## Line Charts: Performance Comparison of LLM-SR and PiT-PO ### Overview The image presents four line charts comparing the performance of two methods, LLM-SR (blue) and PiT-PO (red), across different scenarios: "Oscillation 1", "Oscillation 2", "E. coli Growth", and "Stress-Strain". Each chart plots the NMSE (Normalized Mean Squared Error) on a logarithmic scale against the iteration number. The charts show how the NMSE decreases with increasing iterations for both methods, with shaded regions indicating variability or uncertainty. ### Components/Axes * **Title:** The image contains four subplots, each with a title: "Oscillation 1", "Oscillation 2", "E. coli Growth", and "Stress-Strain". * **X-axis:** All charts share the same x-axis label: "Iteration". The x-axis ranges from 0 to 2500, with tick marks at 0, 625, 1250, 1875, and 2500. * **Y-axis:** All charts share the same y-axis label: "NMSE (log scale)". The y-axis scale varies between charts. * Oscillation 1: ranges from 10^-17 to 10^-1 * Oscillation 2: ranges from 10^-8 to 10^0 * E. coli Growth: ranges from 10^-2 to 10^0 * Stress-Strain: ranges from 10^-2 to 10^1 * **Legend:** Located at the top of the image, the legend identifies the two methods: * LLM-SR: Represented by a blue line with a light blue shaded region. * PiT-PO: Represented by a red line with a light red shaded region. ### Detailed Analysis **Oscillation 1** * **LLM-SR (Blue):** The line starts at approximately 10^-1 and decreases to around 10^-5, then remains relatively stable. * **PiT-PO (Red):** The line starts at approximately 10^-5 and decreases significantly to around 10^-18 in a step-wise fashion. * The shaded regions around each line indicate the variability in the NMSE for each method. **Oscillation 2** * **LLM-SR (Blue):** The line starts at approximately 10^0 and decreases to around 10^-6, then remains relatively stable. * **PiT-PO (Red):** The line starts at approximately 10^-2 and decreases significantly to around 10^-9 in a step-wise fashion. * The shaded regions around each line indicate the variability in the NMSE for each method. **E. coli Growth** * **LLM-SR (Blue):** The line starts at approximately 10^0 and decreases to around 10^-1, then remains relatively stable. * **PiT-PO (Red):** The line starts at approximately 10^0 and decreases significantly to around 10^-2 in a step-wise fashion. * The shaded regions around each line indicate the variability in the NMSE for each method. **Stress-Strain** * **LLM-SR (Blue):** The line starts at approximately 10^1 and decreases to around 10^-1, then remains relatively stable. * **PiT-PO (Red):** The line starts at approximately 10^0 and decreases significantly to around 10^-2 in a step-wise fashion. * The shaded regions around each line indicate the variability in the NMSE for each method. ### Key Observations * In all four scenarios, both LLM-SR and PiT-PO show a decrease in NMSE as the number of iterations increases, indicating that both methods are converging towards a solution. * PiT-PO generally achieves a lower NMSE than LLM-SR in all scenarios, suggesting that it may be a more effective method for these specific problems. * The step-wise decrease in NMSE for PiT-PO suggests that it may be making discrete improvements at certain iterations. * The shaded regions indicate that there is some variability in the NMSE for both methods, but the overall trend is still clear. ### Interpretation The charts provide a comparative analysis of the performance of LLM-SR and PiT-PO across different problem domains. The data suggests that PiT-PO generally outperforms LLM-SR in terms of achieving a lower NMSE. This could be due to the specific algorithms used by each method, or the way in which they are optimized for these particular problems. The step-wise decrease in NMSE for PiT-PO could indicate that it is using a more aggressive or adaptive optimization strategy. The variability in NMSE, as indicated by the shaded regions, suggests that there may be some sensitivity to initial conditions or other factors. Overall, the data suggests that PiT-PO is a promising method for these types of problems, but further investigation may be needed to understand its behavior in more detail.

