https://mlsanity.com/paper/2109.02846/) (2021)
- [Lagrangian Neural Networks](https://mlsanity.com/paper/2003.04630/) (2020)
- [Deep symbolic regression: Recovering mathematical expressions from data via risk-seeking policy gradients](https://mlsanity.com/paper/1912.04871/) (2019)
- [ToRA: A Tool-Integrated Reasoning Agent for Mathematical Problem Solving](https://mlsanity.com/paper/2309.17452/) (2023)
- [Deep Learning for Symbolic Mathematics](https://mlsanity.com/paper/1912.01412/) (2019)
- [How Abilities in Large Language Models are Affected by Supervised Fine-tuning Data Composition](https://mlsanity.com/paper/2310.05492/) (2023)
- [Neural Symbolic Regression that Scales](https://mlsanity.com/paper/2106.06427/) (2021)
- [LLM-SR: Scientific Equation Discovery via Programming with Large Language Models](https://mlsanity.com/paper/2404.18400/) (2024)
- [Formal Mathematical Reasoning: A New Frontier in AI](https://mlsanity.com/paper/2412.16075/) (2024)
- [MathGenie: Generating Synthetic Data with Question Back-translation for Enhancing Mathematical Reasoning of LLMs](https://mlsanity.com/paper/2402.16352/) (2024)
- [Linear algebra with transformers](https://mlsanity.com/paper/2112.01898/) (2021)
- [Learning the greatest common divisor: explaining transformer predictions](https://mlsanity.com/paper/2308.15594/) (2023)
- [Global Lyapunov functions: a long-standing open problem in mathematics, with symbolic transformers](https://mlsanity.com/paper/2410.08304/) (2024)
- [AI Poincar'e 2.0: Machine Learning Conservation Laws from Differential Equations](https://mlsanity.com/paper/2203.12610/) (2022)
- [Transforming the Bootstrap: Using Transformers to Compute Scattering Amplitudes in Planar N = 4 Super Yang-Mills Theory](https://mlsanity.com/paper/2405.06107/) (2024)
- [Constants of motion network](https://mlsanity.com/paper/2208.10387/) (2022)
- [Tree Edit Distance Learning via Adaptive Symbol Embeddings](https://mlsanity.com/paper/1806.05009/) (2018)
- [Discovering Symmetries of ODEs by Symbolic Regression](https://mlsanity.com/paper/2506.19550/) (2025)

## Abstract (arXiv)

The discovery of first integrals is of fundamental scientific importance for understanding conservation laws in dynamical systems. However, existing symbolic computation tools and Large Language Models (LLMs) remain limited on this task because high-quality training data are scarce and successful solutions often depend on mathematical intuition. This paper presents FISolver, an LLM-based solver developed to address this challenge. First, we introduce a "Backward Generation" algorithm that systematically builds large-scale datasets of (differential equation, first integral) pairs by deriving differential equations from sampled integrals, thereby alleviating the data scarcity bottleneck. Second, we apply supervised fine-tuning to a compact mathematical model and further improve its performance through reinforcement learning with a Levenshtein Distance-based shaped reward. In addition, we design data synthesis and blending strategies that support effective adaptation to difficult problem families from sparse examples. Experiments show that FISolver, while requiring substantially lower computational cost, significantly outperforms larger mathematical LLMs and commercial solvers such as Mathematica on challenging benchmarks, indicating a new data-driven route for automated discovery of first integrals.