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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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%% ==============================
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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
%% ==============================
linkStyle default stroke:#4CC9F0,stroke-width:1.618px,opacity:0.6

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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%% ==============================
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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/>$$\mathbf{\Psi}^\theta = [\hat{\psi}_0, \hat{\psi}_1, \hat{\psi}_2]$$")
    trap_pod("L² spatial weighting<br/>$$\mathbf{\Psi}_w = \sqrt{w \Delta x} \odot \mathbf{\Psi}$$")
    SVD("Euclidean SVD<br/>$$\mathbf{\Psi}_w = U \Sigma V^T$$"):::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 $$\sigma_k$$"):::diag
    modes("Physical Basis Functions $$u_k^\text{phys}$$"):::spatial_modes
    overlaps("Overlap matrix $$C_{kn}$$<br/>$$\langle u_k^\text{phys}, \hat{\psi}_n \rangle$$"):::spatial_modes
    temporal_scaling("Temporal mode projection from $$V^T$$"):::temporal_modes
    temporal_alignment("Global U(1) Phase Alignment"):::temporal_modes
    temporal_mode_heatmap("Composition Heatmap (Matrix $$V_{nk}$$)"):::temporal_modes
    temporal_overlap("Orthogonality Check $$\langle v_m, v_n \rangle = \delta_{mn}$$"):::temporal_modes
    cross_temporal("Cross-State Projections $$|\langle \mathbf{e}_n, v_k \rangle|$$"):::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

%% ==============================
%% LINKS (subtle)
%% ==============================
linkStyle default stroke:#4CC9F0,stroke-width:1.618px,opacity:0.6

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.

\[\langle f, g \rangle_{\Delta x, w} = \sum_i w_i f(x_i)^* g(x_i) \Delta x\]