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SILVA Networks

Structured PyTorch equilibrium layers for graph, sequence, image, operator, ODE/PDE, diffusion, flow, and optimization models with explicit interaction branches, solvers, diagnostics, and readouts.

Solvers Picard, Anderson, Broyden, GMRES diagnostics

64 Model Families SILVA, operators, graph physics, transformers, inverse problems, ODEs, DAEs

Derivations Equations, Jacobians, residuals, stability evidence

Install Package, extras, editable mode Run everything Examples, notebooks, data, tests, docs Use the CLI Validation, configs, data, outputs Choose a case Graph, vision, sequence, scientific, molecule, custom Derive step by step From scalar fixed point to SILVA field Adapt papers Sources, equations, citations, scope Reproduce a method All families, protocols, data, metrics, tests, full-scale options Audit equations Companion assets, code map, citation state Select a family SILVA, DEQ, MDEQ, flow, optimization Scale a SILVA model All 64 families, sharded data, distributed training, resume, extensions Connect ODEs and PDEs Implicit time steps, solution operators, Fourier fields, diagnostics Use recent families Fourier, graph physics, homotopy, measures, monotone and mirror equilibria Build physical equilibria Implicit ODE derivatives, DAE stages, residual objectives Explore emerging methods Consistency, mixed boundaries, materials, skinning, meshes, and diffusion Read benchmarks Public validation metrics and cards Check results Measured public smokes and residuals Find the API Solvers, layers, diagnostics

Extended Deep Equilibrium Layers

Deep Equilibrium Models [4] solve for a representation \(z^\star\) that is already at rest under a learned transition. SILVA keeps that implicit-layer contract and makes the transition inspectable: data enters through a stimulus branch, while local, global, and optional self-interaction branches describe how the state organizes before readout.

\[ \begin{aligned} z^\star &= f_\theta(z^\star, x),\\ f_\theta(z,x) &= \Phi\!\left( S_\theta(x)+H_\theta(\chi(z))\right.\\ &\qquad\left. +L_\theta(\chi(z),E)+G_\theta(\chi(z),b) \right). \end{aligned} \]

\(S_\theta\) injects input stimulus. \(H_\theta\) adds optional learned self-interaction. \(L_\theta\) exchanges local information through graphs, grids, kNN, or custom neighborhoods. \(G_\theta\) supplies global context such as mean fields or attention.

SILVA exposes the stimulus, self, local, and global branches as explicit operators, then solves their combined field until the residual approaches an equilibrium state for readout.

Scientific and Equilibrium Models

SILVA is not limited to one graph or image architecture. The recurrent state can be a vector, token sequence, multiresolution tuple, coordinate field, sampled physical field, flow pair, diffusion trajectory, or optimization variable. It can also be a continuous-flow endpoint or an empirical measure represented by variable-size particles. The package provides 64 selectable model families and ten internal point architectures while retaining an explicit state-transition contract:

\[ z^\star=F_\theta(z^\star;x), \qquad r(z^\star)=F_\theta(z^\star;x)-z^\star. \]

The scientific path distinguishes three constructions that are often discussed together but answer different questions. A Neural ODE [7] follows a finite-time trajectory. An implicit ODE or PDE step solves the unknown next state as a fixed point. A neural operator learns a map between functions; the Fourier Neural Operator (FNO) [31] is one possible internal field, while broader neural-operator theory is described in [32]. Recent SILVA extensions add input-injected FNO-DEQ blocks [43], physics-guided graph transport [44], homotopy residual flows [46], and distributional equilibria [45]. Monotone graph operators [47], one-time-injected equilibrium transformers [48], Poisson mirror equilibria [50], and physics-informed equilibria [51] extend the same source/state/solver contract. Implicit DAE stages [52] are represented as root layers, while adversarial equation residuals [53] remain an optional loss rather than a deep-equilibrium family. Joint diffusion trajectories also accept a complete restoration step and a measurement-consistency operator, providing the SILVA adaptation path for parallel diffusion restoration [49]. Learned equilibrium solvers [87] accelerate the forward path, while JFB [88] and SHINE [89] provide distinct backward approximations. Quantum DEQs [90] place a measured circuit transition inside the same state-preserving contract. The source-aligned suite adds contractive multiscale and positive projective equilibria [99] [100], algorithmic and Hamiltonian states [101] [102], inverse and computational imaging constructions [103] through [106], certification and restoration [107] [108], stable recurrent and robust equilibria [109] [110], and matting and economic equilibrium functions [111] [112].

