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Loss Function🪶

The Project 2 loss function is a linear combination of terms that encode: - physical consistency - observational agreement - regularization - spectral ordering.

This is designed so that the weighted summation of these terms defines a family of admissible Hamiltonians to be explored by the model during training. As such, the recovered potential should be interpreted as a single plausible solution within a constrained inverse-problem solution space rather than a unique reconstruction of the ground-truth operator.

Mathematical Formulation🪶

Four Loss Terms

  1. 💙 Schrödinger residual (physics loss)

    \[ \mathcal{L}_\text{TISE} = \sum_{n=1}^N { \left| \hat{H}_\theta\hat{\psi}_n^\theta(x)-E_n^\theta \hat{\psi}_n^\theta(x) \right|^2 } \]

  2. 💚 Scale-aware smoothness

    \[ \mathcal{L}_\text{smooth} = \Big< \frac{\left\| V_\theta''(x) \right\|^2}{\epsilon+\left\| V_\theta(x) \right\|^2} \Big> \]

  3. 🩷 Data mismatch (noisy observables)

    \[ \mathcal{L}_\text{data} = \frac{1}N\sum_{n=1}^N{\big(E_n^\theta-E_n^\text{obs}\big)^2} + \frac{1}N\sum_{n=1}^{N}\frac{1}{M}\sum_{j=1}^M{\Big( |\hat{\psi}_{n}^\theta(x_j)|^2-\rho_n^\text{obs}(x_j)\Big) ^2} \]

  4. 🩵 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 physical law (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()