## 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.

## Chart: 2D Region Plot - Uncertainty Quantification ### Overview The image presents a 2D plot illustrating regions defined by different methods, likely representing uncertainty bounds or confidence intervals in a two-dimensional space defined by variables x1 and x2. The plot compares the regions generated by several models and error estimations. ### Components/Axes * **X-axis:** Labeled "x₁", ranging approximately from -0.5 to 1.5. * **Y-axis:** Labeled "x₂", ranging approximately from -0.4 to 1.6. * **Legend:** Located in the bottom-left corner, containing the following labels and corresponding colors: * "xₛ" (Light Green) * "RneuralODE" (Black) * "ΩResNet" (Blue) * "ΩResNet + Ωₑ" (Red) * "ΩResNet + Error bound from Sander 2022" (Magenta) * **Grid:** A light gray grid is overlaid on the plot area. ### Detailed Analysis The plot displays five distinct regions, each defined by a closed curve. 1. **xₛ (Light Green):** This region is a square, with vertices approximately at (-0.1, 0.7), (0.4, 0.7), (0.4, 1.2), and (-0.1, 1.2). It is the smallest and most contained region. 2. **RneuralODE (Black):** This region is a more complex polygon, enclosing the green region. It has vertices approximately at (-0.2, 0.0), (1.3, 0.0), (1.3, 1.5), and (-0.2, 1.5). The shape is roughly rectangular but with some inward curves. 3. **ΩResNet (Blue):** This region is a polygon, contained within the black region, and appears to be similar in shape to the black region but slightly smaller and more irregular. It has vertices approximately at (0.0, 0.3), (0.8, 0.3), (0.8, 1.3), and (0.0, 1.3). 4. **ΩResNet + Ωₑ (Red):** This region is a polygon, contained within the blue region, and appears to be similar in shape to the blue region but slightly smaller and more irregular. It has vertices approximately at (0.1, 0.4), (0.7, 0.4), (0.7, 1.1), and (0.1, 1.1). 5. **ΩResNet + Error bound from Sander 2022 (Magenta):** This region is the largest and most encompassing, containing all other regions. It is a polygon with vertices approximately at (-0.3, -0.2), (1.4, -0.2), (1.4, 1.6), and (-0.3, 1.6). The regions are nested, with xₛ being the innermost and the Sander 2022 error bound being the outermost. The shapes are not perfectly regular, suggesting the underlying data or model is complex. ### Key Observations * The regions represent increasing levels of uncertainty or confidence intervals. * The "xₛ" region is the most precise, while the Sander 2022 error bound provides the broadest estimate. * The difference between the regions suggests that adding error bounds significantly expands the estimated uncertainty. * The shapes of the regions are not simple, indicating non-linear relationships or complex dependencies. ### Interpretation This chart likely visualizes the results of an uncertainty quantification study. The different regions represent the estimated range of possible values for the variables x₁ and x₂, as determined by different models or methods. * **xₛ:** Could represent a ground truth or a highly accurate estimate. * **RneuralODE:** Represents the region estimated by a Neural ODE model. * **ΩResNet:** Represents the region estimated by a ResNet model. * **ΩResNet + Ωₑ:** Represents the ResNet model's estimate with an additional error component (Ωₑ). * **ΩResNet + Error bound from Sander 2022:** Represents the ResNet model's estimate with an error bound calculated using the method proposed by Sander in 2022. The increasing size of the regions as more error terms are added indicates that the uncertainty in the estimate grows as more factors are considered. The Sander 2022 error bound provides the most conservative (largest) estimate, likely reflecting a higher level of confidence or a more thorough assessment of potential errors. The chart demonstrates the importance of considering uncertainty when making predictions or decisions based on these models. The nested structure highlights how different methods contribute to the overall uncertainty estimate.