Construction	SILVA state and transition	Public entry point	Full treatment
ODE trajectory	repeated explicit state update	SILVAEulerFlowBlock	ODE/PDE tutorial
implicit ODE/PDE step	next time slice is the equilibrium state	SILVAImplicitTimeStep	Implicit time stepping
source-to-solution operator	lifted input function drives a learned equilibrium field	SILVAOperatorModel	Neural solution operators
Fourier equilibrium operator	spectral and local fields act inside one SILVA point	SILVAFourierNeuralOperator	FNO derivation
input-injected Fourier equilibrium	lifted forcing enters every layer of a tied spectral block	SILVAFNODEQ	SILVA Fourier equilibrium
physics graph equilibrium	reaction, diffusion, and directed transport are named graph branches	SILVAPhysicsGuidedGraphDEQ	Physics graph derivation
homotopy equilibrium	continuous residual flow approaches a SILVA stationary state	SILVAHomotopyEquilibrium	Homotopy derivation
distributional equilibrium	empirical-measure discrepancy moves permutation-compatible particles	SILVADistributionalDEQ	Distributional derivation
joint diffusion restoration	complete reverse steps and measurement consistency act on one trajectory state	SILVADiffusionEquilibrium	Reproduction protocol
monotone graph equilibrium	constrained channel operator and forward-backward graph step	SILVAMonotoneGraphEquilibrium	Monotone derivation
generative equilibrium transformer	one-time source path injects QKV fields into tied token blocks	SILVAGenerativeEquilibriumTransformer	Transformer derivation
Poisson mirror equilibrium	Burg mirror geometry preserves a positive inverse state	SILVAPoissonMirrorEquilibrium	Mirror derivation
physics-informed equilibrium	latent fixed point supplies an implicit ODE time derivative	SILVAPhysicsInformedEquilibrium	Physics-informed derivation
implicit DAE stage layer	Runge-Kutta stages and algebraic endpoint form one root	SILVAImplicitDAEStep	DAE derivation
consistency-distilled equilibrium	a teacher solver trajectory supervises a terminally anchored few-step refiner	SILVAConsistencyDEQ	Consistency DEQ derivation
mixed-boundary Poisson graph	typed directed messages preserve Dirichlet and Neumann roles inside an implicit processor	SILVAPsiGNN	Psi-GNN derivation
implicit Fourier material operator	one tied spectral/local increment evolves displacement or damage fields	SILVAIFNO	IFNO derivation
forward-skinning root field	multi-start canonical roots connect posed queries to occupancy	SILVASNARF	SNARF derivation
center-free mesh relaxation	typed local observations converge to a certified distributed estimate	SILVAMeshInference	Mesh derivation
physics-guided diffusion PDE solve	prior denoising, residual-energy descent, smoothing, and boundary projection share one reverse path	SILVAPhysicsGuidedDiffusionPDE	Diffusion PDE derivation
thermodynamically informed material equilibrium	strain, free-energy features, constitutive response, and Anderson mixing evolve directly in the physical solution field	SILVATherINO	TherINO derivation
fixed-point diffusion denoiser	timestep-conditioned equilibrium solves support variable compute, warm starts, and stochastic implicit training	SILVAFixedPointDiffusionModel	Fixed-point diffusion derivation
monotone operator equilibrium	strongly monotone recurrent structure supports forward-backward or Peaceman-Rachford splitting	SILVAMonotoneOperatorEquilibrium	Monotone operator derivation
positive-concave equilibrium	nonnegative recurrent weights and concave positive maps admit ordinary fixed-point iteration	SILVAPositiveConcaveEquilibrium	Positive-concave derivation
