Alright, let's outline the methodology for investigating the geometric structure of the 10-dimensional latent space from our PINN simulations of the 2D Burger's equation. The goal is to understand how the latent representations for different initial conditions (ICs) are structured and related.

**1. Data Preparation and Initial Exploration**

   a.  **Data Loading:**
       You will start by loading the provided data file:
       python
       import numpy as np
       fin = '/Users/fanonymous/Documents/Software/AstroPilot/Project_turbulenceV1/data_for_Paco_turbulence_bundle.npy'
       data_bundle = np.load(fin)
       
       Verify that `data_bundle` has dimensions `(101, 103, 25, 13)`. These dimensions correspond to (x-coordinate, time, initial condition index, features).

   b.  **Latent Space Extraction:**
       The last 10 components of the 4th axis represent the latent space. Extract these:
       `latent_space_data = data_bundle[:, :, :, 3:]`
       This `latent_space_data` array will have dimensions `(101, 103, 25, 10)`. Each of the 25 slices along the third dimension corresponds to a unique initial condition. For each IC, we have `101 * 103 = 10403` latent vectors, each of 10 dimensions.

   c.  **Preliminary Exploratory Data Analysis (EDA):**
       We need to get a feel for the data's basic properties.

       i.  **Global Latent Space Analysis:**
           First, let's look at all latent vectors together. Reshape `latent_space_data` into a 2D array of shape `(101 * 103 * 25, 10)`, let's call this `L_global`.
           -   Calculate the mean vector (10-dim) and the `(10, 10)` covariance matrix of `L_global`.
           -   Perform Principal Component Analysis (PCA) on `L_global`. You'll need to compute the eigenvalues and corresponding eigenvectors.
           -   Record the percentage of variance explained by each principal component (PC) and the cumulative variance explained. We expect to see something like this (these are illustrative values):

             *Hypothetical EDA Table 1: Global Latent Space PCA Summary*
             | Principal Component | Eigenvalue | Variance Explained (%) | Cumulative Variance Explained (%) |
             |---------------------|------------|------------------------|-----------------------------------|
             | PC1                 | 12.5       | 45.0                   | 45.0                              |
             | PC2                 | 7.0        | 25.0                   | 70.0                              |
             | PC3                 | 4.2        | 15.0                   | 85.0                              |
             | PC4                 | 2.0        | 7.0                    | 92.0                              |
             | PC5                 | 1.1        | 4.0                    | 96.0                              |
             | ...                 | ...        | ...                    | ...                               |
             | PC10                | ...        | ...                    | 100.0                             |
             This table will give us an initial estimate of the overall dimensionality required to capture most of the variation in the latent space across all (x, t, IC) points.

       ii. **Per-Initial Condition (IC) Latent Space Analysis:**
           Now, analyze each of the 25 ICs separately. For each IC `k` (from 0 to 24):
           -   Extract its latent vectors: `L_k = latent_space_data[:, :, k, :]`. This results in a `(101, 103, 10)` array.
           -   Reshape `L_k` to `(10403, 10)`, let's call this `L_k_flat`.
           -   For each `L_k_flat`:
               1.  Calculate its 10-dimensional mean vector, `C_k`. This `C_k` is the centroid of the latent vectors for IC `k`. Store all 25 centroids.
               2.  Calculate its `(10, 10)` covariance matrix.
               3.  Perform PCA. Record the eigenvalues and cumulative variance explained for each IC.
           -   Summarize these per-IC PCA results by averaging the variance explained and cumulative variance explained across the 25 ICs. Also, note the standard deviation to understand consistency.

             *Hypothetical EDA Table 2: Per-IC PCA Summary (Averaged over 25 ICs)*
             | Principal Component | Avg. Eigenvalue | Avg. Var. Explained (%) | Avg. Cum. Var. Explained (%) | StdDev of Cum. Var. Expl. (%) |
             |---------------------|-----------------|-------------------------|------------------------------|-------------------------------|
             | PC1_ic              | 8.8             | 55.0                    | 55.0                         | 5.0                           |
             | PC2_ic              | 4.0             | 25.0                    | 80.0                         | 4.0                           |
             | PC3_ic              | 1.9             | 12.0                    | 92.0                         | 3.0                           |
             | ...                 | ...             | ...                     | ...                          | ...                           |
             This tells us about the typical intrinsic dimensionality of the set of latent vectors $\{L(x,t,IC)\}$ for a *single* IC.

       iii. **Analysis of Centroids:**
            Take the 25 centroid vectors `{C_k}` (each 10-dim) collected in the previous step. This forms a `(25, 10)` matrix.
            -   Perform PCA on this matrix of centroids.
            -   Record eigenvalues and cumulative variance explained.

              *Hypothetical EDA Table 3: PCA of IC Centroids*
              | Centroid PC | Eigenvalue | Variance Explained (%) | Cumulative Variance Explained (%) |
              |-------------|------------|------------------------|-----------------------------------|
              | CPC1        | 5.5        | 60.0                   | 60.0                              |
              | CPC2        | 2.6        | 28.0                   | 88.0                              |
              | CPC3        | 0.6        | 7.0                    | 95.0                              |
              | ...         | ...        | ...                    | ...                               |
              This analysis will indicate whether the mean positions of the latent manifolds themselves are structured (e.g., lie on a low-dimensional plane or curve).

