## Line Charts: NMSE vs. Iteration for Different Models and Datasets ### Overview The image contains four line charts arranged in a 2x2 grid. Each chart plots the Normalized Mean Squared Error (NMSE) on a logarithmic scale against the iteration number. The charts compare the performance of two models, LLM-SR (blue) and PiT-PO (red), across four different datasets: Oscillation 1, Oscillation 2, E. coli Growth, and Stress-Strain. Each line is surrounded by a shaded region of the same color, representing the uncertainty or variance in the model's performance. ### Components/Axes * **X-axis (all charts):** Iteration, with tick marks at 0, 625, 1250, 1875, and 2500. * **Y-axis (all charts):** NMSE (log scale). The y-axis scales vary slightly between charts. * Oscillation 1: 10^-5 to 10^-1 * Oscillation 2: 10^-8 to 10^0 * E. coli Growth: 10^-1 to 10^0 * Stress-Strain: 10^-2 to 10^0 * **Titles (each chart):** * Top-left: Oscillation 1 * Top-right: Oscillation 2 * Bottom-left: E. coli Growth * Bottom-right: Stress-Strain * **Legend (top center):** * LLM-SR (blue line and shaded region) * PiT-PO (red line and shaded region) ### Detailed Analysis **Oscillation 1:** * **LLM-SR (blue):** Starts at approximately 10^-2 and remains relatively constant, with a slight decrease, fluctuating between 10^-2 and 10^-3. * **PiT-PO (red):** Starts at approximately 10^-1, then decreases stepwise to approximately 10^-5 by iteration 1875, remaining constant thereafter. * Step 1: From 10^-1 to 10^-3 by iteration 625 * Step 2: From 10^-3 to 10^-5 by iteration 1875 **Oscillation 2:** * **LLM-SR (blue):** Starts at approximately 10^-2 and remains relatively constant, fluctuating between 10^-2 and 10^-3. * **PiT-PO (red):** Starts at approximately 10^-2, then decreases stepwise to approximately 10^-8 by iteration 1250, remaining constant thereafter. * Step 1: From 10^-2 to 10^-3 by iteration 625 * Step 2: From 10^-3 to 10^-8 by iteration 1250 **E. coli Growth:** * **LLM-SR (blue):** Starts at approximately 10^0 and remains relatively constant, fluctuating between 10^-0 and 10^-1. * **PiT-PO (red):** Starts at approximately 10^0, then decreases stepwise to approximately 10^-1 by iteration 1875, remaining constant thereafter. * Step 1: From 10^0 to 10^-1 by iteration 1875 **Stress-Strain:** * **LLM-SR (blue):** Starts at approximately 10^0, then decreases stepwise to approximately 10^-1 by iteration 625, remaining constant thereafter. * Step 1: From 10^0 to 10^-1 by iteration 625 * **PiT-PO (red):** Starts at approximately 10^0, then decreases stepwise to approximately 10^-2 by iteration 625, remaining constant thereafter. * Step 1: From 10^0 to 10^-1 by iteration 625 * Step 2: From 10^-1 to 10^-2 by iteration 625 ### Key Observations * In all four datasets, the PiT-PO model (red) generally achieves a lower NMSE than the LLM-SR model (blue), indicating better performance. * The PiT-PO model exhibits a stepwise decrease in NMSE, suggesting discrete improvements at specific iterations. * The LLM-SR model tends to maintain a more stable NMSE, with less fluctuation. * The shaded regions around the lines indicate the variability in the model's performance across multiple runs or trials. ### Interpretation The charts demonstrate the performance of two different models (LLM-SR and PiT-PO) on four different datasets. The PiT-PO model consistently outperforms the LLM-SR model in terms of NMSE, suggesting that it is a more effective model for these tasks. The stepwise decrease in NMSE for the PiT-PO model may indicate specific iterations where the model learned significant features or adjusted its parameters effectively. The relatively stable NMSE of the LLM-SR model suggests that it may be less sensitive to changes in the data or optimization process. The shaded regions provide insight into the robustness and reliability of each model, with wider regions indicating greater variability in performance. Overall, the data suggests that the PiT-PO model is a better choice for these datasets, but further analysis may be needed to understand the specific factors that contribute to its superior performance.

