# Training Numerical Intelligence via Auto-Diagnosis and Skill Discovery

arXiv: https://arxiv.org/abs/2610.03872 · HTML: https://mlsanity.com/paper/2610.03872/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 58 of its 72 references that are in the map.
- First posted on arXiv 2 Oct 2026.

## Builds on (most cited first, 30 of 58)

- [Training language models to follow instructions with human feedback](https://mlsanity.com/paper/2203.02155/) (2022)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Direct Preference Optimization: Your Language Model is Secretly a Reward Model](https://mlsanity.com/paper/2305.18290/) (2023)
- [Reflexion: Language Agents with Verbal Reinforcement Learning](https://mlsanity.com/paper/2303.11366/) (2023)
- [DAPO: An Open-Source LLM Reinforcement Learning System at Scale](https://mlsanity.com/paper/2503.14476/) (2025)
- [Voyager: An Open-Ended Embodied Agent with Large Language Models](https://mlsanity.com/paper/2305.16291/) (2023)
- [The AI Scientist: Towards Fully Automated Open-Ended Scientific Discovery](https://mlsanity.com/paper/2408.06292/) (2024)
- [SimPO: Simple Preference Optimization with a Reference-Free Reward](https://mlsanity.com/paper/2405.14734/) (2024)
- [ExpeL: LLM Agents Are Experiential Learners](https://mlsanity.com/paper/2308.10144/) (2023)
- [AlphaEvolve: A coding agent for scientific and algorithmic discovery](https://mlsanity.com/paper/2506.13131/) (2025)
- [Agent Laboratory: Using LLM Agents as Research Assistants](https://mlsanity.com/paper/2501.04227/) (2025)
- [ToolRL: Reward is All Tool Learning Needs](https://mlsanity.com/paper/2504.13958/) (2025)
- [Large Language Models as Tool Makers](https://mlsanity.com/paper/2305.17126/) (2023)
- [ReMax: A Simple, Effective, and Efficient Reinforcement Learning Method for Aligning Large Language Models](https://mlsanity.com/paper/2310.10505/) (2023)
- [Language agents achieve superhuman synthesis of scientific knowledge](https://mlsanity.com/paper/2409.13740/) (2024)
- [EvoSkill: Automated Skill Discovery for Multi-Agent Systems](https://mlsanity.com/paper/2603.02766/) (2026)
- [Reinforcement Learning for Self-Improving Agent with Skill Library](https://mlsanity.com/paper/2512.17102/) (2025)
- [AgentRx: Diagnosing AI Agent Failures from Execution Trajectories](https://mlsanity.com/paper/2602.02475/) (2026)
- [Learning to run a Power Network Challenge: a Retrospective Analysis](https://mlsanity.com/paper/2103.03104/) (2021)
- [LLMs for Mathematical Modeling: Towards Bridging the Gap between Natural and Mathematical Languages](https://mlsanity.com/paper/2405.13144/) (2024)
- [SkillForge: Forging Domain-Specific, Self-Evolving Agent Skills in Cloud Technical Support](https://mlsanity.com/paper/2604.08618/) (2026)
- [Automated Hypothesis Validation with Agentic Sequential Falsifications](https://mlsanity.com/paper/2502.09858/) (2025)
- [GenEnv: Difficulty-Aligned Co-Evolution Between LLM Agents and Environment Simulators](https://mlsanity.com/paper/2512.19682/) (2025)
- [SKILLFOUNDRY: Building Self-Evolving Agent Skill Libraries from Heterogeneous Scientific Resources](https://mlsanity.com/paper/2604.03964/) (2026)
- [CodePDE: An Inference Framework for LLM-driven PDE Solver Generation](https://mlsanity.com/paper/2505.08783/) (2025)
- [SkillRevise: Improving LLM-Authored Agent Skills via Trace-Conditioned Skill Revision](https://mlsanity.com/paper/2606.01139/) (2026)
- [Exploration vs Exploitation: Rethinking RLVR through Clipping, Entropy, and Spurious Reward](https://mlsanity.com/paper/2512.16912/) (2025)
- [OPFData: Large-scale datasets for AC optimal power flow with topological perturbations](https://mlsanity.com/paper/2406.07234/) (2024)
- [Paper2Agent: Reimagining Research Papers As Interactive and Reliable AI Agents](https://mlsanity.com/paper/2509.06917/) (2025)
- [Stepwise Guided Policy Optimization: Coloring your Incorrect Reasoning in GRPO](https://mlsanity.com/paper/2505.11595/) (2025)

## Abstract (arXiv)

AI agents are becoming increasingly capable of generating scientific code, but generating code is not the same as improving the algorithms behind it. For numerical solvers, execution feedback can expose poor performance, but rarely reveals its underlying cause and how to address it. We introduce Auto-Diagnosis and Skill Discovery (ADSD), a framework that links numerical diagnosis to reusable solver self-improvement. ADSD follows a diagnosis-first paradigm that first explains why a solver performs poorly, then uses this diagnosis to guide the discovery of appropriate numerical methods. The resulting knowledge is packaged into reusable solver skills, turning solver improvement from trial-and-error editing into a structured process of diagnosis, discovery, and implementation. Across four challenging numerical domains--power flow equation, AC optimal power flow control, stiff ordinary differential equations, and heterogeneous diffusion PDEs--ADSD consistently improves solver accuracy, robustness, and efficiency. On GOC-500 power flow, for example, ADSD reduces mean solver error by nearly 71×, with improvements further transferring to unseen grid topologies and operating regimes.