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The paper introduces Auto-Diagnosis and Skill Discovery (ADSD), a framework for helping AI agents improve numerical solvers. It addresses a limitation of execution feedback: although it can reveal that a solver performs poorly, it usually does not explain the cause or identify an appropriate remedy. ADSD therefore follows a diagnosis-first process that explains the failure, discovers suitable numerical methods, and packages the resulting knowledge as reusable solver skills. Across power-flow equations, AC optimal power-flow control, stiff ordinary differential equations, and heterogeneous diffusion PDEs, the framework improves solver accuracy, robustness, and efficiency. On the GOC-500 power-flow benchmark, it reduces mean solver error by nearly 71 times, with the gains transferring to unseen grid topologies and operating regimes.