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Paper and References

Cite the GitHub repository when using the package, notebooks, examples, or documentation. Cite the SILVA Networks arXiv paper as well when the package is used in connection with the SILVA methodology.

Software Citation

Dr. Jose Luis Silva. SILVA Networks. Version 1.2.2. MIT License.
https://github.com/jseluis/silva-networks
https://doi.org/10.5281/zenodo.21770098

Use the all-versions concept DOI for the current software citation: 10.5281/zenodo.21770098. The historical version 1.0.0 archive remains available at 10.5281/zenodo.21770099.

SILVA Paper Citation

Jose Luis Lima de Jesus Silva. SILVA Networks as Structured Implicit Layers and
Vector Attractors via Dynamic Interaction Fields. 2026. arXiv:2607.28989.
https://arxiv.org/abs/2607.28989

Article PDF: docs/assets/papers/silva-networks-arxiv-2607.28989.pdf.

Article metadata verified from arXiv on August 3, 2026:

Field Value
arXiv ID 2607.28989
Title SILVA Networks as Structured Implicit Layers and Vector Attractors via Dynamic Interaction Fields
Author Jose Luis Lima de Jesus Silva
Submitted July 31, 2026
Primary category Computer Science, Machine Learning
Secondary category Computer Science, Neural and Evolutionary Computing
Version v1
Comment 46 pages, 10 figures
DOI 10.48550/arXiv.2607.28989

BibTeX

BibTeX for the SILVA paper and package is available in docs/assets/bib/silva-networks.bib. That file also includes the core external references used by the documentation: Deep Implicit Layers tutorial chapters, DEQ/MDEQ repositories, TorchDEQ, RAFT, DEQ-Flow, IGNN, DEQ-INR, DEQ-DDIM, solver papers, graph/attention papers, and optimization-layer papers.

@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},
}

@software{silva2026silvanetworkssoftware,
  title   = {SILVA Networks},
  author  = {Silva, Jose Luis},
  year    = {2026},
  version = {1.2.2},
  license = {MIT},
  doi     = {10.5281/zenodo.21770098},
  url     = {https://github.com/jseluis/silva-networks}
}

Author website: https://jsluis.com

GitHub: https://github.com/jseluis

Repository citation metadata is available in CITATION.cff.

Package and Companion Assets

  • Book and Solutions Manual Planned
  • SILVA article PDF: arXiv:2607.28989 article PDF included with the documentation assets.
  • notebooks/: solved progressive notebooks.
  • notebooks/package_api/: package-first API notebooks.
  • examples/: small executable package examples.
  • Research Citation Audit: method-to-paper map for implemented package objects.
  • Method Adaptation Atlas: source-to-SILVA derivations, scope notes, and runnable adaptation checks.

Numbered Reference Registry

Numbered citations are global across the documentation: a marker such as [13] always identifies the same source. Selecting the marker opens the complete entry below. Each entry includes a primary external source that opens in a separate browser tab, while the local entry remains available for continued reading.

