Numerical solvers improve accuracy and efficiency via auto diagnosis
Training Numerical Intelligence via Auto-Diagnosis and Skill Discovery
Artificial IntelligenceComputation and LanguageMachine Learning
Summary
Solvers for complex math problems often struggle without clear reasons for their errors. The authors introduce a system that first figures out why a solver performs poorly, then uses that insight to find better ways of solving the problem. Their approach organizes improvements into reusable skills rather than random trial and error. This method significantly improves solver results in different challenging math problems, even when facing new scenarios.
What this means in practice
- •For power grid engineers: Enhance accuracy and efficiency of numerical solvers used in power flow and optimal power flow control problems.
- •For software developers for scientific computing: Integrate self-improving diagnostic frameworks into numerical simulation software to reduce trial-and-error solver tuning.
Authors
Peter Chen, Wotao Yin
Abstract
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 $71\times$, with improvements further transferring to unseen grid topologies and operating regimes.