**2. Detailed Analysis of Individual Manifolds (Per Initial Condition)**

   Our EDA (specifically, the results similar to Table 2) will suggest an effective dimensionality, `d_k`, for the latent point cloud of each IC `k`. This `d_k` is the number of principal components needed to explain a high percentage (e.g., 95%) of its variance.

   a.  **Characterize Affine Subspace:** For each IC `k`:
       -   Use the centroid `C_k` and the top `d_k` principal component vectors (eigenvectors) `v_k1, v_k2, ..., v_kd_k` obtained from its per-IC PCA. Let `V_k = [v_k1, ..., v_kd_k]` be the matrix of these basis vectors.
       -   The set of latent vectors $\{L(x,t,IC_k)\}$ can be approximated by an affine subspace defined by `C_k + span(V_k)`.
       -   Document `d_k`, `C_k`, and the set of principal vectors `V_k` for each of the 25 ICs. Note the eigenvalues (variances) associated with these principal vectors as they describe the extent of the manifold along these directions.

**3. Comparative Analysis of Latent Space Manifolds Across Initial Conditions**

   The goal here is to understand how these 25 individual manifolds relate to each other.

   a.  **Centroid Configuration:**
       -   Using the PCA results on the centroids (from EDA Table 3), determine the dimensionality `d_C` of the space effectively spanned by `{C_k}`.
       -   Describe the geometric arrangement of these centroids. For instance, if `d_C` is 2, the centroids approximately lie on a 2D plane in the 10-D latent space.

   b.  **Manifold Orientation Comparison:**
       -   We need to compare the orientations of the principal subspaces `span(V_k)`. Let's assume a common intrinsic dimension `d_ic` for all ICs based on EDA Table 2 (e.g., `d_ic=3` if that captures ~92% variance on average).
       -   For each pair of ICs `(k, j)`:
           -   Compare their first principal vectors `v_k1` and `v_j1` by computing the absolute dot product `|v_k1 \cdot v_j1|`. Do this for all pairs and summarize the distribution of these dot products. Repeat for `v_k2` vs `v_j2`, and so on, up to `v_kd_ic` vs `v_jd_ic`. High dot products (close to 1) indicate alignment.
           -   To compare the overall subspaces `span(V_k)` and `span(V_j)`, you can compute principal angles between these subspaces. Alternatively, form orthonormal matrices `M_k` and `M_j` (columns are `v_k1, ..., v_kd_ic` and `v_j1, ..., v_jd_ic` respectively). Then compute `(1/d_ic) * ||M_k^T M_j||_F^2` (average squared cosine of principal angles, normalized).

   c.  **Investigating Simple Transformations:**
       -   **Translations:** If the principal subspaces `span(V_k)` are highly aligned across different `k` (from 3b), but centroids `C_k` vary (from 3a), then the manifolds are primarily translations of a common reference manifold.
       -   **Rotations/Reflections:** If `span(V_k)` are not aligned, check if they might be rotations of each other. This can be inferred if the per-IC PCA eigenvalues (the variances along principal axes) are similar across ICs, even if the eigenvectors `V_k` are different.
           -   You can use Orthogonal Procrustes analysis: for pairs of ICs `(k,j)`, find the optimal rotation `R_kj` that aligns `V_k` to `V_j` (after centering both point clouds). If the residuals after alignment are small, this suggests a rotational relationship.
       -   **Low-dimensional Variation of Orientations:**
           -   Collect all first principal vectors `{v_k1}` for `k=0..24`. This is a set of 25 vectors in $\mathbb{R}^{10}$. Perform PCA on this set. If these vectors lie in a low-dimensional subspace, it suggests the primary orientation of the manifolds varies in a structured way across ICs. Repeat for `{v_k2}`, etc.

**4. Analysis of the Global Latent Space Structure**

   Relate the individual manifold structures to the global structure found in EDA (Table 1).

   a.  **Projection onto Global Subspace:**
       -   Let `d_glob` be the effective dimensionality from the global PCA (e.g., 5 PCs capturing 96% variance). Let `U_glob` be the matrix of these top `d_glob` global principal vectors.
       -   For each IC `k`:
           -   Project its centered latent vectors `(L_k_flat - C_k)` onto this global principal subspace spanned by `U_glob`.
           -   Calculate how much of IC `k`'s variance is captured by this global subspace.
           -   Analyze the coordinates of the projected centroids `C_k` and the orientation of projected `V_k` within this global subspace.

   b.  **Interpretation:**
       -   If `d_ic` (typical per-IC dimension) is significantly smaller than `d_glob`, it implies that individual manifolds are "flatter" than the overall structure, and the variation across ICs explores additional dimensions.
       -   If `d_ic` is similar to `d_glob`, compare the per-IC principal subspaces `span(V_k)` with the global principal subspace `span(U_glob)`. Are they aligned? Or does the global PCA average over differently oriented individual manifolds?

**5. Synthesis: Formulating a Geometric Description**

   Finally, consolidate all findings to describe the latent space structure. Your report should address:
   a.  The typical intrinsic dimensionality `d_ic` of the latent representation $\{L(x,t,IC)\}$ for a single initial condition.
   b.  The extent to which these individual manifolds can be approximated as affine subspaces (flatness, based on PCA variance explained).
   c.  How the manifolds for different ICs are related:
       -   Are they primarily translations of each other? (Similar `V_k`, varying `C_k` that lie on a simple structure).
       -   Are they rotations/transformations of a common template manifold? (Similar per-IC eigenvalues, `V_k` related by structured transformations).
       -   How do the manifold characteristics (centroids `C_k`, orientations `V_k`) vary as a function of the initial condition? Is this variation itself low-dimensional or structured?
   d.  Whether there's an overarching, simpler geometric structure (e.g., a global low-dimensional manifold) that encompasses all the individual IC manifolds. Describe how the initial conditions parameterize movement within or selection of these manifolds.

This systematic approach should allow us to build a comprehensive picture of how the PINN encodes the effects of initial conditions in its latent space. Focus on clear quantification at each step.