silva_recurrent_equilibrium_network Reproduction Dossier

dynamic recurrent model with per-step algebraic equilibria. This dossier connects the source mechanism to its SILVA implementation, compact evidence, replaceable components, and source-scale route. Existing tests and notebooks remain the executable authority.

Evidence boundary

The mechanism is compact-verified in the package suite. The final source-scale stage remains planned until the cited data, complete optimization budget, checkpoints, and evaluation protocol have actually run.

Identity and Sources

Field	Value
Domain	core equilibrium
Task contract	B,T,D sequence and optional initial state -> state, algebraic, and output trajectories
Source relation	paper-adaptation
References	[109]
Repositories	https://arxiv.org/abs/2104.05942
Editable scale plan	experiments/reproduction/configs/silva_recurrent_equilibrium_network.json

Governing Equation

The domain-level state contract is

\[ z^\star=T_\theta(z^\star;x),\qquad y=Q_\psi(z^\star). \]

The implementation registry specializes it operationally as

w_t_star=phi(D11 w_t_star+C1 x_t+D12 u_t); x_(t+1)=A x_t+B1 w_t_star+B2 u_t

Define the root residual

\[ R_\theta(z;x)=z-T_\theta(z;x). \]

At a regular equilibrium, differentiating \(R_\theta(z^\star;x)=0\) gives

\[ \frac{\partial z^\star}{\partial x} = \left(I-\frac{\partial T_\theta}{\partial z}\right)^{-1} \frac{\partial T_\theta}{\partial x}. \]

This identity explains why the forward residual, the conditioning derivative, and the adjoint linear solve must be diagnosed separately from the task metric.

What Is Preserved

• dynamic state recurrence with an algebraic equilibrium at each time index
• bounded algebraic map and independently inspectable state/equilibrium/output trajectories

What Can Be Replaced

Each item below is an explicit control rather than an undocumented modification:

• replace the algebraic map, state update, readout, or initialization
• use structured linear dynamics and custom monotone equilibrium nonlinearities
• sequence length
• dynamic state width
• algebraic width
• per-step tolerance

Constructor and Shape Contract

silva_recurrent_equilibrium_network(input_dim: 'int', state_dim: 'int', equilibrium_dim: 'int', output_dim: 'int', *, contraction: 'float' = 0.7, state_decay: 'float' = 0.8, config: 'SolverConfig | None' = None) -> 'None'

The transition must preserve the declared equilibrium-state shape even when the encoder, branch operators, constraints, solver, and readout are replaced. Test the transition by itself before testing the complete root solve.

Progressive Experiment Ladder

1. Equation and tensor contract

Objective: Make the state, conditioning variables, operator, and readout explicit.

Procedure:

• Write and evaluate the family equation: w_t_star=phi(D11 w_t_star+C1 x_t+D12 u_t); x_(t+1)=A x_t+B1 w_t_star+B2 u_t
• Declare every tensor axis, boundary, mask, graph, or physical unit.
• Check the transition output has exactly the same shape as the equilibrium state.

Acceptance checks:

• finite transition values
• shape-preserving state update
• all conditioning variables affect the intended branch

Evidence target: contract-verified.

2. Primitive mechanism reconstruction

Objective: Build the retained source mechanism from replaceable modules.

Procedure:

• dynamic state recurrence with an algebraic equilibrium at each time index
• bounded algebraic map and independently inspectable state/equilibrium/output trajectories

Acceptance checks:

• primitive modules expose trainable parameters and gradients
• mechanism-specific invariance or constraint check passes
• direct transition evaluation is deterministic under a fixed seed

Evidence target: compact-verified.

3. Public abstraction equivalence

Objective: Verify that the assembled family evaluates the same transition as its primitives.

Procedure:

• Copy the primitive module parameters into the public family constructor.
• Evaluate one transition and one complete equilibrium with identical inputs.
• Compare outputs, residuals, and parameter gradients with declared tolerances.

Acceptance checks:

• transition outputs agree
• equilibrium residual is finite and decreases
• primitive and assembled gradients agree on the compact case

Evidence target: compact-verified.

4. Compact real or analytic task

Objective: Exercise training, evaluation, diagnostics, and serialization end to end.

Procedure:

• Use the cited notebook's deterministic compact fixture before replacing it with source-scale data.

Acceptance checks:

• record task metric
• record fixed-point residual
• record gradient agreement
• record runtime
• checkpoint reload reproduces the recorded prediction
• result record contains data and configuration fingerprints

Evidence target: compact-verified.

5. Official-data subset

Objective: Validate the complete source data path before spending the full budget.

Procedure:

• Acquire the cited data and preserve its official split, preprocessing, units, and metric.
• Freeze preprocessing, split logic, metric code, and checkpoint format.
• Run a deterministic subset large enough to expose batching and memory failures.

