## Chart: Normalized MSE vs Time (equations discovered on clean data) ### Overview The image presents a 3x3 grid of line charts, each displaying the Normalized Mean Squared Error (MSE) against Time. Each chart represents a different equation/system ("barcres", "barmag", "glider", "lv", "predprey", "shearflow", "vdp", "doc", "growth", "RD"). Three different methods ("PySR", "LLM-SR", "KeplerAgent") are compared within each chart. The charts aim to visualize the performance of each method in discovering equations from clean data, with lower MSE indicating better performance. ### Components/Axes * **Title:** "Normalized MSE vs Time (equations discovered on clean data)" - positioned at the top center of the image. * **X-axis Label:** "Time" - common to all subplots, ranging from 0 to 10 (except for "growth" and "RD" which range to 1.5 and 2.0 respectively). * **Y-axis Label:** "Relative MSE" - common to all subplots, with varying scales. * **Legend:** Each subplot has a legend in the top-right corner, identifying the lines representing "PySR" (blue), "LLM-SR" (orange), and "KeplerAgent" (green). * **Subplot Titles:** Each subplot is labeled with the name of the equation/system being analyzed: "barcres", "barmag", "glider", "lv", "predprey", "shearflow", "vdp", "doc", "growth", "RD". ### Detailed Analysis or Content Details Here's a breakdown of each subplot, including approximate data points and trend descriptions: 1. **barcres:** * PySR: Line slopes downward, starting at approximately 0.025 and decreasing to approximately 0.001 at Time = 10. * LLM-SR: Line is relatively flat, around 0.00005 throughout the time range. * KeplerAgent: Line starts at approximately 0.02 and decreases to approximately 0.002 at Time = 10. 2. **barmag:** * PySR: Line is nearly flat, around 0.00002 throughout the time range. * LLM-SR: Line slopes downward, starting at approximately 0.00008 and decreasing to approximately 0.00001 at Time = 10. * KeplerAgent: Line is relatively flat, around 0.00003 throughout the time range. 3. **glider:** * PySR: Line slopes downward, starting at approximately 0.04 and decreasing to approximately 0.005 at Time = 10. * LLM-SR: Line is relatively flat, around 0.015 throughout the time range. * KeplerAgent: Line is relatively flat, around 0.02 throughout the time range. 4. **lv:** * PySR: Line slopes downward, starting at approximately 3.5 and decreasing to approximately 0.1 at Time = 10. * LLM-SR: Line is relatively flat, around 0.5 throughout the time range. * KeplerAgent: Line is relatively flat, around 1.5 throughout the time range. 5. **predprey:** * PySR: Line slopes downward, starting at approximately 12 and decreasing to approximately 0.5 at Time = 10. * LLM-SR: Line is relatively flat, around 2 throughout the time range. * KeplerAgent: Line is relatively flat, around 5 throughout the time range. 6. **shearflow:** * PySR: Line slopes downward, starting at approximately 0.002 and decreasing to approximately 0.00001 at Time = 10. * LLM-SR: Line is relatively flat, around 0.00005 throughout the time range. * KeplerAgent: Line is relatively flat, around 0.0001 throughout the time range. 7. **vdp:** * PySR: Line slopes downward, starting at approximately 0.0001 and decreasing to approximately 0.000005 at Time = 10. * LLM-SR: Line is relatively flat, around 0.00002 throughout the time range. * KeplerAgent: Line is relatively flat, around 0.00003 throughout the time range. 8. **doc:** * PySR: Line slopes downward, starting at approximately 0.00001 and decreasing to approximately 0.000001 at Time = 10. * LLM-SR: Line is relatively flat, around 0.000005 throughout the time range. * KeplerAgent: Line is relatively flat, around 0.000008 throughout the time range. 9. **growth:** * PySR: Line slopes downward, starting at approximately 1.0 and decreasing to approximately 0.05 at Time = 2.0. * LLM-SR: Line is relatively flat, around 0.2 throughout the time range. * KeplerAgent: Line is relatively flat, around 0.5 throughout the time range. 10. **RD:** * PySR: Line slopes downward, starting at approximately 0.1 and decreasing to approximately 0.01 at Time = 2.0. * LLM-SR: Line is relatively flat, around 0.03 throughout the time range. * KeplerAgent: Line is relatively flat, around 0.05 throughout the time range. ### Key Observations * PySR generally exhibits a decreasing MSE over time across all equations, suggesting it improves its equation discovery as time progresses. * LLM-SR tends to have relatively flat MSE curves, indicating consistent performance without significant improvement or degradation over time. * KeplerAgent also shows relatively flat MSE curves, but generally with higher values than LLM-SR. * The scale of the Y-axis varies significantly between subplots, indicating different levels of difficulty in discovering equations for each system. "lv" and "predprey" have the highest MSE values, suggesting these systems are the most challenging. * "barmag" and "doc" have the lowest MSE values, suggesting these systems are the easiest to discover equations for. ### Interpretation The charts demonstrate a comparison of three methods (PySR, LLM-SR, and KeplerAgent) in their ability to discover equations from clean data. PySR consistently shows improvement over time, suggesting it benefits from longer computation or more data. LLM-SR provides stable, but not necessarily optimal, performance. KeplerAgent falls between the two, offering moderate performance without significant improvement. The varying scales of the Y-axis highlight the complexity of the different equations/systems. Systems with higher initial and final MSE values (like "lv" and "predprey") are more difficult to model, while those with lower values (like "barmag" and "doc") are easier. The consistent flatness of the LLM-SR curves could indicate a limitation in its ability to refine its equation discovery process over time, or that it quickly reaches a stable solution. The decreasing trend of PySR suggests an iterative refinement process. The differences in performance across the different systems suggest that the choice of method may depend on the specific characteristics of the data and the complexity of the underlying equation.