Loss Function🪶
Mathematical Formulation🪶
Four Loss Terms🪶
-
Schrödinger residual (physics loss)
\[ \mathcal{L}_\text{TISE} = \sum_{n=1}^N { \left| \hat{H}_\theta\psi_\theta^n(x)-E_n^\theta\psi_n^\theta(x) \right|^2 } \] -
Scale-aware smoothness
\[ \mathcal{L}_\text{smooth} = \Big< \frac{\left\| V_\theta''(x) \right\|^2}{\epsilon+\left\| V_\theta(x) \right\|^2} \Big> \] -
Data mismatch (noisy observables)
\[ \mathcal{L}_\text{data} = \frac{1}N\sum_{n=1}^N{\big(E_n^\theta-E_n^\text{obs}\big)^2} + \frac{1}N\sum_{n=1}^{N}\frac{1}{M}\sum_{j=1}^M{\Big( |\psi_{n}^\theta(x_j)|^2-\rho_n^\text{obs}(x_j)\Big) ^2} \] -
Energy ordering
\[ \mathcal{L}_\text{order} = \frac{1}{N-1}\sum_{n=0}^{N-2}{\bigg[\max{\big(0, E_n^\theta - E_{n+1}^\theta\big)} \bigg]^2} \]
Total Loss🪶
\[ \mathcal{L}_\text{tot} = \lambda_1 \mathcal{L}_\text{TISE} + \lambda_2 \mathcal{L}_\text{smooth} + \lambda_3 \mathcal{L}_\text{data} + \lambda_4 \mathcal{L}_\text{order} \]
Description of Terms🪶
Loss Table
| Loss Term | Soft vs. Hard | Classification | Notes | 🐍 Script | 🐍 Relevant Definitions |
|---|---|---|---|---|---|
| 💙 1. Schrödinger residual (physics loss) | Soft | Physics / consistency loss | Enforces the TISE | physics.py |
tise_residual(), tise_loss() |
| 💚 2. Scale-aware smoothness | Soft | Regularizer / prior | Penalizes curvature in \(V_\theta(x)\). Encourages smoother, physically plausible potentials and stabilizes the inverse problem. | physics.py |
potential_smoothness_loss() |
| 🩷 3. Data mismatch (observables) | Soft | Supervised data-fit loss | Matches learned energies and densities to noisy observed measurements. | inverse.py |
data_mismatch_loss() |
| 🩵 4. Energy ordering | Soft | Ordering penalty / constraint | Encourages \(E_0 < E_1 < \cdots\). Helps prevent state swapping and preserves consistent mode indexing. | physics.py |
energy_ordering_loss() |
| 💜 Total loss | Soft | Composite objective | Weighted sum of the active terms. Static scalar weights are used. | train.py |
train_step() |