Research Citation Audit

This audit maps implemented SILVA Networks package objects to the research lineage that should be cited when the package is used in papers, reports, notebooks, or experiment logs.

The SILVA paper and package should be cited for the package-specific structured interaction field, presets, notebooks, and implementation. Foundational methods should be cited when a result depends on them. The global entries for the SILVA article [1] and software archive [2] begin the numbered registry.

SILVA citation status

The SILVA paper citation is: Jose Luis Lima de Jesus Silva, SILVA Networks as Structured Implicit Layers and Vector Attractors via Dynamic Interaction Fields, 2026, arXiv:2607.28989. Metadata was verified from arXiv on August 3, 2026. BibTeX is available in Paper and References and docs/assets/bib/silva-networks.bib.

Audit Method

The audit was checked against the package source under src/silva_networks:

Source module	Public surface audited
solvers.py	picard, anderson, broyden, gmres, implicit_adjoint_solve
jacobian.py	full Jacobian, VJP/JVP, power iteration, Hutchinson estimates, stability reports
layers.py	stimulus, graph, graph attention, kNN, global attention, channel attention, DEQ wrapper, generic SILVA layer
architectures.py	stacks, graph/image networks, pooling and readout heads
presets.py	graph, vector vision, convolutional vision, molecular presets, energy diagnostic
implicit.py	DEQ/ODE/optimization/MDEQ bridge objects and SILVA-named factories
optimization.py	projected constrained quadratic layers, projections, and optional CVXPYlayers wrapper
deq_engine.py	TorchDEQ-style single-state and multi-state package engine, packed state helpers, variational dropout
flow.py	RAFT/DEQ-Flow-style optical-flow utilities and package-native flow DEQ
datasets.py	tabular, image, pixel-graph, PyG-like, and molecular adapters
frontier.py	input-injected Fourier, physics-guided graph, homotopy-flow, and empirical-measure equilibria
frontier_data.py	equation-checked periodic fields, graph transport systems, affine homotopy roots, and variable-size empirical measures
advanced_equilibria.py	monotone graph and one-time-injected transformer equilibria
physics_informed.py	Poisson mirror equilibrium, physics-informed ODE equilibrium, implicit DAE stages, and residual adversarial training
advanced_data.py	equation-checked chain, image-pair, Poisson, ODE, and DAE teaching batches
scaling.py	all-family execution guides, scale-aware constructors, solver tiers, and distributed model preparation
scaling_data.py	lazy tensor shards and ordinary or distributed data loaders
training.py	supervised evaluation, mixed precision, gradient accumulation, distributed synchronization, and checkpoint resume

