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Project 1 Architecture🪶

%%==================================================================== %% CURVED-CORNER MERMAID WITH SUBGRAPH HEADER %%==================================================================== %%{ init: { "theme": "base", "themeVariables": { "background": "#0d1117", "lineColor": "#14b5ff", "textColor": "whitesmoke", "fontFamily": "'Aclonica', sans-serif", "borderRadius": "16" /* larger radius for more rounded corners */ }, "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; }", "handDrawn": true } }%% %%==================================================================== 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; classDef stage4 fill:#1a6b63,stroke:#00f5db,stroke-width:2px,color:#ffffff,rx:12,ry:12; classDef stage5 fill:#1f4e5f,stroke:#f78166,stroke-width:2px,color:#ffffff,rx:12,ry:12; classDef stage6 fill:#3a2f2a,stroke:#f78166,stroke-width:2px,color:#ffffff,rx:12,ry:12; classDef stage7 fill:#0f2a3d,stroke:#14b5ff,stroke-width:2px,color:#ffffff,rx:12,ry:12; classDef dashed fill:#161b22,stroke:#14b5ff,stroke-dasharray:6 6,color:#ffffff,rx:12,ry:12; %%-------------------------------------------------------------- %% MAIN PIPELINE SUBGRAPH WITH HEADER %%-------------------------------------------------------------- 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.