Source-Aligned Equilibrium Families

This module contains fourteen independently implemented SILVA families that preserve a distinct mechanism from a published equilibrium architecture. Every family exposes its transition, accepts replaceable internal modules where the method permits them, and uses the shared SILVA solver configuration unless the published construction is an equation-residual method rather than a hidden-state root solve.

The compact implementations establish equations, tensor contracts, gradients, constraints, and extension points. Published benchmark numbers require the source dataset, preprocessing, model width, training schedule, checkpoint, and evaluation protocol listed in the corresponding reproduction dossier.

Family	Public class	Solved object	Primary reference
Lipschitz MDEQ	SILVALipschitzMultiscaleEquilibrium	packed multiscale state	[99]
SubDEQ	SILVASubhomogeneousEquilibrium	positive normalized state	[100]
algorithmic reasoner	SILVAAlgorithmicReasoner	graph processor state	[101]
DEQH	SILVAHamiltonianEquilibrium	symmetric Hamiltonian	[102]
inverse imaging	SILVAInverseImagingEquilibrium	reconstructed image	[103]
snapshot compressive imaging	SILVASnapshotCompressiveEquilibrium	video volume	[104]
magnetic-particle imaging	SILVAMagneticParticleEquilibrium	primal/split/dual state	[105]
hyperspectral sparse representation	SILVASparseHyperspectralEquilibrium	sparse code and cube	[106]
serialized smoothing	SILVASerializedSmoothingEquilibrium	noisy-sample equilibria and certificate	[107]
diffusion restoration	SILVADiffusionRestorationEquilibrium	joint reverse trajectory	[108]
recurrent equilibrium network	SILVARecurrentEquilibriumNetwork	algebraic state per time step	[109]
Lipschitz robust equilibrium	SILVALipschitzRobustEquilibrium	logits, bound, and radius	[110]
image matting	SILVAImageMattingEquilibrium	trimap-constrained alpha matte	[111]
dynamic economic equilibrium	SILVADynamicEconomicEquilibrium	feasible policy functions	[112]

Common Result Contract

Most families return a tensor by default and a SILVASourceEquilibriumResult when return_result=True. The expanded result contains the solved state, task output, and complete SolverResult, including residual history and termination information. Specialized results retain additional quantities such as smoothing certificates, recurrent trajectories, robust radii, and economic residuals.

from silva_networks import (
    SILVALipschitzMultiscaleEquilibrium,
    SolverConfig,
)

model = SILVALipschitzMultiscaleEquilibrium(
    input_dim=32,
    scale_dims=(64, 32, 16),
    output_dim=10,
    contraction=0.8,
    config=SolverConfig(
        solver="anderson",
        max_iter=40,
        tol=1e-5,
        backward_mode="implicit",
        backward_solver="gmres",
    ),
)
result = model(features, return_result=True)
scale_states = model.split_state(result.state)

Replaceable Internals

Replacement modules must preserve the documented shape and mathematical contract. For example, inverse imaging accepts any differentiable forward and adjoint pair, snapshot imaging accepts any shape-preserving volumetric prior, and Hamiltonian prediction accepts an invariant or equivariant interaction backbone that returns a square atom-orbital matrix.

model = SILVAInverseImagingEquilibrium(
    channels=2,
    forward_operator=undersampled_fourier,
    adjoint_operator=undersampled_fourier_adjoint,
    prior=multiscale_image_prior,
    step_size=0.2,
    config=solver_config,
)

The Source-Aligned Family Deep Dive derives every transition, gives compact and source-scale routes, and links each family to its executable notebook and reproduction dossier.

API

silva_networks.source_equilibria

Article-backed equilibrium mechanisms adapted to configurable SILVA contracts.

The implementations in this module keep the published mechanism visible while making the transition, physical operator, numerical solver, and readout replaceable. Compact defaults support deterministic tests and teaching; larger experiments can provide the source architecture and data operators directly.

SILVASourceEquilibriumResult dataclass

Common state, output, and numerical record for a source-oriented family.

SILVALipschitzMultiscaleEquilibrium

Bases: Module

Coupled multiscale equilibrium with an explicit contraction bound.

With the states concatenated into z, the transition is

\[ z^\star=\tanh\!\left(S_\theta(x)+\widehat W z^\star\right), \qquad \|\widehat W\|_\infty\leq\rho<1. \]

Splitting the state after the solve exposes every resolution branch while a single normalized recurrent map accounts for all cross-scale communication.

SILVASubhomogeneousEquilibrium

Bases: Module

Positive normalized SubDEQ with configurable subhomogeneity degree.

The default follows the translated-tanh construction

\[ z^\star=\operatorname{norm}_{p}\!\left( [\tanh(Wz^\star)+f_\theta(x)+a]^{q}\right), \qquad a>1,\quad 0<q\leq1. \]

SILVAAlgorithmicReasoner

Bases: Module

Graph equilibrium reasoner whose solved algorithm state is a fixed point.

SILVARadialHamiltonian

Bases: Module

Rotation-invariant pair interaction used by the compact Hamiltonian family.

SILVAHamiltonianEquilibrium

Bases: Module

Self-consistent Hamiltonian equilibrium with a replaceable equivariant backbone.

SILVAResidualImagePrior

Bases: Module

Small residual prior used by compact inverse-imaging examples.

SILVAInverseImagingEquilibrium

Bases: Module

Known-forward-model reconstruction with a learned equilibrium prior.

\[x^\star=D_\theta\!\left(x^\star-eta A^\top(Ax^\star-y)\right).\]

SILVASnapshotCompressiveEquilibrium

Bases: Module

Video snapshot-compressive equilibrium with analytic data consistency.

SILVAMagneticParticleEquilibrium

Bases: Module

ADMM-style magnetic-particle reconstruction with learned consistency.

SILVASparseHyperspectralEquilibrium

Bases: Module

Sparse-code equilibrium with data consistency and a learned cube prior.

SILVASmoothingCertificate dataclass

Predicted classes, probability lower bounds, radii, and sample counts.

SILVASerializedSmoothingEquilibrium

Bases: Module

Serialized randomized smoothing with warm-started equilibrium samples.

SILVADiffusionRestorationEquilibrium

Bases: Module

Joint multivariate diffusion-restoration fixed point with hard data consistency.

SILVARecurrentEquilibriumResult dataclass

Dynamic outputs, explicit states, algebraic equilibria, and solver records.

SILVARecurrentEquilibriumNetwork

Bases: Module

Stable dynamic model with an equilibrium nonlinearity at each time step.

SILVARobustEquilibriumResult dataclass

Equilibrium logits, global bound, margins, and certified input radii.

SILVALipschitzRobustEquilibrium

Bases: Module

Lipschitz-bounded equilibrium with selectable structure-preserving map.

SILVAImageMattingEquilibrium

Bases: Module

Trimap-constrained image-matting equilibrium with a replaceable refiner.

SILVAEconomicEquilibriumResult dataclass

Feasible policies and their resource and Euler-equation residuals.

SILVADynamicEconomicEquilibrium

Bases: Module

Neural equilibrium-function approximation for stochastic growth models.

The policy is trained without labels by minimizing resource and Euler residuals along simulated states, following the dynamic-equilibrium-net construction rather than an implicit hidden-state root solve.

Where to Go Next

Question	Page
How is each transition derived and extended?	Source-Aligned Family Deep Dive
Which data, storage, and compute routes are prepared?	Experiment Protocols
How do I inspect one complete family dossier?	Family Reproduction Dossiers
Where are the full citations?	Paper and References