Method-to-Citation Matrix

Package object or family	What it implements	Cite when used
fixed_point, DEQLayer, SILVALayer, SILVAStack, all equilibrium presets	fixed-point layer \(z^\star=f_\theta(z^\star,x)\) and infinite-depth weight-tied view	SILVA paper/package; Bai, Kolter, and Koltun, Deep Equilibrium Models
implicit_adjoint_solve, implicit-gradient derivations	adjoint solve \((I-J_f^\top)u=g\)	SILVA paper/package; Deep Implicit Layers tutorial; Bai et al. DEQ
picard, damped solver update	damped fixed-point iteration	SILVA paper/package; Banach fixed-point theorem as mathematical background
anderson	Anderson acceleration for fixed-point residuals	Anderson, Iterative Procedures for Nonlinear Integral Equations; Walker and Ni, Anderson Acceleration for Fixed-Point Iterations
broyden	inverse secant quasi-Newton solve for \(F(z)=0\)	Broyden, A Class of Methods for Solving Nonlinear Simultaneous Equations
gmres	Krylov minimal-residual linear solve	Saad and Schultz, GMRES [13]
hutchinson_jacobian_norm, jacobian_regularization_loss	stochastic trace/Frobenius estimator for Jacobian penalties	Hutchinson, A Stochastic Estimator of the Trace; Bai, Koltun, and Kolter, Jacobian Regularization
full_jacobian, vjp, jvp, stability_report	local linearization and product diagnostics	SILVA paper/package; Deep Implicit Layers tutorial
GraphLocal, SILVAGraphLayer, graph preset mean branch	message passing / graph aggregation over edge_index	Kipf and Welling, GCN; Gilmer et al., MPNN; Scarselli et al., GNN model
GraphAttentionLocal, bond-aware molecular local branch	masked neighbor attention over graph edges	Velickovic et al., Graph Attention Networks; Vaswani et al., Attention Is All You Need
MeanFieldGlobal, GatedMeanFieldGlobal, pool_entities	permutation-invariant mean/set pooling and broadcast context	Zaheer et al., Deep Sets; SILVA paper/package for the gated field design
TopKGlobalAttention, ChannelSelfAttentionGlobal, MultiHeadChannelAttentionGlobal	scaled dot-product attention variants, restricted to top-k or hidden channels	Vaswani et al., Attention Is All You Need; Lee et al., Set Transformer for attention on sets
TopKLocal, DynamicChannelLocal, make_knn_edge_index, tabular_to_silva_graph	dynamic nearest-neighbor local graphs	Wang et al., Dynamic Graph CNN; SILVA paper/package for the hidden-channel adaptation
SILVAVisionVectorLayer, SILVAConvVisionClassifier	vector/channel SILVA equilibria and convolutional stimulus front-end	SILVA paper/package; Vaswani attention for global channel attention; Wang et al. DGCNN for dynamic kNN analogy
SILVAMolecularLayer, SILVAMolecularRegressor, molecular_to_silva_graph	bond-aware graph equilibrium and graph-level molecular readout	Gilmer et al., MPNN; Velickovic et al., GAT; cite molecule benchmark/dataset separately
ExplicitEulerODEBlock, SILVAEulerFlowBlock	finite explicit Euler bridge for neural ODE intuition	Chen et al., Neural Ordinary Differential Equations; Deep Implicit Layers tutorial
SILVAFourierOperatorPointArchitecture, fourier_operator	low-mode spectral operator with a local channel projection, used as a shape-preserving SILVA transition field	Li et al., Fourier Neural Operator; Kovachki et al., Neural Operator; SILVA paper/package for its placement inside the structured fixed-point transition
SILVAFNODEQBlock, SILVAFNODEQ	input-injected, tied Fourier transition solved as a steady field equilibrium	Marwah et al., FNO-DEQ [43]; FNO [31]; SILVA paper/package