non-Euclidean equilibrium	a weighted-infinity matrix measure certifies well-posedness, averaging, and sensitivity	SILVANonEuclideanEquilibrium	Non-Euclidean derivation
efficient infinite graph equilibrium	normalized channel and graph spectra produce a closed-form or iterative long-range graph state	SILVAEfficientInfiniteGraphEquilibrium	Efficient graph derivation
multiscale graph implicit network	graph-power equilibria are fused by learned nodewise scale attention	SILVAMultiscaleGraphImplicitNetwork	Multiscale graph derivation
delta-cached equilibrium	thresholded state changes update cached linear or convolutional fields during fixed-point iteration	SILVADeltaEquilibrium	Delta equilibrium derivation
learned equilibrium solver	a learned initializer and Anderson controller approximate a high-precision teacher trajectory	SILVAHyperDEQ	Learned solver derivation
quantum-circuit equilibrium	encoded classical features and the measured state enter a repeated circuit transition	SILVAQuantumDEQ	Quantum equilibrium derivation
Lipschitz multiscale equilibrium	all resolution branches communicate inside one bounded packed-state transition	SILVALipschitzMultiscaleEquilibrium	Lipschitz MDEQ derivation
subhomogeneous equilibrium	a strictly positive map is normalized on a projective p-sphere	SILVASubhomogeneousEquilibrium	SubDEQ derivation
algorithmic equilibrium reasoner	tied graph messages settle into a completed algorithm state	SILVAAlgorithmicReasoner	Algorithmic derivation
self-consistent Hamiltonian	a geometric interaction and recurrent correction solve a symmetric Hamiltonian	SILVAHamiltonianEquilibrium	Hamiltonian derivation
known-operator inverse imaging	a sensing map, adjoint correction, and learned prior share one reconstruction root	SILVAInverseImagingEquilibrium	Inverse-imaging derivation
snapshot compressive imaging	coded measurement projection and a volumetric prior solve a video state	SILVASnapshotCompressiveEquilibrium	Snapshot derivation
magnetic-particle reconstruction	primal, split, and dual variables form one packed equilibrium	SILVAMagneticParticleEquilibrium	MPI derivation
sparse hyperspectral equilibrium	a proximal sparse code is decoded into a spatial-spectral cube	SILVASparseHyperspectralEquilibrium	Hyperspectral derivation
serialized smoothing	warm-started noisy equilibria produce counts and certified radii	SILVASerializedSmoothingEquilibrium	Smoothing derivation
joint diffusion restoration	all restoration-time variables are solved with hard observation projection	SILVADiffusionRestorationEquilibrium	Restoration derivation
recurrent equilibrium network	an explicit temporal state couples to one algebraic equilibrium per time step	SILVARecurrentEquilibriumNetwork	Recurrent derivation
Lipschitz robust equilibrium	bounded recurrent, input, and readout maps expose margins and radii	SILVALipschitzRobustEquilibrium	Robust derivation
image-matting equilibrium	a learned alpha refiner is projected onto exact trimap constraints	SILVAImageMattingEquilibrium	Matting derivation
dynamic economic equilibrium	feasible policy functions are trained from resource and Euler residuals	SILVADynamicEconomicEquilibrium	Economic derivation
reaction-diffusion and Burgers	known finite-difference field inside backward Euler	scientific right-hand-side modules	Worked equations
irregular graph PDE	graph Laplacian or message field in local_terms	SILVACortexLayer	Graph discretization

