Project 2: Low-Fidelity Inverse Schrödinger Problem🪶
Overview
The purpose of project 2 is to study operator features that are identifiable in a low fidelity physics-informed machine learning prototype for the inverse Schrödinger problem.
| Step | Description | Completed? |
|---|---|---|
| 1. Problem formulation | Can a physics-informed neural network recover the unknown potential \(V(x)\) along with the corresponding eigenfunctions from only noisy spectral data and probability-density snapshots? | ✔️ |
| 2. Data collection & curation | - Uniform collocation grid of spatial points $x\in [-5,5]. - Noisy energy and probability density observations. |
✔️ |
| 3. Neural architecture | Two lightweight neural networks with tanh activations (see Project 2 architecture diagram) | ⚠️ Both neural networks are scalar-in, scalar-out, fully differentiable, and deliberately kept shallow to preserve interpretability. |
| 4. Loss function | All terms are soft penalties with static \(\lambda\) weights (see Project 2 loss function). | ✔️ |
| 5. Optimization | - Adam optimizer with a fixed learning rate. - Forward pass training loop that computes loss terms and uses backpropagation to update \(V_\theta\) and \(\psi_n^\theta\). |
❌⚠️ As in project 1, the optimizer is intentionally vanilla; the aim is to expose how the physics prior interacts with noisy data, not to chase maximal performance |
Conceptual Description of What the Model is Learning
What the ML model learns
Neural Ansatz: Let a neural network represent the potential
\[V_\theta(x) \quad \text{small MLP}\]
The model is not learning wavefunctions.
The wavefunctions are constrained by the physics loss.
Each epoch optimizes the networks weights \(\theta\) for \(\{ V_\theta, \psi_n^\theta, E_n^\theta \}\).
This involves minimization of the loss function.
🗝️ Key Points
- Indirect supervision: Project 2 architecture resembles a coupled operator-eigenfunction learning system.
- Proper orthogonal decomposition (POD) is used as a diagnostic tool. This is useful for handling data geometry in an interpretable manner.