Project 1 Architecture🪶

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%%  CURVED-CORNER MERMAID WITH SUBGRAPH HEADER
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        "themeCSS": ".nodeLabel, .edgeLabel, .cluster-label, .cluster-label text, .label, .label text, text, .katex, .katex *, .MathJax, .MathJax *, mjx-container, mjx-container * { color: whitesmoke !important; fill: whitesmoke !important; -webkit-text-fill-color: whitesmoke !important; }",
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flowchart TB

    %%--------------------------------------------------------------
    %%  COLOR RAMP (pseudo-gradient)
    %%--------------------------------------------------------------
    classDef stage0 fill:#0b1c2d,stroke:#14b5ff,stroke-width:2px,color:#ffffff,rx:12,ry:12;
    classDef stage1 fill:#0f2a3d,stroke:#14b5ff,stroke-width:2px,color:#ffffff,rx:12,ry:12;
    classDef stage2 fill:#103b4f,stroke:#00f5db,stroke-width:2px,color:#ffffff,rx:12,ry:12;
    classDef stage3 fill:#124f55,stroke:#00f5db,stroke-width:2px,color:#ffffff,rx:12,ry:12;
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    classDef stage6 fill:#3a2f2a,stroke:#f78166,stroke-width:2px,color:#ffffff,rx:12,ry:12;
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    %%--------------------------------------------------------------
    %%  MAIN PIPELINE SUBGRAPH WITH HEADER
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    subgraph PIML["PIML Framework"]
        direction TB
        B["1️⃣ Collocation Points"]:::stage1
        C["2️⃣ Neural Ansatz"]:::stage2
        D["3️⃣ Automatic Differentiation"]:::stage3
        E["4️⃣ Variational Physics Loss"]:::stage4
        F["5️⃣ Total Loss"]:::stage5
        G["6️⃣ Optimizer (Adam)"]:::stage6

        B --> C
        C --> D
        D --> E
        E --> F
        F --> G
        G -- training loop --> C
    end

    %%--------------------------------------------------------------
    %%  CONTEXT & DIAGNOSTICS
    %%--------------------------------------------------------------
    A["0️⃣ Define SHO Dynamics"]:::stage0
    H["7️⃣ Diagnostics & Sanity Checks"]:::stage7

    A --> PIML:::dashed
    PIML --> H

    click C "assets/phase_space.png" "View figure"

Mathematical Mapping🪶

Step	Component	Mathematical Description	Interpretation	Importance in Pipeline
0️⃣	Problem setup	SHO Lagrangian and equations of motion	Defines the physical system.	Provides the exact DE the network must respect.
1️⃣	Collocation points	\(t\in [0, 2\pi]\)	Synthetic "data" for physics enforcement.	Keeps the pipeline purely physics-driven.
3️⃣	Automatic differentiation	\(p_\theta = \dot{q}_\theta \quad \text {and} \quad \ddot{q}_\theta\)	Recovers velocity and acceleration	Provides the quantities needed for the physics residual.
4️⃣	Physics loss	\(\mathcal{L}_\text{phys} = \langle (\ddot{q}_\theta + \omega^2 q_\theta)^2\rangle\)	Encodes Euler-Lagrange structure	Low fidelity is maintained since the network only needs to reduce the residual (not satisfy it exactly).
5️⃣	Total loss	\(\mathcal{L}_\text{total} = \mathcal{L}_\text{phys}\)	Low-fidelity PINN objective function	Highlights how pure physics can drive learning.
6️⃣	Optimization	\(\theta_{k+1} = \theta_k - \eta \nabla_\theta \mathcal{L}_\text{total}\)	Gradient-based learning	Standard gradient descent; the dynamics of convergence reveal interpretability cure.
7️⃣	Diagnostics	\(H_\theta(t) = H(q_\theta, p_\theta)\)	Sanity checks and structure validation	Makes failure modes explicit for analysis.