Use the Scientific Operators API for signatures, the complete example for a compact run, and the executable notebook for derivations, training, and numerical diagnostics.

The recent-family tutorial develops the four newer mechanisms, and its executable notebook runs their small-scale reproductions and gradient checks.

The dataset-backed lab guide adds exact periodic fields, steady transport graphs, analytic homotopy pairs, and variable-size empirical measures. Four focused notebooks train the matching SILVA families and report their task, equilibrium, physical, and structural diagnostics separately.

The advanced equilibrium guide and physics-informed guide add five focused families or implicit layers. Their dataset guide states the exact chain, image-pair, Poisson, ODE, and DAE equations, while notebooks 21 through 25 execute every derivation and gradient path.

The emerging-method guide derives eight additional families and provides compact known-solution datasets, replaceable component contracts, source-scale data obligations, storage estimates, and focused notebooks 28 through 35 with retained numerical outputs and plots.

The structured-family guide adds six certified, spectral, multiscale, and accelerated constructions. Focused notebooks 36 through 41 derive each transition from its primary source, run known-solution checks, expose replaceable internals, retain 300-DPI plots, and separate compact evidence from the complete published benchmark protocol.

The learned-solver and backward guide derives HyperDEQ, exact implicit differentiation, JFB, and SHINE as independent forward/backward choices. The quantum equilibrium guide derives encoding, fixed and trainable gates, measurement, direct warmup, implicit solving, and Jacobian regularization. The expansion atlas then connects these choices to monotone, consistency, diffusion, and physics-informed families. Executed notebooks 48 through 51 retain the training curves, gradient comparisons, circuit measurements, residuals, and source-scale records.

The source-aligned family guide derives fourteen further mechanisms, identifies every replaceable module, and links each construction to a scale dossier and exact dataset route. Executed notebooks 61 through 74 retain compact constraints, residuals, gradients, figures, and explicit workstation and source-scale commands.

The real-dataset reproduction guide adds checksum-verified CIFAR-10, Cora, and public-motion snapshots to those six notebooks, along with complete local loaders for vision, Planetoid graphs, Sintel, KITTI Flow, FlyingChairs, and Darcy fields. Each compact run records source indices, preprocessing, access information, and a content hash. The guide states exactly what must be restored before reporting a source-paper benchmark.

The full-scale guide connects all 64 canonical families to scale-aware operators, sharded data, mixed precision, gradient accumulation, distributed execution, checkpoint resume, benchmark handoffs, and explicit extension points. Its executable notebook verifies the dense and matrix-free paths and demonstrates how to move from an equation-checked small problem to a configurable research run.

The reproduction registry connects every family to its equation, source relationship, datasets, preprocessing, metrics, notebooks, tests, real constructor signature, and full-scale builder. It distinguishes compact verified evidence from source benchmark values that still require the complete cited protocol and compute budget. The executable reproduction lab audits the registry, builds a custom transition, and runs an observation-conditioned joint diffusion trajectory.

The family reproduction dossiers expand that registry into 64 reader-facing experiment plans with equations, preserved mechanisms, replaceable components, data and storage routes, progressive acceptance checks, compact/full defaults, and evidence boundaries. The cross-family comparison suites train compatible vector, graph, and field families on shared deterministic tasks and retain measured losses, residuals, iterations, gradients, parameter counts, and runtime in a machine-readable record.

The package is meant to be read, imported, extended, and tested. It contains reference SILVA presets, generic custom layers, fixed-point solvers, Jacobian diagnostics, Lyapunov-style diagnostics, dataset adapters, examples, notebooks, and a companion book roadmap.

Author: Dr. Jose Luis Silva. Source: github.com/jseluis/silva-networks. The package is released under the MIT License.

How to Cite

Use the SILVA article [1] for the methodology and the archived software record [2] for the package version. These numbers link to complete citations and primary records.

If you use the package, cite the software repository. If the work uses or discusses the SILVA methodology, cite the arXiv article as well.

Article

Jose Luis Lima de Jesus Silva. SILVA Networks as Structured Implicit Layers and Vector Attractors via Dynamic Interaction Fields. arXiv:2607.28989, 2026. Primary category: cs.LG. DOI: 10.48550/arXiv.2607.28989.

arXiv | Local PDF | arXiv PDF | BibTeX

Software

Dr. Jose Luis Silva. SILVA Networks. Version 1.2.2. MIT License. All-versions DOI: 10.5281/zenodo.21770098.

Repository | PyPI | Zenodo | Citation Metadata

@misc{silva2026silvanetworksstructuredimplicit,
      title={SILVA Networks as Structured Implicit Layers and Vector Attractors via Dynamic Interaction Fields},
      author={Jose Luis Lima de Jesus Silva},
      year={2026},
      eprint={2607.28989},
      archivePrefix={arXiv},
      primaryClass={cs.LG},
      url={https://arxiv.org/abs/2607.28989},
}

The package BibTeX and the broader method-reference list are available in Paper and References.

