{
  "image0": {
    "name": "centroid_pca_scatter_3D_3_20250604-135958.png",
    "caption": "Three-dimensional scatter plot showing the projection of the 25 per-initial condition (IC) latent manifold centroids onto their first three principal components (CPC1, CPC2, and CPC3). Each point represents the centroid for a unique initial condition and is colored according to its corresponding IC index (0 to 24). The plot demonstrates that the centroids are arranged along a predominantly one-dimensional path, strongly aligned with CPC1, indicating that the primary effect of varying the initial condition is to translate the latent manifold along a specific direction."
  },
  "image1": {
    "name": "intrinsic_dim_dist_plot_2_20250604-135804.png",
    "caption": "Distribution of the intrinsic dimensionality for the latent representations of each of the 25 initial conditions (ICs). Intrinsic dimensionality is defined as the minimum number of principal components required to capture over 95\\% of the variance for each IC's latent vectors. The plot shows that all 25 ICs result in latent manifolds with an intrinsic dimensionality of 3."
  },
  "image2": {
    "name": "subspace_similarity_heatmap_4_20250604-140218.png",
    "caption": "Subspace similarity between 3D latent manifolds for different initial conditions. The heatmap shows the average squared cosine of the principal angles between the subspaces spanned by the top three principal components for each pair of initial conditions ($IC_k$ and $IC_l$). High values (bright yellow) indicate strong alignment. The consistently high similarity across all pairs demonstrates that the 3D latent manifolds associated with different initial conditions are highly parallel."
  },
  "image3": {
    "name": "variance_captured_by_global_subspace_5_20250604-140503.png",
    "caption": "Percentage of the intrinsic variance for each initial condition (IC) latent manifold captured by the 6-dimensional global principal subspace. The consistently high values demonstrate that the individual 3D manifolds are effectively embedded within this common global subspace."
  },
  "image4": {
    "name": "centroid_pca_scatter_2D_3_20250604-135958.png",
    "caption": "\\caption{Initial condition (IC) manifold centroids projected onto their first two principal components (CPC1 and CPC2). Each point represents the centroid for a specific IC, labeled and colored by its index (0-24). The points form a clear, near-linear trajectory predominantly along CPC1, indicating that changing the IC primarily translates the corresponding latent manifold along a dominant direction.}"
  },
  "image5": {
    "name": "pca_of_principal_vectors_4_20250604-140218.png",
    "caption": "Principal Component Analysis (PCA) of the sets of per-initial condition (IC) principal vectors. Top row shows scree plots for the collection of first ($v_{k1}$), second ($v_{k2}$), and third ($v_{k3}$) per-IC principal vectors across all 25 ICs, indicating high variance capture by the first component in each set. Bottom row shows the 2D projection of these vector sets onto their respective first two principal components, colored by IC index, revealing a structured, low-dimensional variation in the orientation of the 3D per-IC manifolds."
  },
  "image6": {
    "name": "projected_centroids_2D_5_20250604-140503.png",
    "caption": "Projection of per-initial condition latent manifold centroids onto the first two global principal components. Each point, labeled and colored by initial condition index, reveals a near-linear arrangement predominantly along the first global component. This indicates that the PINN encodes variations due to initial conditions primarily by translating the corresponding latent manifolds along a structured, low-dimensional trajectory within the global latent space."
  }
}