graph_convection_diffusion, SILVAGraphConvectionDiffusion, SILVAPhysicsGuidedGraphDEQ	separated graph source, reaction, normalized diffusion, and directed transport branches	Rodrigo-Bonet and Deligiannis, physics-guided graph DEQ [44]; graph convolution literature; SILVA paper/package
SILVAHomotopyTransition, SILVAHomotopyEquilibrium	continuous residual path \(\dot z=T(z;x)-z\) toward a SILVA fixed point	Ding et al., HomoODE [46]; Neural ODEs [7]; SILVA paper/package
distributional_discrepancy, SILVADistributionalTransition, SILVADistributionalDEQ	empirical-measure discrepancy and differentiable particle descent with variable-size masks	Geuter et al., DDEQ [45]; SILVA paper/package
SILVAInjectedSelfAttention	manual, fused, or query-chunked injected attention with one shared equation	GET [48]; scaled dot-product attention [54]; SILVA paper/package
SILVAMonotoneGraphTransition(operator_rank=...)	factorized constrained channel map with the same monotonicity lower bound	MIGNN [47]; SILVA paper/package
SILVAPhysicsInformedEquilibrium(derivative_mode="matrix_free")	JVP/GMRES implicit time derivative without a dense latent Jacobian	PIDEQ [51]; JVP API [57]; GMRES [13]
SILVAImplicitDAEStep(linear_solver="gmres")	Newton-Krylov implicit Runge-Kutta stage root	DAE-PINN [52]; GMRES [13]; SILVA package
SILVAConsistencyDEQ, silva_consistency_loss, update_consistency_ema	terminally anchored solver-trajectory consistency and one/few-step refinement	C-DEQ [59]; cite WikiText-103 [65], OGB [66], or ImageNet [67] when used
SILVAPsiGNNProcessor, SILVAPsiGNN	typed mixed-boundary Poisson graph equilibrium, finite-element residual, and Jacobian stabilization	Psi-GNN [60]; Gmsh [72] when its mesh generator is used; SILVA package
SILVAIFNOIncrement, SILVAIFNO	shared-depth Fourier residual integration for displacement or damage fields	IFNO [61]; cite the selected simulator or DIC source separately; SILVA package
SILVABlendWeightField, SILVACanonicalOccupancy, SILVASNARF	multi-start forward-skinning roots and canonical occupancy	SNARF [62]; AMASS [68], D-FAUST [69], CAPE [70], and SMPL [71] according to the assets used
SILVAMeshInference	typed directed relaxation, centralized identity check, and M-matrix convergence certificate	Mesh Inference [63]; SILVA package
SILVAPhysicsGuidedDiffusionPDE, gaussian_smooth_2d	frozen-prior reverse inference with PDE-energy guidance and hard boundary projection	physics-guided diffusion PDE [64]; cite the numerical generator used for the field archive; SILVA package
SILVAThermodynamicEncoder, SILVAThermodynamicUpdate, SILVATherINO	strain, stress, energy, and macroscopic-loading channels iterated in physical solution space	TherINO [73]; cite the selected constitutive solver and microstructure archive for source-scale studies; SILVA package
SILVATimestepFixedPointBlock, SILVAFixedPointDenoiser, SILVAFixedPointDiffusionModel	timestep-wise fixed-point denoising, iteration allocation, state reuse, and stochastic Jacobian-free training	fixed-point diffusion models [74]; cite the selected diffusion prior and image dataset separately; SILVA package
builders in emerging_data	deterministic exact or known-solution preflights for the eight emerging families	cite the corresponding family source [59]-[64] and [73]-[74], the builder name, seed, shape, and residual tolerance; do not present these fixtures as source benchmark archives
SILVAMonotoneDenseOperator, SILVAMonotoneOperatorEquilibrium	strongly monotone parameterization with forward-backward or Peaceman-Rachford splitting	monDEQ [75]; cite the selected task dataset separately; SILVA package