\n ## 2D Plot: Comparison of Set Bounds and Data Distribution ### Overview The image is a 2D scatter plot with overlaid geometric shapes representing different sets or bounds. It appears to compare the coverage or containment of a data distribution (represented by a scatter of points) against various theoretical or computed regions. The plot is set against a white background with a light gray grid. ### Components/Axes * **Axes:** * **Horizontal Axis (x1):** Labeled `x1`. Scale ranges from approximately -0.5 to 1.5, with major tick marks at -0.5, 0, 0.5, 1, and 1.5. * **Vertical Axis (x2):** Labeled `x2`. Scale ranges from approximately -0.4 to 1.6, with major tick marks at -0.4, -0.2, 0, 0.2, 0.4, 0.6, 0.8, 1, 1.2, 1.4, and 1.6. * **Legend:** Located at the bottom-center of the plot area. It contains five entries, each with a colored line sample and a label: 1. **Green Line:** `X_s` 2. **Black Line:** `R_{neuralODE}` 3. **Blue Line:** `Ω_{ResNet}` 4. **Red Line:** `Ω_{ResNet} + Ω_ε` 5. **Magenta Line:** `Ω_{ResNet} + Error bound from Sander 2022` * **Data Series & Shapes:** * **Scatter Points:** A dense cluster of black points, with a smaller number of blue points interspersed, concentrated in the central region of the plot. * **Geometric Shapes (from outermost to innermost):** 1. **Magenta Polygon:** A large, irregular hexagon forming the outermost boundary. 2. **Green Rectangle:** A smaller, axis-aligned rectangle nested inside the magenta shape. 3. **Red Polygon:** An irregular pentagon/hexagon nested inside the green rectangle. 4. **Blue Polygon:** A smaller irregular shape nested inside the red polygon. 5. **Black Polygon:** The innermost irregular shape, closely surrounding the main cluster of scatter points. ### Detailed Analysis * **Spatial Layout & Containment:** The shapes exhibit a clear nested hierarchy. The magenta polygon is the largest and contains all other elements. The green rectangle is fully contained within the magenta polygon. The red, blue, and black polygons are successively nested within the green rectangle, with the black polygon being the innermost bound. * **Approximate Coordinates of Shape Vertices (Estimated from Grid):** * **Magenta Polygon (`Ω_{ResNet} + Error bound...`):** Vertices are approximately at (-0.4, -0.3), (-0.2, -0.3), (-0.4, 1.4), (1.0, 1.4), (1.2, 1.2), and (1.2, 0.2). * **Green Rectangle (`X_s`):** Appears to be an axis-aligned rectangle with corners approximately at (0.2, 0.3) and (0.6, 0.8). * **Red Polygon (`Ω_{ResNet} + Ω_ε`):** Vertices are approximately at (0.25, 0.35), (0.25, 0.75), (0.35, 0.8), (0.55, 0.7), (0.55, 0.4), and (0.45, 0.35). * **Blue Polygon (`Ω_{ResNet}`):** Vertices are approximately at (0.3, 0.4), (0.3, 0.7), (0.4, 0.75), (0.5, 0.65), (0.5, 0.45), and (0.4, 0.4). * **Black Polygon (`R_{neuralODE}`):** Vertices are approximately at (0.35, 0.45), (0.35, 0.65), (0.45, 0.7), (0.5, 0.6), (0.5, 0.5), and (0.45, 0.45). * **Scatter Point Distribution:** The black points form a dense, roughly elliptical cluster centered near (0.4, 0.6). The blue points are scattered within and slightly around this main cluster, with a few appearing near the boundary of the blue polygon. ### Key Observations 1. **Containment Hierarchy:** There is a strict visual containment: Magenta ⊃ Green ⊃ Red ⊃ Blue ⊃ Black ≈ Data Cluster. 2. **Tightness of Bounds:** The black polygon (`R_{neuralODE}`) appears to be the tightest bound around the primary data cluster. The blue polygon (`Ω_{ResNet}`) is slightly larger, the red polygon (`Ω_{ResNet} + Ω_ε`) larger still, and the green rectangle (`X_s`) is a much looser, axis-aligned bound. The magenta polygon is a very loose, outer bound. 3. **Legend-Color Correspondence:** The colors in the legend match the shapes in the plot exactly. The magenta line corresponds to the outermost polygon, the green line to the rectangle, and so on, down to the black line for the innermost polygon. 4. **Data vs. Bounds:** The majority of the black scatter points are contained within the innermost black polygon. The blue scatter points are also mostly within the black and blue polygons, though a few are near or slightly outside the blue boundary. ### Interpretation This plot likely visualizes and compares different methods for computing **invariant sets, reachable sets, or error bounds** for a dynamical system or a neural network model (suggested by terms like `ResNet` and `neuralODE`). * **`X_s` (Green):** Represents a simple, possibly predefined, safe set or state space region (an axis-aligned box). * **`Ω_{ResNet}` (Blue):** Represents a set computed or guaranteed by a Residual Network (ResNet) based method. It is tighter than `X_s`. * **`Ω_{ResNet} + Ω_ε` (Red):** Likely represents the ResNet set augmented with an additional error term (`Ω_ε`), resulting in a slightly larger, more conservative bound. * **`R_{neuralODE}` (Black):** Represents a set computed by a Neural Ordinary Differential Equation (Neural ODE) method. It provides the tightest fit to the empirical data distribution (the scatter points), suggesting it may be a more accurate model of the system's true behavior or reachable states. * **`Ω_{ResNet} + Error bound...` (Magenta):** Represents a very conservative, possibly theoretical, error bound from a cited work (Sander 2022) applied to the ResNet set. Its large size indicates it is a loose, worst-case guarantee. **The core message** is a comparison of **set computation techniques**. The Neural ODE method (`R_{neuralODE}`) produces a set that most closely matches the observed data distribution, while the ResNet-based methods (`Ω_{ResNet}`) produce larger, more conservative sets. The plot demonstrates the trade-off between the tightness of a computed bound and its conservatism, with the Neural ODE approach appearing superior in this specific case for approximating the data's support. The magenta bound serves as a reference for a known, highly conservative theoretical limit.