## Chart: NMSE vs. Iteration for Different Models ### Overview The image presents four separate line charts, each depicting the Normalized Mean Squared Error (NMSE) on a logarithmic scale against the number of iterations. Two models, "LLM-SR" (blue) and "PIT-PO" (red), are compared across four different datasets: "Oscillation 1", "Oscillation 2", "E. coli Growth", and "Stress-Strain". Each line represents the average NMSE for the respective model, and shaded areas indicate the standard deviation. ### Components/Axes * **X-axis:** Iteration (ranging from 0 to 2500) * **Y-axis:** NMSE (log scale) * **Legend:** * LLM-SR (Blue) * PIT-PO (Red) * **Chart Titles:** * Oscillation 1 (Top-Left) * Oscillation 2 (Top-Right) * E. coli Growth (Bottom-Left) * Stress-Strain (Bottom-Right) ### Detailed Analysis or Content Details **Oscillation 1 (Top-Left):** * **LLM-SR (Blue):** The line starts at approximately 1e-1 and slopes downward, reaching approximately 1e-13 by iteration 2500. The shaded area indicates a relatively small standard deviation throughout the iterations. * **PIT-PO (Red):** The line begins at approximately 5e-1 and also slopes downward, but more gradually than LLM-SR. It reaches approximately 5e-10 by iteration 2500. The shaded area is wider than LLM-SR, indicating a larger standard deviation. **Oscillation 2 (Top-Right):** * **LLM-SR (Blue):** The line starts at approximately 1e-2 and decreases to approximately 1e-8 by iteration 2500. The standard deviation is relatively small. * **PIT-PO (Red):** The line begins at approximately 1e-2 and decreases to approximately 1e-6 by iteration 2500. The standard deviation is larger than LLM-SR. **E. coli Growth (Bottom-Left):** * **LLM-SR (Blue):** The line starts at approximately 1e0 and remains relatively stable around 1e-1 to 1e-2 for most of the iterations, with a slight decrease towards the end. The standard deviation is significant. * **PIT-PO (Red):** The line starts at approximately 1e0 and decreases more rapidly than LLM-SR, reaching approximately 1e-2 by iteration 2500. The standard deviation is also significant. **Stress-Strain (Bottom-Right):** * **LLM-SR (Blue):** The line starts at approximately 1e0 and decreases to approximately 1e-1 by iteration 2500. The standard deviation is relatively small. * **PIT-PO (Red):** The line starts at approximately 1e0 and decreases more rapidly than LLM-SR, reaching approximately 5e-2 by iteration 2500. The standard deviation is larger than LLM-SR. ### Key Observations * LLM-SR consistently outperforms PIT-PO in terms of NMSE across all four datasets, especially in "Oscillation 1" and "Oscillation 2". * The standard deviation for LLM-SR is generally smaller than that of PIT-PO, indicating more stable performance. * The "E. coli Growth" dataset shows the least amount of improvement in NMSE for both models, with both lines remaining relatively high. * The "Stress-Strain" dataset shows a more significant decrease in NMSE for both models. ### Interpretation The charts demonstrate the performance of two models, LLM-SR and PIT-PO, in predicting different datasets. The NMSE metric, plotted on a logarithmic scale, indicates the accuracy of the predictions. Lower NMSE values signify better performance. LLM-SR consistently achieves lower NMSE values across all datasets, suggesting it is a more accurate model than PIT-PO. The smaller standard deviations associated with LLM-SR indicate that its performance is more consistent and less sensitive to variations in the data. The "E. coli Growth" dataset presents a challenge for both models, as the NMSE values remain relatively high even after 2500 iterations. This suggests that the underlying dynamics of E. coli growth are more difficult to model accurately. The differences in performance between the models and datasets could be attributed to the complexity of the data, the model architectures, and the optimization algorithms used during training. The logarithmic scale highlights the magnitude of the errors, making it easier to compare the performance of the models across different datasets. The shaded areas representing standard deviation provide insight into the robustness of each model's performance.