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  99. Sato, Naoki, and Iiduka, Hideaki. Lipschitz Multiscale Deep Equilibrium Models: A Theoretically Guaranteed and Accelerated Approach. AISTATS, 2026. Primary source. Research repository. BibTeX: sato2026lipschitzmdeq.
  100. Sittoni, Pietro, and Tudisco, Francesco. Subhomogeneous Deep Equilibrium Models. arXiv:2403.00720, 2024. Primary source. BibTeX: sittoni2024subhomogeneous.
  101. Georgiev, Dobrik; Wilson, J. J.; Buffelli, Davide; and Liò, Pietro. Deep Equilibrium Algorithmic Reasoning. Advances in Neural Information Processing Systems 37, 2024. Primary source. Research repository. CLRS benchmark repository. BibTeX: georgiev2024dear.
  102. Wang, Zun; Liu, Chang; Zou, Nianlong; Zhang, He; Wei, Xinran; Huang, Lin; Wu, Lijun; and Shao, Bin. Infusing Self-Consistency into Density Functional Theory Hamiltonian Prediction via Deep Equilibrium Models. Advances in Neural Information Processing Systems 37, 2024. Primary source. Research repository. BibTeX: wang2024deqh.
  103. Gilton, Davis; Ongie, Gregory; and Willett, Rebecca. Deep Equilibrium Architectures for Inverse Problems in Imaging. arXiv:2102.07944, 2021. Primary source. BibTeX: gilton2021inverse.
  104. Zhao, Yaping; Zheng, Siming; and Yuan, Xin. Deep Equilibrium Models for Snapshot Compressive Imaging. Proceedings of the AAAI Conference on Artificial Intelligence 37(3), 3642–3650, 2023. DOI: 10.1609/aaai.v37i3.25475. Primary source. Research repository. BibTeX: zhao2023deqsci.
  105. Güngör, Alper; Askin, Baris; Soydan, Damla Alptekin; Top, Can Barış; Saritas, Emine Ulku; and Çukur, Tolga. DEQ-MPI: A Deep Equilibrium Reconstruction with Learned Consistency for Magnetic Particle Imaging. IEEE Transactions on Medical Imaging, 2023. DOI: 10.1109/TMI.2023.3300704. Primary source. Research repository. BibTeX: gungor2023deqmpi.
  106. Gkillas, Alexandros; Ampeliotis, Dimitris; and Berberidis, Kostas. Connections between Deep Equilibrium and Sparse Representation Models with Application to Hyperspectral Image Denoising. IEEE Transactions on Image Processing 32, 1513–1528, 2023. DOI: 10.1109/TIP.2023.3245323. Primary source. BibTeX: gkillas2023hyperspectral.
  107. Gao, Weizhi; Hou, Zhichao; Xu, Han; and Liu, Xiaorui. Certified Robustness for Deep Equilibrium Models via Serialized Random Smoothing. Advances in Neural Information Processing Systems 37, 2024. Primary source. Research repository. BibTeX: gao2024serialized.
  108. Cao, Jiezhang; Shi, Yue; Zhang, Kai; Zhang, Yulun; Timofte, Radu; and Van Gool, Luc. Deep Equilibrium Diffusion Restoration with Parallel Sampling. IEEE/CVF Conference on Computer Vision and Pattern Recognition, 2024. Primary source. Research repository. BibTeX: cao2024deqir.
  109. Revay, Max; Wang, Ruigang; and Manchester, Ian R. Recurrent Equilibrium Networks: Flexible Dynamic Models with Guaranteed Stability and Robustness. IEEE Transactions on Automatic Control, 2023. Primary source. BibTeX: revay2023ren.
  110. Havens, Aaron; Araujo, Alexandre; Garg, Siddharth; Khorrami, Farshad; and Hu, Bin. Exploiting Connections between Lipschitz Structures for Certifiably Robust Deep Equilibrium Models. Advances in Neural Information Processing Systems 36, 2023. Primary source. Research repository. BibTeX: havens2023lipschitz.
  111. Liu, Xinshuang, and Zhao, Yue. Image Matting Based on Deep Equilibrium Models. In Artificial Neural Networks and Machine Learning – ICANN 2024, 379–391. Springer, 2024. DOI: 10.1007/978-3-031-72335-3_26. Primary source. Research repository. BibTeX: liu2024deqmatt.
  112. Azinovic, Marlon; Gaegauf, Luca; and Scheidegger, Simon. Deep Equilibrium Nets. International Economic Review 63(4), 1471–1525, 2022. DOI: 10.1111/iere.12575. Primary source. Research repository. BibTeX: azinovic2022deep.

