DEQ Engine Bridge

Run:

python examples/deq_engine_bridge.py

This example uses the package-native DEQ engine for a compact fixed-point system. It is the smallest bridge from a user-defined transition to the general silva_deq interface.

Equation

The state is

\[ z\in\mathbb R^{4\times 5}, \qquad x\in\mathbb R^{4\times 3}. \]

The transition is

\[ f_\theta(z,x) = \tanh(W_xx+W_zz). \]

The engine solves

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

In code:

def transition(z):
    return torch.tanh(input_proj(x) + state_proj(z))

result = silva_deq(
    transition,
    z0,
    config=SILVADEQConfig(forward_solver="anderson", forward_max_iter=8, alpha=0.7),
    return_result=True,
)

What to Inspect

The printed dictionary reports:

Field	Meaning
`device`	resolved CPU, CUDA, or MPS device
`state_shape`	equilibrium state tensor shape
`iterations`	solver iterations used
`residual`	final fixed-point residual
`residual_ratio`	final residual divided by initial residual
`has_grad`	whether gradients reached `input_proj`

Why It Matters

This example demonstrates the general contract:

\[ \text{user transition} \quad\to\quad \text{fixed-point solve} \quad\to\quad \text{diagnostics} \quad\to\quad \text{PyTorch gradients}. \]

Use DEQ Engine API for the full engine object map.

Citations

Cite the SILVA package [2] for this implementation, Deep Equilibrium Models [4] for the fixed-point framing, and TorchDEQ [35] when discussing the general DEQ-engine interface lineage.

Direct links and BibTeX keys are collected in Equilibrium and Implicit Layers.

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 the latent vector or tensor z, the condition is the injected observation x, and the repeated map is the tied map f_theta(z, x).

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 state shape, iterations, residual ratio, and gradient availability. The invariants that must remain true are state shape and a decreasing or bounded residual.

Run the Complete Example

python examples/deq_engine_bridge.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', 'state_shape': (4, 5), 'iterations': 8, 'residual': 0.0011139592388644814, 'residual_ratio': 0.0005245877954072668, 'has_grad': True}

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: deq-engine-bridge
  state: the latent vector or tensor z
  condition: the injected observation x
  repeated_transition: the tied map f_theta(z, x)
  invariant_checks: state shape and a decreasing or bounded residual
  compact_evidence: state shape, iterations, residual ratio, and gradient availability
  scale_axes: latent width, solver tolerance, and iteration budget
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 chosen implicit model with forward and backward solver sweeps. Increase only one of latent width, solver tolerance, and iteration budget 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 does the general engine connect to SILVA and optical flow?	DEQ Engine and Optical Flow
Which engine state contracts are public?	DEQ Engine API
How does exact implicit backward work?	Implicit Backward Guide