SILVAPositiveConcaveTransition, SILVAPositiveConcaveEquilibrium	nonnegative recurrent maps with positive-concave activations and fixed-point iteration	pcDEQ [76]; cite the selected image benchmark separately; SILVA package
SILVANonEuclideanDenseOperator, SILVANonEuclideanEquilibrium	weighted-infinity one-sided Lipschitz certificate, averaged iteration, and sensitivity bound	NEMON [77]; cite the perturbation protocol and dataset separately; SILVA package
silva_normalized_gram, SILVAEfficientInfiniteGraphEquilibrium	normalized channel Gram map with closed-form or iterative infinite graph equilibrium	EIGNN [78]; cite the graph dataset and split separately; SILVA package
SILVAMultiscaleGraphImplicitNetwork	parallel graph-power equilibria and nodewise scale attention	MGNNI [79]; cite the node or graph benchmark separately; SILVA package
SILVADeltaOperator, SILVADeltaEquilibrium	cached linear/convolutional updates from thresholded state deltas	DeltaDEQ [80]; cite the INR image or optical-flow benchmark and base model separately; SILVA package
builders in structured_data	deterministic known-solution checks for six structured families	cite the corresponding source [75]-[80], the builder name, seed, shape, and residual tolerance; do not report these fixtures as source benchmark results
runtime_for_tier, prepare_silva_model, fit_supervised scale controls	mixed precision, process distribution, accumulation, and resumable training for the unchanged SILVA model	PyTorch distributed and precision APIs [55] [56]; SILVA package
make_periodic_elliptic_dataset, make_graph_transport_dataset, make_affine_homotopy_dataset, make_variable_measure_dataset	deterministic teaching problems with exact spectral, graph, fixed-point, or empirical-moment checks	FNO and neural operators [31] [32], FNO-DEQ [43], physics-guided graph DEQ [44], DDEQ [45], and HomoODE [46], according to the builder used
QuadraticOptimizationLayer, SILVAQuadraticOptimizationLayer	differentiable quadratic argmin and fixed-point KKT solve	Amos and Kolter, OptNet; Agrawal et al., Differentiable Convex Optimization Layers; Deep Implicit Layers tutorial
SILVAProjectedQPLayer, silva_projected_qp_layer, SILVAConstrainedQuadraticLayer, silva_cvxpy_layer	projected constrained quadratic programs and optional CVXPYlayers bridge	OptNet for optimization-layer framing; Agrawal et al. differentiable convex optimization layers; CVXPYlayers when the optional bridge is used
ToyMultiscaleDEQBlock, SILVAMultiscaleDEQBlock	coupled low/high equilibrium state	Bai, Koltun, and Kolter, Multiscale Deep Equilibrium Models
SILVADEQEngine, silva_deq, pack_state, unpack_state, SILVAVariationalDropout	general package-native DEQ engine for one or multiple tensor states	SILVA package; TorchDEQ; DEQ
SILVADEQFlow, silva_deq_flow, SILVAOpticalFlowDEQ, silva_all_pairs_correlation, silva_flow_warp, silva_local_correlation_lookup	compact optical-flow fixed-point estimator with RAFT-style correlation and DEQ-Flow framing	Teed and Deng, RAFT; Bai et al., Deep Equilibrium Optical Flow Estimation; DEQ-Flow repository; SILVA package
silva_endpoint_error, silva_flow_smoothness_loss	optical-flow evaluation and regularization utilities	cite the optical-flow benchmark/dataset and RAFT/DEQ-Flow when used with the flow DEQ
dataset loaders and adapters	source data, standardization, graph conversion	cite the dataset source, UCI page, torchvision dataset, PyG dataset, or molecular benchmark used in the experiment

