Stacked Architecture

examples/stacked_architecture.py builds a graph-level classifier with three SILVA equilibrium layers, mixed solvers, and one custom local branch.

python examples/stacked_architecture.py

Equation

The stack computes

\[ z_1^\star=f_{\theta_1}(z_1^\star,x), \qquad z_2^\star=f_{\theta_2}(z_2^\star,z_1^\star), \qquad z_3^\star=f_{\theta_3}(z_3^\star,z_2^\star). \]

For graph classification, node states are pooled:

\[ h_g = \frac{1}{|\mathcal V_g|} \sum_{i\in\mathcal V_g}z_{3,i}^\star, \qquad \hat y_g=R_\phi(h_g). \]

Model

The example uses three hidden widths and three solver configurations:

model = SILVAGraphNetwork(
    in_dim=6,
    hidden_dims=[16, 16, 12],
    out_dim=2,
    task="graph",
    pooling="mean",
    config=[
        SolverConfig(solver="picard", max_iter=8, alpha=0.5),
        SolverConfig(solver="anderson", max_iter=8, alpha=0.5, history=3),
        SolverConfig(solver="broyden", max_iter=8, alpha=0.5),
    ],
    local=lambda dim, index: SignedLocal(dim) if index == 1 else "graph",
    global_term="mean",
)

The second local branch is replaced by a custom module:

\[ L_\psi(Z)_i = \sum_{j\in\mathcal N(i)} W_\psi z_j. \]

The first and third layers keep the built-in graph local branch.

Device Handling

The script selects the available PyTorch device:

device = resolve_device("auto")
batch = move_to_device(batch, device)
model = model.to(device)

This is the same path used for CPU, CUDA, and MPS validation. The printed solvers list confirms that every equilibrium layer used its configured solver.

The input has shape (entities, 6). The three solved states have hidden widths 16, 16, and 12; mean pooling then produces one (graphs, 12) matrix for the classifier. Inspect all three residual trajectories and convergence flags, because the final loss does not establish convergence of an earlier point.

See Stacking, Solvers, and Devices for the full contract and Solvers and Linear Algebra for the numerical-method sources.

Complete Worked Study

The short construction above identifies the main API. A complete study must also distinguish the state equation, task objective, numerical residual, gradient path, and scale transfer. In this example, the equilibrium state is one image tensor per resolution or linked SILVA point, the condition is image features and per-scale source injections, and the repeated map is shape-preserving convolutional, U-Net, attention, or multiscale fusion blocks.

Derivation From Transition to Reported Result

The forward solve is defined by

\[ z^\star = T_\theta(z^\star,x). \]

The task output and task objective are separate from convergence:

\[ \widehat y = R_\phi(z^\star), \qquad \mathcal L_{\mathrm{task}}=\ell(\widehat y,y). \]

For a computed state \(z_K\), the normalized fixed-point residual is

\[ r_K = \frac{\lVert T_\theta(z_K,x)-z_K\rVert_2} {\lVert z_K\rVert_2+\varepsilon}. \]

A small task loss does not imply a small \(r_K\), and a small \(r_K\) does not establish task quality. Both belong in the result. For implicit training, the parameter sensitivity follows

\[ \frac{\mathrm d z^\star}{\mathrm d\theta} = \left(I-\partial_z T_\theta(z^\star,x)\right)^{-1} \partial_\theta T_\theta(z^\star,x). \]

This is why the example checks gradients in addition to forward convergence. The reader-facing evidence for this route is logit shape, pointwise solvers, loss, and gradient flow across the stack. The invariants that must remain true are channel/spatial shape at every scale and deterministic fusion.

Run the Complete Example

python examples/stacked_architecture.py

Measured Compact Output

The following output was produced by the executable program in the current repository. Floating-point values may vary slightly across devices and library builds, while shapes, finite values, invariants, and declared tolerances must remain stable.

device cpu
logits_shape (2, 2)
solvers ['picard', 'anderson', 'broyden']
final_loss 0.586849570274353

Interpret the Output

Evidence	What it answers	What would require investigation
Tensor shapes	Did every source, state, branch, and readout preserve its declared contract?	A changed entity, channel, token, or spatial dimension
Task metric	Did the compact task execute and produce finite evidence?	Non-finite loss, a missing mask, or a metric computed on the wrong split
Fixed-point residual	Did the returned state satisfy the repeated transition to the requested tolerance?	A residual plateau, rising trajectory, or convergence flag inconsistent with the value
Iteration or trajectory data	How much numerical work was required?	Solver effort that grows sharply under a small input or resolution change
Gradient evidence	Can the loss reach every trainable component through the selected backward mode?	Missing, non-finite, or implausibly large gradients
Domain invariant	Did the method retain positivity, feasibility, boundary values, permutation behavior, or another structural requirement?	A task metric that looks acceptable while the structural contract fails

The compact output is a mechanism check, not a paper-scale benchmark claim. It shows that data enter the intended construction, the transition executes, the solver returns diagnostics, and differentiation reaches trainable parameters.

Add a Solver and Scale Sweep

The next run should hold model parameters and data fixed while changing one numerical control at a time. A complete experiment record can use this schema:

experiment:
  example: stacked-architecture
  state: one image tensor per resolution or linked SILVA point
  condition: image features and per-scale source injections
  repeated_transition: shape-preserving convolutional, U-Net, attention, or multiscale fusion blocks
  invariant_checks: channel/spatial shape at every scale and deterministic fusion
  compact_evidence: logit shape, pointwise solvers, loss, and gradient flow across the stack
  scale_axes: image resolution, channels, scales, internal depth, and batch size
solver_sweep:
  methods: [picard, anderson, broyden]
  tolerances: [1.0e-4, 1.0e-6, 1.0e-8]
  maximum_iterations: [25, 50, 100]
report:
  - task_metric
  - fixed_point_residual
  - backward_linear_residual
  - iterations
  - wall_time
  - peak_memory
  - gradient_norm

At full scale, move toward the complete stacked architecture with independently budgeted points. Increase only one of image resolution, channels, scales, internal depth, and batch size at a time. Retain this compact run as a regression test, preserve the source split and preprocessing receipt, archive the resolved configuration and checkpoint, and report convergence failures rather than discarding them.

Where to Go Next

Question	Page
How do multiple fixed points differ from depth inside one point?	Stacking and Devices
How are heterogeneous points linked?	Cortex Hierarchies
Which architecture containers are public?	Architectures API