## 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" (represented by a blue line) and "PIT-PO" (represented by a red line), are compared across four different scenarios: Oscillation 1, Oscillation 2, E. coli Growth, and Stress-Strain. Shaded regions around each line indicate the standard deviation or confidence interval. ### Components/Axes * **X-axis:** Iteration, ranging from 0 to 2500. * **Y-axis:** NMSE (log scale). The scale varies for each chart, but generally spans several orders of magnitude. * **Legend:** Located at the top-right of the image, it identifies the lines: * LLM-SR (Blue) * PIT-PO (Red) * **Chart Titles:** Each subplot has a title indicating the scenario: * Oscillation 1 (Top-Left) * Oscillation 2 (Top-Right) * E. coli Growth (Bottom-Left) * Stress-Strain (Bottom-Right) * **Gridlines:** Present in all charts to aid in reading values. ### Detailed Analysis or Content Details **Oscillation 1 (Top-Left):** * Both lines start at approximately 10^-1. * The blue line (LLM-SR) shows a consistent downward trend, decreasing to approximately 10^-5 by iteration 2500. * The red line (PIT-PO) also decreases, but more erratically, ending at approximately 10^-3 by iteration 2500. * The shaded region around the blue line is relatively narrow, indicating lower variance. The red line's shaded region is wider, suggesting higher variance. **Oscillation 2 (Top-Right):** * Both lines start around 10^-2. * The blue line (LLM-SR) initially decreases, then plateaus around 10^-5. * The red line (PIT-PO) fluctuates around 10^-2, with some dips and rises. * The shaded regions are relatively narrow for both lines. **E. coli Growth (Bottom-Left):** * Both lines start around 10^0. * The blue line (LLM-SR) decreases in steps, reaching approximately 10^-1 by iteration 2500. * The red line (PIT-PO) decreases more rapidly and in larger steps, reaching approximately 10^-2 by iteration 2500. * The shaded regions are wider, indicating higher variance. **Stress-Strain (Bottom-Right):** * Both lines start around 10^-1. * The blue line (LLM-SR) decreases gradually to approximately 10^-2. * The red line (PIT-PO) initially decreases, then increases sharply around iteration 1250, reaching approximately 10^-1 before decreasing again to approximately 10^-2. * The shaded regions are relatively narrow for the blue line, but wider for the red line, especially during the increase around iteration 1250. ### Key Observations * LLM-SR generally exhibits smoother and more consistent decreases in NMSE compared to PIT-PO. * PIT-PO shows more variability and, in the Stress-Strain scenario, a notable increase in NMSE at iteration 1250. * The E. coli Growth scenario shows the most significant reduction in NMSE for both models. * The logarithmic scale emphasizes the relative changes in NMSE, making it easier to compare performance across different scenarios. ### Interpretation The charts compare the performance of two models, LLM-SR and PIT-PO, in terms of their ability to minimize the Normalized Mean Squared Error (NMSE) across four different dynamic systems. The consistent downward trend of LLM-SR in most scenarios suggests it is more stable and reliable in reducing error. The fluctuations and occasional increases in NMSE for PIT-PO indicate it may be more sensitive to the specific dynamics of the system or require more careful tuning. The E. coli Growth scenario shows the largest error reduction, potentially indicating that the models are well-suited for modeling biological systems. The Stress-Strain scenario, with PIT-PO's increase in NMSE, suggests a potential instability or limitation of that model under certain conditions. The shaded regions around the lines represent the uncertainty or variance in the model's performance, providing a measure of confidence in the results. The use of a logarithmic scale is crucial for visualizing the wide range of NMSE values, allowing for a clear comparison of performance even when errors are very small. The charts collectively demonstrate the importance of model selection and parameter tuning for achieving optimal performance in different dynamic systems.

\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.

## Line Graphs: NMSE Performance Across Tasks ### Overview The image contains four line graphs arranged in a 2x2 grid, comparing the performance of two methods (LLM-SR and PiT-PO) across four tasks: Oscillation 1, Oscillation 2, E. coli Growth, and Stress-Strain. All graphs share identical axes labels and scales, with iterations (0–2500) on the x-axis and NMSE (log scale) on the y-axis. The legend identifies blue lines as LLM-SR and red lines as PiT-PO, with shaded regions representing uncertainty intervals. --- ### Components/Axes - **X-axis**: Iteration (0–2500, linear scale). - **Y-axis**: NMSE (log scale, 10⁻¹ to 10⁰). - **Legend**: - Blue: LLM-SR - Red: PiT-PO - **Shaded Regions**: Confidence intervals (wider at lower iterations, narrowing over time). --- ### Detailed Analysis #### Oscillation 1 - **LLM-SR (Blue)**: Starts at ~10⁻¹, drops sharply to ~10⁻³ by iteration 625, then plateaus. - **PiT-PO (Red)**: Starts at ~10⁻³, drops to ~10⁻⁵ by iteration 625, then plateaus. - **Trend**: PiT-PO consistently outperforms LLM-SR after iteration 625. Both methods stabilize by iteration 1250. #### Oscillation 2 - **LLM-SR (Blue)**: Starts at ~10⁻², drops to ~10⁻⁴ by iteration 625, then plateaus. - **PiT-PO (Red)**: Starts at ~10⁻³, drops to ~10⁻⁵ by iteration 625, then plateaus. - **Trend**: Similar to Oscillation 1, PiT-PO achieves lower NMSE earlier and maintains superiority. #### E. coli Growth - **LLM-SR (Blue)**: Starts near 10⁰, drops to ~10⁻¹ by iteration 625, then plateaus. - **PiT-PO (Red)**: Starts near 10⁰, drops to ~10⁻¹ by iteration 625, then plateaus. - **Trend**: Both methods converge at ~10⁻¹ by iteration 625, with PiT-PO showing slightly lower NMSE throughout. #### Stress-Strain - **LLM-SR (Blue)**: Starts at ~10⁻¹, drops to ~10⁻² by iteration 625, then plateaus. - **PiT-PO (Red)**: Starts at ~10⁻², drops to ~10⁻³ by iteration 625, then plateaus. - **Trend**: PiT-PO achieves lower NMSE earlier and maintains a consistent advantage. --- ### Key Observations 1. **Early Performance Drop**: All tasks show a sharp NMSE reduction (~1–2 orders of magnitude) around iteration 625 for both methods. 2. **Convergence**: By iteration 1250, NMSE values stabilize for both methods across all tasks. 3. **Method Comparison**: PiT-PO (red) consistently achieves lower NMSE than LLM-SR (blue) after iteration 625 in all tasks. 4. **Uncertainty**: Shaded regions (confidence intervals) are widest at early iterations, narrowing significantly by iteration 1250. --- ### Interpretation The data demonstrates that **PiT-PO outperforms LLM-SR** in reducing NMSE across diverse tasks (oscillations, biological growth, mechanical stress). The early drop in NMSE (~iteration 625) suggests a critical adaptation phase where PiT-PO’s methodology (e.g., parameter tuning, model architecture) becomes more effective. The plateauing NMSE after iteration 1250 implies diminishing returns from further iterations. The narrowing confidence intervals indicate increasing model stability over time. These results highlight PiT-PO’s robustness in handling varied dynamical systems, with potential applications in predictive modeling where low NMSE is critical.