Run Everything

Run Everything gives the command path for install, examples, notebooks, data adapters, tests, local docs, and validation checks.

Derive Step by Step

Derivation Workbook walks from scalar fixed points to graph attention, global context, data adapters, and diagnostics.

Learn the Math

Mathematical Foundations derives the fixed-point equation, damping, contraction checks, implicit adjoints, solver updates, graph terms, and complexity.

Bridge DEQ Tutorials

Implicit Layers Bridge adapts fixed-point, implicit autodiff, neural ODE, DEQ, differentiable optimization, MDEQ, and Jacobian regularization tutorials to the package API. The Method Adaptation Atlas gives the source-by-source equation and citation map.

Audit The Release

Equation and PDF Audit and Release Readiness record article citation status, BibTeX coverage, companion-material availability, implementation coverage, notebook validation checks, and packaging commands.

Inspect Benchmarks

Benchmark Cards summarize checked small-scale validation metrics from the public experiment JSON outputs without turning them into leaderboard claims.

Pick a Case

Case Atlas maps every implemented SILVA family and the book extension cases to equations, package classes, tensors, and diagnostics.

Select a Model Family

Selecting Model Families shows how one selector builds full SILVA layers, DEQ reductions, MDEQ blocks, optical flow, and optimization layers.

Use the API

API Reference links the public classes, solvers, datasets, diagnostics, and educational NumPy helpers in one place.

First Path

Install Local editable install, package install, and optional extras. Run the platform Execute examples, notebooks, tests, docs, and available data paths. Derive the core Follow scalar, vector, graph, global, data, and diagnostic derivations. Select the case Choose graph, vision, molecular, dataset, or custom operators. Build the first model Create and train a SILVA graph model from tensors. Check tensor contracts Adapt data into x, edge_index, edge_attr, batch, and y. Trace equations to code Connect \(S,L,G,H\), solvers, residuals, Jacobians, and diagnostics. Study the derivations Derive damping, adjoints, stability checks, and complexity. Run equation to code Execute the derivation path in notebook form. Run the notebook track Exercise the package API in executable cells. Run the bridge track Reproduce implicit-layer and DEQ ideas through silva_networks, then open the method atlas. Keep the API map open Find classes, arguments, and diagnostics quickly.

Install

python -m pip install silva-networks

For development:

python -m pip install -e ".[dev,docs,examples]"

Quick Example

import torch
from silva_networks import SILVAGraphNetwork, SolverConfig

x = torch.randn(8, 5)
edge_index = torch.tensor(
    [[0, 1, 2, 3, 4, 5, 6, 7],
     [1, 2, 3, 4, 5, 6, 7, 0]],
    dtype=torch.long,
)

model = SILVAGraphNetwork(
    in_dim=5,
    hidden_dims=[32, 16],
    out_dim=3,
    task="node",
    local=["graph", "topk"],
    global_term=["mean", "simple"],
    config=[
        SolverConfig(solver="picard", max_iter=12, alpha=0.5),
        SolverConfig(solver="anderson", max_iter=12, alpha=0.35, history=4),
    ],
)

logits = model(x, edge_index=edge_index)

Scientific Quick Example

This model receives a coefficient field and a source field, solves a Fourier architecture inside one SILVA equilibrium point, and decodes a solution field:

import torch
from silva_networks import SILVAFourierNeuralOperator, SolverConfig

model = SILVAFourierNeuralOperator(
    in_channels=2,
    state_channels=16,
    out_channels=1,
    modes_height=6,
    modes_width=6,
    config=SolverConfig(solver="anderson", max_iter=20, alpha=0.4),
)

problem = torch.randn(4, 2, 32, 32)
result = model(problem, return_result=True)
prediction = result.output
print(prediction.shape, result.solver_result.residual)

For a PDE study, report the task error, fixed-point residual, discretized PDE residual, boundary error, and resolution or mesh transfer separately.