Claim-to-Citation Guide

Use the narrowest citation set that supports the claim.

Claim in a report	Minimum citation set
"SILVA Networks were used."	SILVA paper/package
"The model is an equilibrium / infinite-depth layer."	SILVA paper/package and Bai et al. DEQ
"Gradients use implicit differentiation / adjoint solve."	Deep Implicit Layers tutorial and Bai et al. DEQ
"Anderson acceleration was used."	Anderson 1965 and Walker-Ni 2011
"Broyden acceleration was used."	Broyden 1965
"GMRES was used for the adjoint linear system."	Saad-Schultz 1986
"A Fourier neural operator was used inside a SILVA point."	Li et al. FNO, Kovachki et al. neural operator, and SILVA paper/package
"The forcing is reinjected in a tied Fourier equilibrium transition."	FNO-DEQ [43], FNO, and SILVA paper/package
"Convection-diffusion physics is represented inside a graph equilibrium."	physics-guided graph DEQ [44], the graph discretization source, and SILVA paper/package
"A continuous residual path approaches the SILVA fixed point."	HomoODE [46], Neural ODEs, and SILVA paper/package
"The SILVA state is an empirical measure optimized by particle descent."	DDEQ [45] and SILVA paper/package
"Injected attention used a fused scaled-dot-product kernel."	GET when using its injected equilibrium mechanism, PyTorch scaled dot-product attention [54], and SILVA paper/package
"Physics derivatives or DAE Newton steps were matrix free."	the corresponding PIDEQ or DAE-PINN source, GMRES, JVP documentation [57], and SILVA package
"Training used mixed precision or distributed data parallelism."	PyTorch AMP or DDP [55] [56], plus the SILVA package version and resolved runtime configuration
"A generated SILVA teaching dataset was used."	SILVA paper/package, the corresponding method citation, and the builder name, seed, shape, and equation-check tolerance; do not present it as an external benchmark
"A Jacobian Frobenius penalty was used."	Hutchinson 1989 and Bai-Koltun-Kolter 2021
"The graph local operator is attention-based."	GAT and Attention Is All You Need
"The molecular model is message-passing-like."	MPNN and GAT, plus SILVA
"The global branch is permutation invariant."	Deep Sets, plus SILVA for gated/top-k branch design
"The model uses top-k or channel attention."	Attention Is All You Need; Set Transformer for set-attention framing
"The hidden-channel local branch uses dynamic kNN."	Dynamic Graph CNN as related dynamic graph literature, plus SILVA for the hidden-channel adaptation
"The optimization layer solves a differentiable argmin."	OptNet or differentiable convex optimization layers
"The ODE bridge connects continuous-depth models to equilibria."	Neural ODEs and Deep Implicit Layers tutorial
"The engine handles multi-state DEQ systems."	TorchDEQ as related interface lineage; DEQ; SILVA package for this implementation
"The optical-flow module uses all-pairs correlation and recurrent refinement."	RAFT; DEQ-Flow if framed as an equilibrium optical-flow solve

Package-Specific Contributions

The following are SILVA/package-specific compositions. Cite the SILVA paper and repository, then cite the relevant building blocks only as background:

SILVA/package object	Why it is package-specific
structured \(S+H+L+G\) field	combines stimulus, self, local, and global operators into one equilibrium transition
gated mean-field global branch	uses attention-style scoring over a graph/set mean, then broadcasts a gated context
hidden-channel dynamic kNN for vector vision	adapts dynamic-neighborhood graph ideas to channels inside a sample
quadratic interaction energy	diagnostic proxy for state/field alignment; not a theorem of global Lyapunov stability by itself
graph, vision, convolutional, and molecular SILVA presets	package-level reference configurations and tensor contracts
package implicit bridge notebooks	tutorials that express DEQ, ODE, optimization, MDEQ, and Jacobian regularization through silva_networks APIs
method adaptation atlas	source-by-source translation from external implicit-layer, DEQ, ODE, optimization, and optical-flow methods into package equations, APIs, and scope notes
package-native DEQ engine	TorchDEQ-style convenience interface implemented through SILVA solvers and state packing
package-native optical-flow DEQ	compact RAFT/DEQ-Flow-inspired implementation using SILVA solvers
package-native projected QP layer	projected fixed-point QP implementation with selectable nonnegative, box, simplex, and affine constraints
recent equilibrium family adaptations	places Fourier input injection, graph-physics branches, homotopy flow, and empirical-measure descent inside canonical SILVA state and transition contracts
recent equilibrium teaching datasets	provides deterministic, typed examples whose targets satisfy the documented spectral, graph, fixed-point, or empirical-moment relation
advanced graph and transformer equilibria	maps monotone operator splitting and one-time QKV injection into canonical SILVA source/state contracts
positive Poisson mirror equilibrium	places Burg mirror geometry, Poisson KL, forward/adjoint operators, and learned regularizer gradients inside SILVA
physics-informed equilibrium	combines latent fixed points, implicit-function time derivatives, initial conditions, ODE residuals, and Jacobian regularization
implicit DAE stage layer	solves one- or multistage Runge-Kutta DAE equations as a differentiable SILVA root layer
adversarial equation-residual objective	exposes separate generator/discriminator losses and records that the cited DEQGAN abbreviation means Differential Equation
advanced equilibrium teaching datasets	provides deterministic targets with directly evaluated graph, teacher-map, intensity, ODE, or DAE equations
all-family scale contract	maps all 64 canonical SILVA families to data, literature, benchmark, numerical controls, runtime tiers, and extension points
SILVAHyperDEQ, SILVAHyperAndersonController, silva_hyper_deq_loss	learned initialization, residual compression, Anderson coefficients, mixing, and teacher-trajectory supervision
SolverConfig(backward_mode="jfb")	one final differentiable transition approximates the implicit inverse by identity
BroydenInverseEstimate, SolverConfig(backward_mode="shine")	retains and optionally refines the forward Broyden inverse estimate for the adjoint
SILVAQuantumDEQ, SILVAStatevectorQuantumCircuit	encoded circuit transition, Pauli-Z measurement, direct/warmup/implicit routes, and Jacobian regularization
package tensor-shard format	stores aligned tensor samples in lazy independently loadable shards with an atomic JSON manifest

