Project 1: Low-Fidelity Harmonic OscillatorðŠķ
Overview
Create a foundational teaching model in PIML design, emphasizing interpretability and parsimony over raw accuracy.
- Simulates a 1D simple harmonic oscillator (SHO) with an unknown frequency while softly enforcing the equation of motion for the SHO and analyzing energy conservation.
- Prioritizes geometric intuition and visibility of failure modes over benchmark performance.
- Motivated by the perspective taken by Kutz & Brunton (2022) that parsimony itself is a powerful regularizer in PIML architectures.
The 1D simple harmonic oscillator is interpreted as a system defined by the physical structure $$ \mathcal{S} = { \text{Euler-Lagrange equation, energy conservation, symplectic symmetry} }$$ rather than a simple curve-fitting of the observed trajectory \(q(t)\).
| State Space \(\mathcal{X}\) | Structure \(\mathcal{S}\) | Admissible Set \(\mathcal{M}_\mathcal{S}\) |
|---|---|---|
| Phase Space, \(\mathbb{R}^2\) | \(\big\{ \ddot{q} + \omega^2 q = 0 \, , \, \dot{E} = 0 \big\}\) | \(q(t) \in \big\{ \mathbb{R}^2 : \mathcal{S} \, \text{holds} \big\}\) |
PIML Design
| Step | Description | Completed? |
|---|---|---|
| 1. Problem formulation | Can a neural function approximator recover physically meaningful motion via minimization of a variational residual, rather than fitting observed data? | â |
| 2. Data collection & curation | - Intentionally minimal (i.e., no observable trajectories). - Collocation points in time serve as synthetic "data" that serves to embed physics into training . | â |
| 3. Neural architecture | - Low-depth MLP, scalar input \(\rightarrow\) scalar output, tanh activations. - No convolutions, recurrences, or unnecessary inductive biases. - Physics enters through the loss function, not the architecture |
â ïļ |
| 4. Loss function | $$ L_\text{phys}=\big\langle (\ddot{q} + \omega^2 q)^2 \big\rangle $$ Encodes Euler-Lagrange structure, second-order dynamics, and physical consistency. | â |
| 5. Optimization strategy | - Standard Adam optimizer with fixed learning rate. - Optimization is intented to reveal physical structure, rather than fully customize for performance. | ââ ïļ |
Designed to be low fidelity and interpretible. This means
- Physical structure is encouraged via a soft penalty term in the loss function.
- Constraints are not directly enforced.
- We do not seek to design a strict symplectic integrator.
ðĄ Take-Home Message The overall model is biased towards a Hamiltonian structure without being strictly symplectic.