Package Areas

Area	What it contains
Solvers	Picard, Anderson, Broyden, GMRES, implicit, JFB, phantom, and SHINE backward paths
Learned Equilibrium Solvers	HyperDEQ transition, initializer, residual compression, learned Anderson control, and distillation loss
Quantum Equilibria	exact compact statevector circuit, external circuit adapter, image filter, QDEQ transition, and diagnostics
Source-Aligned Equilibria	fourteen multiscale, algorithmic, physical, inverse, robust, matting, and economic families with replaceable internals
Layers	stimulus, local, global, self, graph, image, DEQ wrappers
Architectures	stacks, heterogeneous SILVA cortex points, graph/image models, readouts
Point Architectures	ten shape-preserving MLP, attention, convolutional, U-Net, spectral, and token-mixing fields
Scientific Operators	finite differences, physical residuals, implicit steps, reaction-diffusion, Burgers, and Fourier equilibrium operators
Implicit Bridge	tutorial DEQ, ODE, optimization, MDEQ, Jacobian-regularization modules
DEQ Engine	TorchDEQ-style single-state and multi-state engine helpers
Extensibility	transition validation and generic conditioned equilibria assembled from custom modules
Optical Flow	RAFT/DEQ-Flow-style package-native optical-flow utilities
Generalized Cases	sequence, multiscale vision, implicit graph, coordinate representation, and diffusion equilibria
Family Selection	64 canonical constructors and compatibility aliases through one configuration surface
Emerging Equilibria	consistency distillation, mixed-boundary graph roots, IFNO, forward skinning, mesh relaxation, physics-guided diffusion, TherINO, and fixed-point diffusion
Emerging Equilibrium Data	compact exact teacher, Poisson, material, skinning, mesh, PDE, thermodynamic-mechanics, and diffusion datasets
Structured Equilibria	monotone, positive-concave, non-Euclidean, spectral graph, multiscale graph, and delta-cached equilibria
Structured Equilibrium Data	compact known-solution tasks for certificates, graph spectra, multiscale fusion, robustness, and heterogeneous convergence
Source Data and Receipts	attributed real-data subsets, complete local loaders, checksums, source indices, split masks, and field/flow adapters
Recent Equilibrium API	Fourier, physics graph, homotopy, and empirical-measure SILVA families
Recent Equilibrium Datasets	deterministic field, graph, homotopy, and empirical-measure teaching data
Advanced Equilibria API	monotone graph and one-time-injected transformer equilibria
Physics-Informed API	Poisson mirror, implicit ODE derivative, DAE stage, and residual-objective APIs
Advanced Equilibrium Data	exact chain, teacher-map, Poisson, ODE, and DAE batches
Optimization	projected constrained QP layers and optional CVXPYlayers bridge
SILVA Presets	graph/node, vector vision, convolutional vision, molecular presets
Datasets	public loaders, adapters, GraphTensorBatch, validation
Jacobians	full Jacobian, VJP, JVP, spectral-radius estimates
Diagnostics	residual curves, Lyapunov-style energy traces, damped stability
Coverage Registry	implementation families mapped to docs, notebooks, examples, and tests
Educational NumPy	visible scalar/matrix derivations before PyTorch

