Project 2 Architecture🪶
Conceptual Overview🪶
Project 2 employs a coupled physics-informed architecture that jointly learns: $$ V_\theta(x) \, , \qquad \psi_n^\theta(x) \, , \qquad E_n^\theta \,. $$
This results in a model that behaves as a constrained operator-learning system whose goal is to identify physically consistent Hamiltonian structure rather than simply interpolate observed data.
Overall Architecture🪶
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%% ==============================
%% NODES
%% ==============================
potential(["$$V_\theta$$"])
psi0_raw(["$$ \psi_0^\theta $$"]):::psiraw
psi1_raw(["$$\psi_1^\theta$$"]):::psiraw
psi2_raw(["$$\psi_2^\theta$$"]):::psiraw
psi0(["$$\hat{\psi}_0^\theta$$"]):::psinorm
psi1(["$$\hat{\psi}_1^\theta$$"]):::psinorm
psi2(["$$\hat{\psi}_2^\theta$$"]):::psinorm
E0(["$$E_0^\theta$$"]):::energy
E1(["$$E_1^\theta$$"]):::energy
E2(["$$E_2^\theta$$"]):::energy
%% ==============================
%% PIPELINE
%% ==============================
subgraph PIML["PIML Framework"]
direction LR
subgraph synthetic_data["1️⃣ Synthetic Data"]
direction LR
subgraph domain_norm["Uniform Grid"]
spatial_grid["$$x\in[-5, 5]$$"]:::spatial_stencil
deltax["$$\Delta x$$"]:::spatial_stencil
end
trap["Trapezoidal weights"]
observed_data("$$\rho_n^\text{obs}, \ E_n^\text{obs}$$")
end
subgraph neural_ansatz["2️⃣ Neural Ansatz"]
direction TB
subgraph energy_init["Linear Energy Initialization"]
E_init["$$E_n = \text{linspace}(0.5, n-0.5, n)$$"]:::energy
end
subgraph energy_eigenvalues["Energy Eigenvalues"]
E0
E1
E2
end
E_init --> energy_eigenvalues
subgraph PINN["2️⃣ PINN"]
direction LR
subgraph raw_wavefunctions["Raw Learned Wavefunctions"]
direction TB
psi0_raw
psi1_raw
psi2_raw
end
MLP1("MLP"):::MLP --> potential
MLP2("MLP"):::MLP --> psi0_raw
MLP3("MLP"):::MLP --> psi1_raw
MLP4("MLP"):::MLP --> psi2_raw
end
end
subgraph normalization["3️⃣A Orthonormalization"]
direction TB
apply_trapezoidal_weights["$$\int|\psi_n^\theta |^2 dx$$"]:::norm
gs["GS+re-norm"]:::norm
eps("$$+\epsilon \ \text{stability}$$"):::norm
end
subgraph normalized_wavefunctions["Normalized Wavefunctions"]
direction TB
psi0
psi1
psi2
end
subgraph finite_difference["3️⃣B Finite difference"]
direction TB
eigenfunction_stencil("$$\frac{\partial^2}{dx^2}\hat{\psi}_n^\theta$$"):::stencil
potential_stencil("$$V_\theta''(x)$$"):::stencil
end
subgraph residual["4️⃣ Physics Residual"]
direction LR
R["$$R_n(x)=-\frac{1}{2}\nabla^2\hat{\psi} + (V-E)\hat{\psi}$$"]:::physics
end
style R stroke-width:5px;
subgraph loss["5️⃣ Total Loss"]
direction TB
Lp["Physics loss"]:::loss_terms
Ln["Order"]:::loss_terms
Ls["Smoothness"]:::loss_terms
Ld["Data mismatch"]:::loss_terms
end
opt["6️⃣ Optimizer (Adam)"]:::opt
%% FLOW
spatial_grid --> MLP1 & MLP2 & MLP3 & MLP4
deltax & trap --> apply_trapezoidal_weights
psi0_raw & psi1_raw & psi2_raw --> gs
apply_trapezoidal_weights & eps --> gs
gs --> psi0 & psi1 & psi2
psi0 & psi1 & psi2 --> eigenfunction_stencil & R
potential --> potential_stencil & R
eigenfunction_stencil --> R
E0 & E1 & E2 --> R & Ln & Ld
R --> Lp
potential & potential_stencil --> Ls
psi0 & psi1 & psi2 --> Ls & Ld
observed_data --> Ld
Lp & Ln & Ls & Ld --> opt
opt ==> MLP1
opt ==> MLP2
opt ==> MLP3
opt ==> MLP4
opt ==> E0
opt ==> E1
opt ==> E2
end
%% ==============================
%% DIAGNOSTICS
%% ==============================
subgraph diagnostics["7️⃣ Diagnostics"]
direction TB
sanity["sanity checks"]:::diag
pod["7️⃣ POD analysis"]:::pod
sanity ~~~ pod
end
step0["0️⃣ Define TISE dynamics"]:::step0 --> PIML
psi0 & psi1 & psi2 --> sanity & pod
E0 & E1 & E2 --> sanity
potential --> sanity
deltax & trap --> pod
%% ==============================
%% LINKS
%% ==============================
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End-to-end PIML pipeline
The diagram illustrates the physics-informed machine learning pipeline for solving the Schrödinger equation implemented in project 2.