## Line Chart with Geometric Shapes: Nested Regions and Error Bounds ### Overview The image depicts a 2D coordinate system with axes labeled **x₁** (horizontal, -0.5 to 1.5) and **x₂** (vertical, -0.4 to 1.6). It contains five geometric shapes and data series, each associated with a legend label. The outermost element is a magenta rectangle, enclosing progressively smaller nested shapes (green rectangle, red hexagon, blue hexagon) and scattered black points. The legend clarifies the meaning of each color and label. --- ### Components/Axes - **Axes**: - **x₁**: Horizontal axis, range [-0.5, 1.5], labeled at intervals of 0.5. - **x₂**: Vertical axis, range [-0.4, 1.6], labeled at intervals of 0.2. - **Legend** (bottom-right corner): - **Green**: **Xₛ** (smallest green rectangle). - **Black**: **R_neuralODE** (scattered points). - **Blue**: **Ω_ResNet + Ω_ε** (inner blue hexagon). - **Red**: **Ω_ResNet + Ω_ε + Error bound from Sander 2022** (outer red hexagon). - **Magenta**: **Ω_ResNet + Error bound from Sander 2022** (outermost magenta rectangle). --- ### Detailed Analysis 1. **Magenta Rectangle** (Ω_ResNet + Error bound from Sander 2022): - Forms a closed loop with vertices at approximately: - (-0.5, -0.2) → (-0.5, 1.4) → (1.5, 1.4) → (1.5, 0.8) → (1, 0.8) → (1, -0.2) → (-0.5, -0.2). - Encloses all other elements. 2. **Green Rectangle** (Xₛ): - Centered at (0.5, 0.5), with sides at x₁ = 0.3–0.7 and x₂ = 0.3–0.7. - Contains all inner shapes. 3. **Red Hexagon** (Ω_ResNet + Ω_ε + Error bound from Sander 2022): - Inscribed within the green rectangle, with vertices forming a hexagon. - Slightly offset from the green rectangle’s edges. 4. **Blue Hexagon** (Ω_ResNet + Ω_ε): - Inscribed within the red hexagon, with vertices forming a smaller hexagon. - Contains all black points. 5. **Black Points** (R_neuralODE): - Scattered within the blue hexagon, densely clustered near (0.5, 0.5). --- ### Key Observations - **Nested Structure**: The magenta rectangle encloses the green rectangle, which contains the red hexagon, followed by the blue hexagon. This suggests hierarchical error bounds or confidence intervals. - **Data Distribution**: Black points (R_neuralODE) are concentrated in the blue hexagon, indicating a relationship between the neural ODE model and the ResNet error bounds. - **Error Bound Overlap**: The red and magenta lines share the "Error bound from Sander 2022" label, implying additive error contributions from ResNet and Sander’s theoretical framework. --- ### Interpretation The chart visualizes nested error bounds or confidence regions for a neural network model (ResNet) combined with a neural ODE. The magenta rectangle represents the total error bound from Sander (2022), while the green, red, and blue shapes likely represent progressively tighter approximations or components of the error. The black points (R_neuralODE) suggest that the neural ODE model’s predictions align closely with the ResNet error bounds (blue hexagon). The overlapping labels for red and magenta lines indicate that the error bound from Sander 2022 is a cumulative effect of ResNet’s inherent error (Ω_ResNet) and additional theoretical contributions (Ω_ε). This structure may reflect a multi-stage analysis of model uncertainty or robustness.