Project 2: Low-Fidelity Inverse Schrödinger Problem🪶

Overview

🥅 Goal🗝️ Key Points:eigenote: Physical Structure

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

5️⃣ 5 Brunton Steps🌍 Global Design Choices🔢 Numerical Methods

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

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        A("️1️⃣ Define known TISE <br> dynamics for <br> 1-D Harmonic <br> Oscillator") --> B("2️⃣ Generate <br> noisy <br> observations")
        B --> C("3️⃣ Learn potential <br> and <br> associated <br> eigenstates")
        C --> D("4️⃣ Constrain with <br> physics residual")
        D --> E("5️⃣ Regularize with <br> smoothness")
        E --> F("6️⃣ Analyze learned <br> geometry via POD")
        F --> G("7️⃣ Determine what <br> information remains <br> identifiable?")

Conceptual description of what the model is learning (step 3️⃣)

The neural architecture jointly parameterizes:

\[V_\theta(x)\, , \quad \psi_n^\theta(x) \, , \quad E_n^\theta\]

for $n=0,1,2$ using lightweight differentiable neural networks.

The eigenfunctions $\psi_n^\theta(x)$ are not freely learned fields.

The learned eigenstates 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.
In the training loop, orthonormalization (via Gram-Schmidt and l2 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. ✨
- Sequential Gram-Schmidt orthogonalization ensures that the learned eigenfunctions satisfy orthogonality requirements of Hamiltonian eigenstates.
- The trapezoidal rule provides a discrete approximation to the continuous Hilbert-space inner product on the uniform spatial grid.
- 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.

Note on POD usage in Project 2

Proper orthogonal decomposition (POD) is treated as a diagnostic probe of learned basis geometry rather as 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?