Method Citation Matrix

Package area Main package objects Research lineage
equilibrium layers fixed_point, DEQLayer, SILVALayer, presets DEQ, implicit layers, SILVA
nonlinear solvers picard, anderson, broyden fixed-point iteration, Anderson acceleration, Broyden quasi-Newton
linear adjoints implicit_adjoint_solve, gmres implicit differentiation, GMRES
Jacobian diagnostics full_jacobian, vjp, jvp, Hutchinson estimators implicit layers, Hutchinson trace estimation, Jacobian-regularized DEQ
graph local terms GraphLocal, GraphAttentionLocal GNNs, GCN, GAT, MPNN
global set context MeanFieldGlobal, GatedMeanFieldGlobal, pool_entities Deep Sets, attention, SILVA global field
top-k/channel attention TopKGlobalAttention, channel attention modules scaled dot-product attention, set attention
dynamic kNN TopKLocal, DynamicChannelLocal, make_knn_edge_index dynamic graph / EdgeConv literature
molecular presets SILVAMolecularLayer, SILVAMolecularRegressor MPNN, GAT, dataset-specific molecular benchmarks
implicit bridge SILVAFixedPointBlock, SILVAEulerFlowBlock, SILVAQuadraticOptimizationLayer, SILVAMultiscaleDEQBlock DEQ, Neural ODEs, differentiable optimization, MDEQ
optimization layers SILVAProjectedQPLayer, silva_projected_qp_layer, silva_cvxpy_layer OptNet, differentiable convex optimization layers, CVXPYlayers
general DEQ engine SILVADEQEngine, silva_deq, pack_state, SILVAVariationalDropout TorchDEQ, DEQ, SILVA
SILVA DEQ flow SILVADEQFlow, silva_deq_flow, all-pairs correlation, flow warping RAFT, DEQ-Flow, SILVA
sequence and multiscale cases SILVASequenceDEQ, SILVAMultiscaleDEQ and task heads DEQ, MDEQ, SILVA
graph, INR, and diffusion cases SILVAImplicitGraphNetwork, SILVAImplicitNeuralRepresentation, SILVADiffusionEquilibrium IGNN, DEQ-INR, DEQ-DDIM, DeqIR, SILVA
scientific operators and implicit PDE steps SILVAOperatorModel, SILVAFourierNeuralOperator, SILVAImplicitTimeStep, scientific residual helpers Neural ODEs, FNO, neural operators, SILVA
steady neural operators SILVAFNODEQ, silva_fno_deq FNO-DEQ, Fourier neural operators, SILVA input injection
physics graph equilibria SILVAGraphConvectionDiffusion, SILVAPhysicsGuidedGraphDEQ pGCN-DEQ, graph convection-diffusion operators, SILVA
continuous equilibrium paths SILVAHomotopyEquilibrium HomoODE, continuous deep equilibria, conditioned ODE flows, SILVA residual flow
empirical-measure equilibria SILVADistributionalTransition, SILVADistributionalDEQ DDEQ, Wasserstein gradient flows, MMD, energy distance, SILVA
recent equilibrium teaching data make_periodic_elliptic_dataset, make_graph_transport_dataset, make_affine_homotopy_dataset, make_variable_measure_dataset FNO/FNO-DEQ, pGCN-DEQ, homotopy equilibrium, or DDEQ according to the generated problem; SILVA for the typed equation checks
coupled RAFT/DEQ-Flow SILVARAFTDEQ, correlation pyramid, update block, correction loss RAFT, DEQ-Flow, SILVA
scalable SILVA execution build_scaled_silva, full_scale_solver_config, runtime_for_tier, prepare_silva_model SILVA families, implicit differentiation, scaled dot-product attention, distributed data parallelism, mixed precision
source-aware reproduction silva_reproduction_spec, build_silva_reproduction, silva_family_signature SILVA article and all cited family adaptations with explicit evidence boundaries
lazy sharded data SILVAShardedTensorDataset, write_silva_tensor_shards, make_silva_dataloader package-native tensor-shard contract and PyTorch data loading
consistency acceleration SILVAConsistencyDEQ, teacher trajectories, local/global consistency loss C-DEQ solver-time distillation and SILVA transitions
mixed-boundary Poisson graphs SILVAPsiGNN, SILVAPsiGNNProcessor, make_psi_poisson_grid Psi-GNN and SILVA typed graph equilibria
implicit material operators SILVAIFNO, SILVAIFNOIncrement, make_ifno_material_dataset IFNO tied Fourier residual integration and SILVA field contracts
articulated implicit shapes SILVASNARF, canonical weight/occupancy fields SNARF forward skinning, multi-start root search, and SILVA solvers
typed distributed relaxation SILVAMeshInference, M-matrix certificate Mesh Inference linear-Gaussian mechanism and SILVA fixed points
physics-guided field diffusion SILVAPhysicsGuidedDiffusionPDE, Poisson energy and boundary projector reverse diffusion with inference-time PDE guidance
thermodynamic material equilibria SILVATherINO, SILVAThermodynamicEncoder, SILVAThermodynamicUpdate, make_therino_elastic_dataset TherINO physical-strain iteration, constitutive encoding, and SILVA solvers
fixed-point diffusion denoisers SILVAFixedPointDenoiser, SILVAFixedPointDiffusionModel, timestep transition and compute allocation FPDM timestep-conditioned roots, stochastic Jacobian-free training, reuse, and SILVA diagnostics
learned equilibrium solvers SILVAHyperDEQ, SILVAHyperInitializer, SILVAHyperAndersonController HyperDEQ learned initialization, learned Anderson updates, and SILVA transition replacement
backward approximations backward_mode="jfb", backward_mode="shine", BroydenInverseEstimate, shine_adjoint_solve JFB identity approximation and SHINE forward-inverse reuse
quantum-circuit equilibria SILVAQuantumDEQ, statevector circuit, image filter, circuit adapter QDEQ direct/warmup/implicit execution and SILVA solver diagnostics
contractive multiscale and positive projective equilibria SILVALipschitzMultiscaleEquilibrium, SILVASubhomogeneousEquilibrium Lipschitz MDEQ [99] and SubDEQ [100]
algorithmic and Hamiltonian equilibria SILVAAlgorithmicReasoner, SILVAHamiltonianEquilibrium, SILVARadialHamiltonian DEAR [101] and DEQH [102]
inverse and computational imaging equilibria SILVAInverseImagingEquilibrium, SILVASnapshotCompressiveEquilibrium, SILVAMagneticParticleEquilibrium, SILVASparseHyperspectralEquilibrium inverse-imaging DEQ [103], DEQSCI [104], DEQ-MPI [105], and sparse hyperspectral DEQ [106]
certified and restoration equilibria SILVASerializedSmoothingEquilibrium, SILVADiffusionRestorationEquilibrium serialized randomized smoothing [107] and DeqIR [108]
stable recurrent and robust equilibria SILVARecurrentEquilibriumNetwork, SILVALipschitzRobustEquilibrium recurrent equilibrium networks [109] and structure-preserving Lipschitz DEQs [110]
matting and economic equilibrium functions SILVAImageMattingEquilibrium, SILVADynamicEconomicEquilibrium DEQ-Matt [111] and Deep Equilibrium Nets for economics [112]