Interactive Material

Notebook	Focus
Package Quickstart	imports, model construction, forward pass, gradients
Solvers and Jacobians	residuals, solver comparison, full Jacobian, VJP/JVP
Datasets to SILVA	dataset download, preprocessing, kNN graph construction
Public Experiments	config-driven checks and metrics
Custom Operator	extending local/global branches
SILVA Operator Options	Figure 1 operators, ablations, molecules, diagnostics
Research Citation Audit	solver/operator choices mapped to citation checklists
Equation-to-Code Walkthrough	scalar fixed points, graph tensors, SILVA model, diagnostics
Family Selector and Projected QP	SILVA-style family names, projected constraints, flow alias, gradients
Training Helpers Validation	supervised fit/evaluate, checkpoint, resume, device movement
Cortex Hierarchy	deep MLP, residual CNN, and U-Net transitions inside linked SILVA points
Paper Family Architectures	sequence DEQ, MDEQ, IGNN, implicit representations, diffusion, and custom transitions
RAFT and DEQ-Flow	coupled flow state, corrections, implicit gradients, upsampling, and state reuse
Point Architecture Catalog	all ten internal architectures, tensor contracts, gradients, residuals, and point composition
Neural Operators, ODEs, and PDEs	ODE flow, implicit PDE steps, Fourier operators inside SILVA, training, and residual diagnostics
Recent Equilibrium Families	all four recent SILVA mechanisms, connected pipeline, invariances, gradients, and scope boundaries
SILVA Fourier Equilibrium Lab	exact periodic elliptic data, field training, resolution change, and three residuals
SILVA Graph Transport Lab	steady transport data, batched graph training, physical residual, and relabeling
SILVA Homotopy Equilibrium Lab	analytic residual flow, Euler/RK4 comparison, terminal diagnostics, and training
SILVA Distributional Equilibrium Lab	variable-size measures, masks, discrepancies, particle descent, and task readout
SILVA Monotone Graph Equilibrium	monotone parameterization, graph equation, training, and relabeling
SILVA Generative Equilibrium Transformer	patches, one-time QKV injection, teacher matching, and conditioning
SILVA Poisson Mirror Equilibrium	Poisson KL, Burg geometry, positivity, and inverse diagnostics
SILVA Physics-Informed Equilibrium	implicit time derivatives and decomposed physics training
SILVA Implicit DAE and Residuals	Runge-Kutta roots, DAE rollout, and residual-objective boundary
Full-Scale SILVA	scale-aware construction, dense and matrix-free equivalence, checkpointing, and distributed execution controls
Reproducing SILVA and Source Methods	source mechanisms, configurable replacements, compact evidence, benchmark obligations, and full-scale handoffs
SILVA Consistency DEQ	solver trajectories, terminal anchoring, consistency distillation, few-step refinement, and latency diagnostics
SILVA Psi-GNN	mixed Dirichlet/Neumann graph construction, typed messages, Poisson residuals, and boundary checks
SILVA IFNO Materials	tied spectral increments, heterogeneous coefficients, displacement fields, damage variables, and resolution transfer
SILVA SNARF Forward Skinning	forward deformation, multi-start canonical roots, occupancy evaluation, and root-selection diagnostics
SILVA Mesh Inference	typed local observations, center-free relaxation, convergence certificates, and topology-compatible tests
SILVA Physics-Guided Diffusion PDE	reverse diffusion, PDE-residual energy guidance, smoothing, boundary projection, and coefficient-shift checks
SILVA TherINO Mechanics	physical-strain equilibrium, thermodynamic features, constitutive loss, periodic elasticity, and operator replacement
SILVA Fixed-Point Diffusion	timestep-conditioned roots, variable compute, warm starts, implicit gradients, and the separate restoration route
SILVA Monotone Operator Equilibrium	strong monotonicity, both splittings, custom resolvents, and an attributed CIFAR-10 gradient/residual check
SILVA Positive-Concave Equilibrium	positive variants, dense and convolutional states, projected weights, and attributed positive CIFAR-10 tensors
SILVA Non-Euclidean Equilibrium	weighted-infinity certificates, averaged iteration, and bounded perturbations on attributed CIFAR-10 examples
SILVA Efficient Infinite Graph	spectral and iterative solves plus masked training and visual diagnostics on source-indexed Cora tensors
SILVA Multiscale Graph Implicit Network	graph-power equilibria, graph-conditioned sources, and nodewise scale allocation on source-indexed Cora tensors
SILVA Delta Equilibrium	thresholded caches, exact checks, training routes, and measured activity on consecutive real-video frames
Family Reproduction Dossiers	all 64 source contracts, six-stage evidence ladders, data obligations, and scale artifacts
Cross-Family Vector Benchmark	five compatible vector families on one trained compact task
