## 6 Results and Discussions
### Performance Comparison on ODE Problem  
The ODE 1D optimization task serves as a compact, low-dimensional testbed to assess convergence behavior, generalization, and efficiency under controlled complexity. Despite its simplicity, the presence of deceptive local optima and sparse reward gradients still challenges heuristic consistency and stability. Table 4 summarizes the comparative performance across all methods, and Figure 5 provides visual comparison. **Core Performance Metrics. ** EoH-MR again achieves the best result with the **highest objective value** of \(-\)**0.034**, outperforming all other LLM-driven variants and traditional baselines. Although all scores are negative due to the problem formulation, a less negative value indicates better solution quality. EoH-M and EoH achieve slightly inferior results of \(-\)0.179 and \(-\)0.258, respectively, while FunSearch and HillClimb perform worst, converging to local minima at \(-\)2.401 and \(-\)1.278. As visualized in Figure 5, this confirms that even in low-dimensional smooth settings, co-evolution with memory and refinement provides tangible benefits. **Stability and Robustness. ** EoH and EoH-MR both attain a perfect success rate of **1.000**, while FunSearch and HillClimb exhibit much weaker feasibility (0.6667 and 0.5714). In terms of normalized stability score, EoH-MR again leads with the highest score (**0.513**), followed closely by EoH-M (0.476), as illustrated in Figure 5. These results underscore the importance of refinement and evolution memory even in simplified domains. **Efficiency Metrics. ** EoH demonstrates the most efficient behavior, requiring only **6** LLM calls and **3456** tokens, and completing in **309.302** s. This reflects the advantage of simpler strategies in tractable domains. However, both EoH-M and EoH-MR strike a better trade-off between quality and cost, with moderate resource usage and improved convergence. HillClimb, despite its simplicity, consumes the most time (1242.08 s), under-performing on both axes. **Diversity and Exploration. ** Among all methods, EoH-MR achieves the highest solution diversity score (**1.636**), indicating that its search generates a wider variety of viable candidates. This is closely followed by EoH-M and HillClimb. However, traditional methods like FunSearch suffer from limited diversity (0.590), likely due to fixed operators and a \begin{table}
\begin{tabular}{l c c c c c} \hline \hline
**Metric Category** & **FunSearch** & **HillClimb** & **EoH** & **EoH-M** & **EoH-MR** \\ \hline \multicolumn{6}{l}{Core Performance Metrics} \\ \hline Best Score & \(-\)2.401 & \(-\)1.278 & \(-\)0.258 & \(-\)0.179 & \(-\)**0.034** \\ Standard Deviation & 10.662 & 18.628 & 9.780 & 11.879 & 11.151 \\ \hline \multicolumn{6}{l}{Stability and Robustness Metrics} \\ \hline Success Rate & 0.6667 & 0.5714 & **1.000** & 0.833 & **1.000** \\ Stability Score & 0.501 & 0.426 & 0.291 & 0.476 & **0.513** \\ \hline \multicolumn{6}{l}{Efficiency Metrics} \\ \hline Total LLM Calls & 20 & 33 & **6** & 10 & 15 \\ Total Tokens & 13,533 & 37,557 & **3456** & 9321 & 16,025 \\ Total Evaluations & 21 & 21 & **6** & 8 & 12 \\ Computation Time (s) & 536.78 & 1242.08 & **309.302** & 609.196 & 1133.604 \\ \hline \multicolumn{6}{l}{Solution Diversity Metrics} \\ \hline Total Unique Scores & 5 & 11 & 6 & 7 & 10 \\ Score Range & 27.13 & 66.68 & 23.74 & 25.06 & 27.20 \\ Solution Diversity Score & 0.590 & 1.006 & 0.997 & 1.051 & **1.636** \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Comprehensive comparison results of the ODE 1D scientific discovery task. Figure 5: Performance metrics for the ODE symbolic discovery task. (**Left**): Best fitting score (less negative is better). (**Middle**): Success rate over evaluation runs. (**Right**): Diversity score measuring variety of discovered formulas. lack of feedback-driven mutation dynamics. While EoH has moderate diversity (0.997), its simpler design leads to less exploratory power compared to EoH-MR. Figure 5 highlights these trends clearly.