Equilibrium and Implicit Layers

DEQ Engines and Optical Flow

The package bridge notebooks in notebooks/implicit_bridge/ and the Method Adaptation Atlas connect these methods to SILVA equations and public silva_networks APIs. Each method is paired with its primary paper or repository. The SILVA article PDF is included with the documentation as the package's companion paper.

The optical-flow implementation in silva_networks.flow is also package-native. It expresses all-pairs correlation, recurrent update fields, local correlation lookup, and fixed-point flow solving through SILVA APIs.

Point Architecture Sources

The built-in point-architecture catalog uses compact, shape-preserving adaptations of the following architecture patterns:

The catalog translates these patterns into compact, shape-preserving transition modules designed for use inside SILVA points.

Solvers and Linear Algebra

Graphs, Attention, and Messages

Citation Rules for Reports

When writing up package results:

  1. Cite the SILVA paper and package for the structured interaction field, presets, implementation, and package-native notebooks.
  2. Cite DEQ/implicit-layer literature when the claim is about equilibrium states, infinite-depth weight tying, or implicit differentiation.
  3. Cite the numerical method actually used: Anderson, Broyden, or GMRES.
  4. Cite graph/attention/set/molecular papers only when the corresponding package branch is used or discussed.
  5. Cite dataset sources separately from model-method citations.

The four builders in silva_networks.frontier_data generate deterministic teaching problems rather than redistributing an external benchmark. Report the builder, seed, tensor shape, physical or statistical parameters, and residual tolerance. When replacing generated data with a published benchmark, also cite that benchmark and report its official split and metric protocol. The complete mapping is in Dataset-Backed Equilibrium Labs.

Public dataset sources:

Reference policy: the SILVA article PDF is included with the documentation assets. Third-party papers and upstream repositories are cited through canonical links.

Where to Go Next

Question Page
How should the article and package be cited? How to Cite
How does each cited method connect to SILVA? Method Adaptation Atlas
Which identifiers and records have been audited? Research Citation Audit