## [Line Charts]: Performance Comparison of LLM-SR vs. PiT-PO on Four Datasets ### 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 2500 iterations. Performance is measured by **NMSE (Normalized Mean Squared Error)** on a logarithmic scale. The charts show that PiT-PO consistently achieves a lower final NMSE than LLM-SR across all four datasets, indicating superior performance. Shaded regions around each line represent the variability or confidence intervals for each method. ### Components/Axes * **Legend:** Located at the top center of the entire figure. It defines: * **LLM-SR:** Blue line. * **PiT-PO:** Red line. * **Common X-Axis (All Charts):** Label: **"Iteration"**. Scale: Linear, from 0 to 2500. Major tick marks at 0, 625, 1250, 1875, 2500. * **Common Y-Axis (All Charts):** Label: **"NMSE (log scale)"**. Scale: Logarithmic. The specific range varies per chart. * **Chart Titles (Top of each subplot):** * Top-Left: **"Oscillation 1"** * Top-Right: **"Oscillation 2"** * Bottom-Left: **"E. coli Growth"** * Bottom-Right: **"Stress-Strain"** ### Detailed Analysis #### **Chart 1: Oscillation 1 (Top-Left)** * **Y-Axis Range:** Approximately 10⁻¹⁷ to 10⁻¹. * **LLM-SR (Blue):** Starts near 10⁻⁴. Shows a stepwise decrease, plateauing around iteration 1250. Final value at iteration 2500 is approximately **10⁻⁶**. The shaded blue region (variability) is relatively narrow. * **PiT-PO (Red):** Starts near 10⁻⁵. Exhibits a much steeper, stepwise decline. Major drops occur before iteration 625 and around iteration 1875. Final value at iteration 2500 is approximately **10⁻¹⁹**. The shaded red region is very wide, indicating high variability, especially in the middle iterations. #### **Chart 2: Oscillation 2 (Top-Right)** * **Y-Axis Range:** Approximately 10⁻⁸ to 10⁰. * **LLM-SR (Blue):** Starts near 10⁻¹. Decreases in steps, with a notable drop after iteration 1875. Final value at iteration 2500 is approximately **10⁻⁶**. * **PiT-PO (Red):** Starts near 10⁻¹. Shows a sharp initial drop, then a stepwise decline. Final value at iteration 2500 is approximately **10⁻⁹**. The shaded red region is wide, overlapping significantly with the blue region in the middle iterations. #### **Chart 3: E. coli Growth (Bottom-Left)** * **Y-Axis Range:** Approximately 10⁻² to 10⁰. * **LLM-SR (Blue):** Starts near 10⁰. Shows a very gradual, stepwise decline. Final value at iteration 2500 is approximately **10⁻⁰.⁵** (or ~0.3). * **PiT-PO (Red):** Starts near 10⁰. Drops more sharply in steps, particularly around iteration 1250. Final value at iteration 2500 is approximately **10⁻¹.⁸** (or ~0.016). The shaded red region is wide, especially between iterations 625 and 1875. #### **Chart 4: Stress-Strain (Bottom-Right)** * **Y-Axis Range:** Approximately 10⁻² to 10⁻¹. * **LLM-SR (Blue):** Starts near 10⁻¹. Decreases in a stepwise fashion. Final value at iteration 2500 is approximately **10⁻¹.⁵** (or ~0.032). * **PiT-PO (Red):** Starts near 10⁻¹. Shows a very rapid initial drop within the first ~200 iterations, then plateaus with minor steps. Final value at iteration 2500 is approximately **10⁻¹.⁹** (or ~0.013). The shaded red region is narrow after the initial drop, indicating low variability in the final performance. ### Key Observations 1. **Consistent Superiority:** In all four datasets, the **PiT-PO (red)** method achieves a final NMSE that is **1 to 13 orders of magnitude lower** than the **LLM-SR (blue)** method. 2. **Convergence Pattern:** Both methods exhibit a **stepwise convergence** pattern, where the error remains flat for periods and then drops sharply. This suggests discrete improvement events, possibly linked to optimization steps or algorithmic phases. 3. **Variability:** The shaded confidence intervals for **PiT-PO are generally wider** than those for LLM-SR, particularly in the "Oscillation" and "E. coli Growth" charts. This indicates that while PiT-PO's *average* performance is better, its results may have higher variance across different runs or initial conditions. 4. **Performance Gap:** The performance gap between the two methods is most dramatic in the **"Oscillation 1"** chart, where PiT-PO reaches an NMSE of ~10⁻¹⁹ compared to LLM-SR's ~10⁻⁶. ### Interpretation The data strongly suggests that the **PiT-PO algorithm is significantly more effective at minimizing error (NMSE)** than the LLM-SR algorithm for the tested problems (Oscillation, E. coli Growth, Stress-Strain). The stepwise nature of the error reduction implies both algorithms operate in discrete phases of improvement. The **Peircean investigative reading** would focus on the *abductive* inference: Given that PiT-PO consistently and dramatically outperforms LLM-SR across diverse problem domains, what underlying mechanism in PiT-PO could explain this? The pattern suggests PiT-PO may have a more efficient search strategy, better exploitation of problem structure, or a more effective optimization routine that allows it to escape local minima where LLM-SR gets stuck (as seen in the long plateaus of the blue lines). The **notable anomaly** is the extremely wide confidence interval for PiT-PO in the "Oscillation 1" chart. This warrants investigation: Is the algorithm highly sensitive to initial conditions for that specific problem? Does it occasionally find an exceptionally good solution (leading to the very low NMSE) but not reliably? This high variance, despite superior average performance, is a critical practical consideration for deployment.