Example Citation Checklist

For a graph node model with GAT local branch, gated mean global branch, Anderson solver, and Jacobian diagnostics, cite:

1. SILVA paper/package.
2. Deep Equilibrium Models.
3. Graph Attention Networks.
4. Attention Is All You Need, if discussing attention scores explicitly.
5. Deep Sets, if discussing permutation-invariant graph/set pooling.
6. Anderson 1965 and Walker-Ni 2011.
7. Hutchinson 1989 and/or Jacobian-regularized DEQs if using Hutchinson Jacobian penalties or Frobenius diagnostics.

For a molecular regressor, cite:

1. SILVA paper/package.
2. Deep Equilibrium Models.
3. Neural Message Passing for Quantum Chemistry.
4. Graph Attention Networks.
5. The dataset or benchmark source used for the molecule task.

For an implicit bridge optimization experiment, cite:

1. SILVA package for the tutorial implementation.
2. Deep Implicit Layers tutorial.
3. OptNet or differentiable convex optimization layers.
4. Deep Equilibrium Models if the optimization layer is solved through the package fixed-point solver.

For an optical-flow DEQ validation experiment, cite:

1. SILVA package for the package-native implementation.
2. RAFT for all-pairs correlation and recurrent flow refinement lineage.
3. Deep Equilibrium Optical Flow Estimation for the DEQ-flow framing.
4. The optical-flow dataset or benchmark if external data is used.

Audit Findings

Finding	Status
SILVA article metadata is recorded.	Uses arXiv:2607.28989 with BibTeX.
The package already cited several papers in docstrings, but docs did not have a single method-level citation matrix.	Fixed by this audit page.
The derivation pages explained equations but did not always name the originating method literature beside the formula.	Cross-links added from foundations and implementation derivations.
Some SILVA operators are related to published mechanisms but are not exact reproductions.	Labeled as package-specific compositions above.
Flow and DEQ-engine modules were public through __init__ and needed their own citation/API coverage.	Added RAFT, DEQ-Flow, and TorchDEQ lineage.
Dataset adapters cannot be fully cited without knowing the downstream dataset actually used.	Docs instruct users to cite each dataset source separately.
Small generated problems could be mistaken for published benchmark reproductions.	Dataset labs label them as deterministic teaching problems and document the separate benchmark handoff, source, split, and metric requirements.
External tutorials and repositories needed a clear method lineage.	Added the Method Adaptation Atlas and notebook as SILVA-native translations with upstream citations.
The optimization bridge could be misread as one object with one scope.	Scope note added: the implicit tutorial object is an unconstrained quadratic bridge; the optimization module adds projected constrained QP layers and an optional CVXPYlayers wrapper.

Where to Go Next

Question	Page
Where is the maintained bibliography?	Paper and References
How should methods and results be cited in reports?	Citation-Aware Reporting
Which generated datasets support the recent family labs?	Dataset-Backed Equilibrium Labs
Where is the front-page citation presented?	Home