Acceptance checks:

• dataset receipt and checksum are stored
• resume and evaluation paths reproduce the same subset metric
• memory and runtime are measured rather than estimated

Evidence target: subset-verified.

6. Source-scale reproduction or declared extension

Objective: Run the cited protocol, or change it explicitly as a SILVA extension.

Procedure:

• Acquire the cited data and preserve its official split, preprocessing, units, and metric.
• Build the same SILVA family with source-aligned task modules and scale controls.
• Run forward, loss, backward, checkpoint resume, and metric validation on a small shard before the complete experiment.
• sequence dataset, sampling interval, state normalization, initialization, horizon, optimizer, solver, seeds, rollout metric, stability, runtime, and memory

Acceptance checks:

• all required artifacts are archived
• reported metrics use the cited evaluation protocol
• every architectural or training deviation is listed
• claims match the achieved evidence status

Evidence target: planned.

Data, Access, and Storage

Candidate datasets:

• system identification
• nonlinear dynamics
• sequence prediction

Authoritative routes:

• https://arxiv.org/abs/2104.05942

Access obligations:

• Follow the cited repository and dataset terms, then record source revisions and archive checksums.

Storage planning:

• Measure one processed sample, estimate the complete split, and budget raw data, processed shards, checkpoints, optimizer state, and diagnostics separately.

Preprocessing record:

• record dataset version, split, normalization, shape convention, and seed
• preserve masks, graph indices, boundaries, or physical units required by the domain

Metrics and Current Evidence

Required metrics:

• task metric
• fixed-point residual
• gradient agreement
• runtime
• memory

This family is verified through its listed mechanism tests and executed notebook. It is not included in a same-task comparison when another family does not share its input, state, output, and loss contract. The absence of a comparison row is therefore a scope decision, not missing implementation evidence.

Executed notebook paths:

• notebooks/package_api/71_silva_recurrent_equilibrium_network.ipynb

Mechanism tests:

• tests/test_source_equilibria.py

Compact Defaults

Option	Value
tier	'smoke'
config	SolverConfig(solver='anderson', max_iter=12, tol=1e-05, alpha=1.0, history=3, ridge=0.0001, beta=1.0, stop_mode='relative', relative_eps=1e-08, anderson_batch_dims=1, track_residuals=True, reengage=True, backward_mode='implicit', backward_solver='gmres', backward_max_iter=20, backward_tol=1e-05, backward_stop_mode='relative', backward_relative_eps=1e-08, phantom_steps=1, phantom_tau=1.0, neumann_terms=5, shine_refine_steps=0, indexing=(), return_best=True)

Full Defaults

Option	Value
tier	'full'
config	SolverConfig(solver='anderson', max_iter=60, tol=1e-05, alpha=1.0, history=6, ridge=0.0001, beta=1.0, stop_mode='relative', relative_eps=1e-08, anderson_batch_dims=1, track_residuals=True, reengage=True, backward_mode='implicit', backward_solver='gmres', backward_max_iter=80, backward_tol=1e-05, backward_stop_mode='relative', backward_relative_eps=1e-08, phantom_steps=1, phantom_tau=1.0, neumann_terms=5, shine_refine_steps=0, indexing=(), return_best=True)

Defaults establish a starting budget; the cited source protocol takes precedence whenever reproduction is the claim.

Source-Scale Checklist

• Acquire the cited data and preserve its official split, preprocessing, units, and metric.
• Build the same SILVA family with source-aligned task modules and scale controls.
• Run forward, loss, backward, checkpoint resume, and metric validation on a small shard before the complete experiment.

Benchmark-specific requirements:

• sequence dataset, sampling interval, state normalization, initialization, horizon, optimizer, solver, seeds, rollout metric, stability, runtime, and memory

Required archived artifacts:

• machine-readable model and solver configuration
• dataset receipt with source revision, split, license, and checksum
• preprocessing and normalization record
• seeded training and evaluation log
• checkpoint and optimizer-resume state for trained experiments
• task metrics and equilibrium diagnostics in a machine-readable result
• runtime, peak-memory, device, precision, and dependency record
• declared deviations from the cited protocol

Reporting Rule

Report the achieved evidence status, not the intended one. A compact or subset run may validate the implementation and data path, but only a completed cited protocol supports a source-scale reproduction statement. Modified operators are valuable SILVA extensions when every deviation is named and measured.

Where to Go Next

Question	Page
Where are all family dossiers?	Family Dossier Index
How is a custom family assembled?	Advanced Extension Handbook
How are experiment stages represented in the API?	Research-Depth API
Which lab inspects every dossier?	Family Dossier Lab