\n ## Line Charts with Confidence Intervals: Performance Comparison of LLM-SR vs. PiT-PO ### Overview The image displays a 2x2 grid of four line charts. Each chart compares the performance of two methods, **LLM-SR** (blue line) and **PiT-PO** (red line), across a series of iterations. Performance is measured by **NMSE (Normalized Mean Squared Error)** on a logarithmic scale. Shaded regions around each line represent confidence intervals or variability. The overall purpose is to demonstrate the convergence behavior and final error of the two methods on four different tasks or datasets. ### Components/Axes * **Legend:** Located at the top center of the entire figure. * **Blue Line:** LLM-SR * **Red Line:** PiT-PO * **Common Axes:** * **X-axis (All Charts):** Label: `Iteration`. Scale: Linear, from 0 to 2500. Major tick marks at 0, 625, 1250, 1875, 2500. * **Y-axis (All Charts):** Label: `NMSE (log scale)`. Scale: Logarithmic (base 10). The range varies per subplot. * **Subplot Titles:** * Top-Left: `Oscillation 1` * Top-Right: `Oscillation 2` * Bottom-Left: `E. coli Growth` * Bottom-Right: `Stress-Strain` ### Detailed Analysis #### 1. Oscillation 1 (Top-Left) * **Y-axis Range:** Approximately 10⁻⁵ to 10⁻¹. * **LLM-SR (Blue):** Starts near 10⁻². Shows a very gradual, step-wise decrease, plateauing just above 10⁻³ by iteration 2500. The blue shaded confidence interval is relatively narrow. * **PiT-PO (Red):** Starts near 10⁻². Exhibits a rapid, step-wise descent, reaching approximately 10⁻⁵ by iteration 1250 and remaining stable thereafter. The red shaded confidence interval is wider than LLM-SR's, especially in the early iterations (0-1250). * **Trend:** PiT-PO converges to a significantly lower error (by about two orders of magnitude) much faster than LLM-SR. #### 2. Oscillation 2 (Top-Right) * **Y-axis Range:** Approximately 10⁻⁹ to 10⁰ (1). * **LLM-SR (Blue):** Starts near 10⁰. Drops quickly to around 10⁻² within the first ~200 iterations, then plateaus with a very slight downward trend, ending near 10⁻³. Confidence interval is narrow. * **PiT-PO (Red):** Starts near 10⁰. Follows a similar initial drop to ~10⁻². Then, around iteration 1250, it experiences a dramatic, sharp drop to approximately 10⁻⁹, where it remains. The red shaded region is very wide between iterations 625 and 1250, indicating high variance before the final convergence. * **Trend:** PiT-PO achieves an extremely low final error (10⁻⁹), which is about six orders of magnitude lower than LLM-SR's final error (~10⁻³). The convergence is discontinuous, marked by a single massive improvement. #### 3. E. coli Growth (Bottom-Left) * **Y-axis Range:** Approximately 10⁻¹ to 10⁰ (1). * **LLM-SR (Blue):** Starts just below 10⁰. Shows a very slow, almost flat decline, ending slightly above 10⁻¹. Confidence interval is narrow. * **PiT-PO (Red):** Starts at a similar point to LLM-SR. Remains close to LLM-SR until approximately iteration 1250, after which it begins a step-wise descent, reaching a final value near 10⁻¹. Its confidence interval becomes notably wide after iteration 1250. * **Trend:** Both methods show limited improvement. PiT-PO eventually achieves a slightly lower error than LLM-SR, but the difference is less than one order of magnitude. This task appears more challenging for both methods. #### 4. Stress-Strain (Bottom-Right) * **Y-axis Range:** Approximately 10⁻² to 10⁰ (1). * **LLM-SR (Blue):** Starts near 10⁰. Decreases in a step-wise fashion, plateauing around 10⁻¹ by iteration 1250 and remaining there. Confidence interval is moderate. * **PiT-PO (Red):** Starts near 10⁰. Drops more rapidly than LLM-SR, reaching a plateau near 10⁻² by iteration 625. It maintains this low error for the remainder of the iterations. Its confidence interval is wide during the initial descent (0-625). * **Trend:** PiT-PO converges faster and to a lower final error (10⁻²) compared to LLM-SR (10⁻¹), a difference of one order of magnitude. ### Key Observations 1. **Consistent Superiority:** In all four tasks, the **PiT-PO** method (red) achieves a lower final NMSE than the **LLM-SR** method (blue). 2. **Convergence Speed:** PiT-PO generally converges faster, often showing dramatic drops in error at specific iteration points (e.g., ~1250 in Oscillation 2, ~625 in Stress-Strain). 3. **Magnitude of Improvement:** The performance gap varies significantly by task. It is most extreme in **Oscillation 2** (10⁻⁹ vs. 10⁻³) and least pronounced in **E. coli Growth**. 4. **Variance:** The shaded confidence intervals for PiT-PO are frequently wider than those for LLM-SR, particularly during periods of rapid change. This suggests PiT-PO's performance may be more variable or sensitive during its optimization process before stabilizing. 5. **Task Difficulty:** The **E. coli Growth** task shows the least improvement for both methods, with final errors remaining relatively high (near 10⁻¹), indicating it may be a more complex or noisy problem. ### Interpretation The data strongly suggests that the **PiT-PO** optimization or learning method is more effective than **LLM-SR** for the class of problems represented by these four tasks. Its ability to reach orders-of-magnitude lower error, especially in the oscillation problems, indicates a superior capability for finding high-precision solutions. The step-wise convergence patterns, particularly the dramatic drop in Oscillation 2, are characteristic of optimization processes that escape local minima or undergo phase transitions in learning. The wider confidence intervals for PiT-PO during these transitions imply that while the method is powerful, its path to the solution may be less predictable or more dependent on initial conditions compared to the steadier, but less effective, LLM-SR. The stark difference in performance between tasks (e.g., Oscillation 2 vs. E. coli Growth) highlights that the relative advantage of PiT-PO is problem-dependent. It excels dramatically on certain physical or mathematical systems (oscillations, stress-strain) but offers a more modest gain on the biological growth model. This could inform which types of scientific or engineering problems would benefit most from applying the PiT-PO methodology.