Project 2: Low-Fidelity Inverse Schrödinger Problem🪶
Overview
- Study which physical structures of an unknown Hamiltonian remain identifiable under low-fidelity discretization, noisy observations, and constrained Hilbert-space geometry in the inverse Schrödinger problem.
- It is not exact reconstruction of the ground-truth potential \(V(x)\), but rather to determine which operator structures remain stable and recoverable under limited information.
- Indirect supervision: the architecture behaves as a coupled operator-eigenfunction learning system*.
- Proper orthogonal decomposition (POD) is used as a geometry-aware probe for studying basis conditioning, variance concentration, mode alignment, and potential mode mixing within the learned eigenstate manifold.
The inverse Schrödinger problem is interpreted as a problem of recovering elements of the physical structure $$ \mathcal{S} = { \text{eigenvalue constraints, orthogonality, normalization, operator geometry} }$$ rather than reconstructing a unique potential function \(V(x)\).
| State Space \(\mathcal{X}\) | Structure \(\mathcal{S}\) | Admissible Set \(\mathcal{M}_\mathcal{S}\) |
|---|---|---|
| Hilbert Space, \(\mathcal{H}\) | \(\big\{\hat{H}\psi=E\psi : \langle\psi_i, \psi_j\rangle = \delta_{ij} \, , \, \|\psi_i\|_{L^2}=1 \big\}\) | \(\psi \in \big\{ \mathcal{H} : \mathcal{S} \, \text{holds} \big\}\) |
PIML Design
| Step | Description | Completed? |
|---|---|---|
| 1. Problem formulation | Can a physics-informed neural network infer the unknown potential \(V(x)\) and its associated eigenfunctions along from noisy spectral and probability-density observations, and if not, which physical structures remain identifiable? | ✔️ |
| 2. Data collection & curation | - Uniform collocation grid of spatial points $x\in [-5,5]. - Noisy energy and probability density observations. |
✔️ |
| 3. Neural architecture | Coupled operator architecture with weighted wavefunction normalization, sequential Gram-Schmidt orthogonalization, and shared-potential eigenstate constraints | ⚠️ Both neural networks are scalar-in, scalar-out, fully differentiable, and deliberately kept shallow to preserve interpretability. |
| 4. Loss function | Composite loss with physics, smoothness, ordering, and data-consistency terms defined over a weighted discrete Hilbert-space geometry. | ✔️ |
| 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 (step 3️⃣)
The neural architecture jointly parameterizes:
for \(n=0,1,2\) using lightweight differentiable neural networks.
Note on Learned Wavefunctions
The learned eigenfunctions \(\psi_n^\theta(x)\) are not freely learned fields. They are constrained by multiple coupled structures:
- The time-independent Schrödinger equation (TISE).
- Weighted \(L^2\) normalization.
- Sequential Grahm-Schmidt orthogonalization.
- Shared dependence on the learned potential \(V_\theta(x)\).
Consequently, the architecture behaves as a constrained operator-eigenfunction learning system rather than a collection of independently learned functions.
- A central difference stencil is used to approximate the \(\partial_{xx}\) operator.
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In the training loop, orthonormalization (via Gram-Schmidt and \(L^2\) inner product with trapezoidal weighting) is performed before the loss function is calculated. Thus, the Schrödinger residual is computed using orthonormalized eigenfunctions,
$$ \hat{H}_\theta \hat{\psi}_n^\theta \approx E_n^\theta \hat{\psi}_n^\theta \, ,$$ and the deviation from this eigenvalue equation is minimized during training. ✨
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Consistent quadrature weighting throughout training and POD analysis ensures that the learned geometry and the diagnostic geometry are defined with respect to the same inner product.
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Proper orthogonal decomposition (POD) is used as a diagnostic probe of learned basis geometry (rather than a computational tool for dimension reduction).
♾️ Note on Ill-posedness
The inverse Schrödinger problem is fundamentally ill-posed: multiple potentials may reproduce nearly identical spectral measurements and probability-density observations.
Thus, exact recovery of the ground-truth potential is generally impossible from limited and noisy data alone.
The central question of Project 2 is therefore: Which potential structures of the underlying Hamiltonian remain stable and identifiable despite non-uniqueness, discretization error, and measurement noise?