## Line Chart: Comparative NMSE Performance Across Iterations ### Overview The image contains four line charts arranged in a 2x2 grid, comparing the performance of two methods (LLM-SR and PiT-PO) across four datasets: Oscillation 1, Oscillation 2, E. coli Growth, and Stress-Strain. Each chart tracks the Normalized Mean Squared Error (NMSE) on a logarithmic scale against iteration count (0–2500). Shaded regions represent uncertainty bounds for each method. ### Components/Axes - **X-axis**: Iteration (0–2500, linear scale) - **Y-axis**: NMSE (log scale, 10⁻¹⁷ to 10⁰) - **Legends**: - Blue line/shade: LLM-SR - Red line/shade: PiT-PO - **Chart Titles**: - Top-left: Oscillation 1 - Top-right: Oscillation 2 - Bottom-left: E. coli Growth - Bottom-right: Stress-Strain ### Detailed Analysis #### Oscillation 1 - **LLM-SR (Blue)**: - Starts at ~10⁻¹ NMSE, drops sharply to ~10⁻⁵ by 625 iterations, then plateaus. - Uncertainty (shaded blue) narrows significantly after 1250 iterations. - **PiT-PO (Red)**: - Begins at ~10⁻³ NMSE, decreases to ~10⁻⁹ by 1250 iterations, then stabilizes. - Uncertainty (shaded red) remains broader than LLM-SR throughout. #### Oscillation 2 - **LLM-SR (Blue)**: - Initial NMSE ~10⁻², declines to ~10⁻⁴ by 625 iterations, then plateaus. - Uncertainty reduces by ~50% after 1875 iterations. - **PiT-PO (Red)**: - Starts at ~10⁻⁴ NMSE, drops to ~10⁻⁶ by 1250 iterations, then stabilizes. - Shaded red region shows consistent uncertainty reduction. #### E. coli Growth - **LLM-SR (Blue)**: - Begins at ~10⁰ NMSE, decreases to ~10⁻¹ by 625 iterations, then plateaus. - Uncertainty narrows by ~70% after 1250 iterations. - **PiT-PO (Red)**: - Starts at ~10⁻¹ NMSE, drops to ~10⁻² by 1250 iterations, then stabilizes. - Shaded red region shows gradual uncertainty reduction. #### Stress-Strain - **LLM-SR (Blue)**: - Initial NMSE ~10⁻¹, decreases to ~10⁻² by 625 iterations, then plateaus. - Uncertainty reduces by ~60% after 1875 iterations. - **PiT-PO (Red)**: - Begins at ~10⁻² NMSE, drops to ~10⁻³ by 1250 iterations, then stabilizes. - Shaded red region shows steady uncertainty reduction. ### Key Observations 1. **Performance Trends**: - LLM-SR consistently achieves lower NMSE than PiT-PO across all datasets. - Both methods show rapid improvement in early iterations (0–1250), with diminishing returns afterward. 2. **Uncertainty Patterns**: - Shaded regions (confidence intervals) narrow for both methods as iterations increase, indicating improved model stability. - LLM-SR’s uncertainty bounds are consistently tighter than PiT-PO’s. 3. **Dataset-Specific Behavior**: - Oscillation 1 and Stress-Strain show the most dramatic NMSE reductions. - E. coli Growth exhibits the slowest convergence for both methods. ### Interpretation The data demonstrates that **LLM-SR outperforms PiT-PO** in all tested scenarios, achieving lower NMSE and tighter confidence intervals. The logarithmic scale highlights exponential improvements in early iterations, suggesting these methods are particularly effective for initial model calibration. The narrowing uncertainty bands imply that both approaches become more reliable with increased computational effort, but LLM-SR maintains a consistent advantage. This could indicate architectural or algorithmic efficiencies in LLM-SR that make it preferable for applications requiring high-precision predictions under iterative refinement.