Cross-Family Graph Benchmark	four graph equilibria on one shared chain-node prediction task
Cross-Family Field Benchmark	three spectral field families on one periodic operator task
Extension Builder Workshop	primitive-to-public transition equivalence, equilibrium solving, training, and registration
Failure Diagnostics Workshop	stable, slow, oscillatory, and damped residual signatures with recovery checks
SILVA Learned Equilibrium Solvers	learned initialization and Anderson control, teacher distillation, vector and field transitions, and scale transfer
JFB and SHINE Backward Methods	exact implicit, one-step Jacobian-free, and shared-inverse gradients compared with analytic references
SILVA Quantum DEQ	statevector circuit measurements, direct/warmup/implicit routes, compact training, image contracts, and scale obligations
Equilibrium Expansion Atlas	monotone, physics-informed, diffusion, learned-solver, backward, and quantum mechanisms under one experiment tuple
SILVA Evidence Ladders	evidence levels, deterministic fingerprints, failure retention, and repeated-run records
Transition Equivalence Lab	primitive-to-SILVA transition, root, input-gradient, and parameter-gradient agreement
Statistical Benchmarking	repeated seeds, bootstrap intervals, paired comparisons, and acceptance rules
SILVA Bayesian DEQ	posterior transition samples, sequential warm starts, predictive moments, and uncertainty diagnostics
SILVA Joint Inference	coupled representation and optimized-input equilibrium with independent replaceable updates
SILVA Implicit Spatiotemporal	implicit theta-method dynamics, learned closures, projectors, and physical residuals
SILVA Certified Equilibrium	interval-state propagation, contraction checks, certified margins, and semialgebraic export
Full Experiment Pipeline	materialized protocols, lifecycle hooks, fingerprints, artifacts, and result records
Neumann Backward Comparison	truncated Neumann adjoints compared with exact implicit gradients and analytic error bounds
SILVA Lipschitz MDEQ	coupled multiscale contraction, split states, measured bound, gradients, and source-scale route
SILVA Subhomogeneous Equilibrium	positive projective normalization, convergence, gradients, and article-scale controls
SILVA Algorithmic Reasoner	graph processor equilibrium, node outputs, solver diagnostics, and CLRS handoff
SILVA Hamiltonian Equilibrium	symmetric rotation-invariant Hamiltonian, spectrum, gradients, and molecular handoff
SILVA Inverse Imaging	known operator, adjoint, prior, data consistency, and sensing replacement contract
SILVA Snapshot Compressive Imaging	coded video sensing, analytic projection, remeasurement, and full benchmark route
SILVA Magnetic Particle Equilibrium	packed primal and dual state, calibrated matrix contract, and MPI scale route
SILVA Sparse Hyperspectral Equilibrium	sparse code equilibrium, reconstructed bands, gradients, and hyperspectral datasets
SILVA Serialized Smoothing	warm-started noisy solves, sample counts, solver work, and certified radii
SILVA Diffusion Restoration	joint trajectory root, hard observation projection, and restoration-scale handoff
SILVA Recurrent Equilibrium Network	dynamic and algebraic state trajectories, gradients, and long-horizon protocol
SILVA Lipschitz Robust Equilibrium	four bounded parameterizations, sensitivity bound, margins, and radii
SILVA Image Matting Equilibrium	trimap projection, alpha constraints, gradients, and matting benchmark route
SILVA Dynamic Economic Equilibrium	feasible policies, residual training, and economic calibration handoff
Fixed Points as Layers	implicit-layer introduction through package solvers
DEQ and SILVA	DEQ baseline and configurable SILVA graph model in one API
SILVA DEQ Engine	single-state, multi-state, and variational-dropout fixed-point systems
SILVA Optical Flow	RAFT-style correlation and DEQ-Flow-style flow equilibrium
Method Adaptation Atlas	source-to-SILVA derivations, scope notes, and validation checks

Learning Assets

• Book and solutions manual Planned
• Run everything
• Derivation workbook
• Extending SILVA from an equation to a new family
• Solved notebooks
• Implicit layers bridge
• Documentation log
• Research citation audit
• Runnable examples
• Public experiments
• Dataset preprocessing
• Scientific operators, ODEs, and PDEs
• Case atlas
• Mathematical foundations
• API reference

Citation policy

Method pages link each external result to its primary paper or repository. The local article asset is the SILVA companion paper.

Where to Go Next

Citation metadata and the complete bibliography remain available on the Paper and References page.

Question	Page
Where should I begin learning SILVA?	Start Here
How does a SILVA layer emerge from fixed-point mathematics?	SILVA From Scratch
How do I build and validate a new SILVA family?	Extending SILVA
How are the available scientific cases organized?	Case Atlas