| Step # | Descripition |
|---|---|
| 0️⃣ | Definition of the physical system (TISE). |
| 1️⃣ | Synthetic data generation on a uniform grid with trapezoidal weights. |
| 2️⃣ | Neual ansatz comprising MLPs for the potential \(V_\theta\) and wavefunctions \(\psi_n^\theta\), alongside trainable energy eigenvalues \(E_n^\theta\). |
| 3️⃣ | Wavefunction orthonormalization via Gram-Schmidt and finite-difference derivative calculation. |
| 4️⃣ | Calculation of the physics residual \(R_n(x)\) where the governing law is enforced. |
| 5️⃣ | Total loss computation combining physics, smoothness, and data mismatch terms. |
| 6️⃣ | Optimization via Adam, feeding back into the trainable parameters (thick arrows). |
| 7️⃣ | Parallel analysis including sanity checks and POD. |
POD Diagnostics🪶
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%% ==============================
%% INPUT BLOCKS
%% ==============================
subgraph synthetic_data["1️⃣ Synthetic Data"]
direction LR
subgraph domain_norm["Uniform Grid"]
spatial_grid["$$x \in [-5, 5]$$"]:::spatial_stencil
deltax["$$\Delta x$$"]:::spatial_stencil
end
trap["Trapezoidal weights"]
end
subgraph normalized_wavefunctions["Normalized Wavefunctions"]
direction TB
psi0(["$$\hat{\psi}_0^\theta$$"]):::psinorm
psi1(["$$\hat{\psi}_1^\theta$$"]):::psinorm
psi2(["$$\hat{\psi}_2^\theta$$"]):::psinorm
end
%% ==============================
%% POD DIAGRAM
%% ==============================
subgraph pod["7️⃣ POD analysis"]
PsiMat("Snapshot matrix construction<br/>$$\boldsymbol{\Psi}^\theta = [\hat{\psi}_0, \hat{\psi}_1, \hat{\psi}_2]$$")
trap_pod("L² spatial weighting<br/>$$\boldsymbol{\Psi}_w = \sqrt{w \Delta x} \odot \boldsymbol{\Psi}$$")
SVD("Euclidean SVD<br/>$$\boldsymbol{\Psi}_w = U \Sigma V^T$$"):::diag
podscale("Physical mode recovery<br/>$$u_k^{\text{phys}} = u_k / \sqrt{w \Delta x}$$"):::spatial_modes
align("Phase/Sign Alignment<br/>(Relative to Ground Truth)"):::spatial_modes
spec("Singular values spectrum<br/>$$\sigma_k$$"):::diag
modes("Physical Basis Functions<br/>$$u_k^{\text{phys}}$$"):::spatial_modes
overlaps("Overlap matrix $$C_{kn}$$<br/>$$\langle u_k^{\text{phys}}, \hat{\psi}_n \rangle$$"):::spatial_modes
temporal_scaling("Temporal mode projection<br/>$$V^T$$"):::temporal_modes
temporal_alignment("Global $$U(1)$$ Phase Alignment"):::temporal_modes
temporal_mode_heatmap("Composition Heatmap<br/>Matrix $$V_{nk}$$"):::temporal_modes
temporal_overlap("Orthogonality Check<br/>$$\langle v_m, v_n \rangle = \delta_{mn}$$"):::temporal_modes
cross_temporal("Cross-State Projections<br/>$$|\langle \mathbf{e}_n, v_k \rangle|$$"):::temporal_modes
%% POD Companion Panels
PsiMat --> trap_pod
trap_pod --> SVD
SVD --> spec
SVD --> podscale
SVD --> temporal_scaling
podscale --> align
align --> modes
PsiMat --> overlaps
temporal_scaling --> temporal_alignment
temporal_alignment --> temporal_mode_heatmap
temporal_alignment --> temporal_overlap
temporal_alignment --> cross_temporal
PsiMat --> cross_temporal
end
%% ==============================
%% CONNECTIONS
%% ==============================
psi0 --> PsiMat
psi1 --> PsiMat
psi2 --> PsiMat
deltax --> trap_pod
trap --> trap_pod
%% ==============================
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%% ==============================
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✅ To Do
- Make sure \(\sqrt{w_i\Delta x}\) weighting is properly applied at all relevant steps.
Proper Orthogonal Decomposition (POD) Diagnostics
This figure details the weighted POD pipeline used for diagnostic verification.
- Snapshot Matrix: Formed from normalized wavefunctions.
- Weighting: Scaling with \(\sqrt{w_i \Delta x}\) ensures the L2 inner product maps to a Euclidean dot product for SVD.
- Physical Scaling: Rescaling SVD modes back to physical space.
-
Overlaps: Calculation of the overlap matrix \(C_{kn}\) to assess mode orthogonality and alignment with learned states.
-
Write the mathematical expression for the inner product with the subscripts used.
\[\langle f, g \rangle_{\Delta x, w} = \sum_i w_i f(x_i)^* g(x_i) \Delta x\]