## Chart: Reachability Sets Comparison ### Overview The image is a 2D plot comparing different reachability sets for a system, likely related to neural network verification or control. It shows the state space boundaries estimated by different methods, including neural ODEs and ResNets, along with error bounds. The plot visualizes the tightness and accuracy of these different approximations. ### Components/Axes * **X-axis:** x1, ranging from -0.5 to 1.5 with increments of 0.5. * **Y-axis:** x2, ranging from -0.4 to 1.6 with increments of 0.2. * **Legend (bottom-right):** * Green: $\mathcal{X}_s$ * Black: $\mathcal{R}_{neuralODE}$ * Blue: $\Omega_{ResNet}$ * Red: $\Omega_{ResNet} + \Omega_{\epsilon}$ * Magenta: $\Omega_{ResNet}$ + Error bound from Sander 2022 ### Detailed Analysis * **$\mathcal{X}_s$ (Green):** This set forms the outermost bound among the inner sets. It's a rectangular shape with rounded corners, centered around (0.5, 0.6). The x-axis ranges from approximately 0.1 to 0.8, and the y-axis ranges from approximately 0.3 to 0.85. * **$\mathcal{R}_{neuralODE}$ (Black):** This set is contained within all other sets. It is an irregular polygon formed by the densest cluster of points. The points are concentrated around x1 = 0.4 and x2 = 0.6. * **$\Omega_{ResNet}$ (Blue):** This set is slightly larger than the black set. It is a polygon with vertices around (0.25, 0.4), (0.6, 0.4), (0.7, 0.7), and (0.3, 0.8). * **$\Omega_{ResNet} + \Omega_{\epsilon}$ (Red):** This set is larger than the blue set. It is a polygon with vertices around (0.2, 0.35), (0.75, 0.35), (0.8, 0.75), and (0.2, 0.8). * **$\Omega_{ResNet}$ + Error bound from Sander 2022 (Magenta):** This set forms the outermost bound. It is a polygon with vertices around (-0.5, -0.3), (1.4, -0.3), (1.4, 1.4), and (-0.5, 1.4). ### Key Observations * The black set ($\mathcal{R}_{neuralODE}$) is the tightest bound, closely following the shape of the data points. * The green set ($\mathcal{X}_s$) provides a larger, more conservative bound. * The magenta set ($\Omega_{ResNet}$ + Error bound from Sander 2022) is the most conservative, providing the largest bound. * The blue and red sets ($\Omega_{ResNet}$ and $\Omega_{ResNet} + \Omega_{\epsilon}$) fall between the black and green sets in terms of tightness. ### Interpretation The plot compares the accuracy and conservativeness of different methods for estimating the reachable set of a system. The neural ODE approach (black) provides the tightest estimate, suggesting it's the most accurate. However, it may not be guaranteed to be a strict bound. The other methods (green, blue, red, magenta) provide outer bounds with varying degrees of conservativeness. The magenta set, based on Sander 2022, provides the most conservative bound, ensuring that the true reachable set is contained within it, but at the cost of being less precise. The plot highlights the trade-off between accuracy and safety in reachability analysis.