@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}
}

@misc{kolter2020deepimplicitlayers,
  title  = {Deep Implicit Layers},
  author = {Kolter, Zico and Duvenaud, David and Johnson, Matt},
  year   = {2020},
  note   = {NeurIPS tutorial},
  url    = {https://implicit-layers-tutorial.org/}
}

@online{kolter2020deepimplicitlayersintroduction,
  title  = {Deep Implicit Layers: Chapter 1, Introduction},
  author = {Kolter, Zico and Duvenaud, David and Johnson, Matt},
  year   = {2020},
  url    = {https://implicit-layers-tutorial.org/introduction}
}

@online{kolter2020deepimplicitlayersimplicitfunctions,
  title  = {Deep Implicit Layers: Chapter 2, Implicit Functions and Automatic Differentiation},
  author = {Kolter, Zico and Duvenaud, David and Johnson, Matt},
  year   = {2020},
  url    = {https://implicit-layers-tutorial.org/implicit_functions}
}

@online{kolter2020deepimplicitlayersneuralodes,
  title  = {Deep Implicit Layers: Chapter 3, Neural Ordinary Differential Equations},
  author = {Kolter, Zico and Duvenaud, David and Johnson, Matt},
  year   = {2020},
  url    = {https://implicit-layers-tutorial.org/neural_odes}
}

@online{kolter2020deepimplicitlayersdeq,
  title  = {Deep Implicit Layers: Chapter 4, Deep Equilibrium Models},
  author = {Kolter, Zico and Duvenaud, David and Johnson, Matt},
  year   = {2020},
  url    = {https://implicit-layers-tutorial.org/deep_equilibrium_models}
}

@online{kolter2020deepimplicitlayersoptimization,
  title  = {Deep Implicit Layers: Chapter 5, Differentiable Optimization},
  author = {Kolter, Zico and Duvenaud, David and Johnson, Matt},
  year   = {2020},
  url    = {https://implicit-layers-tutorial.org/differentiable_optimization}
}

@inproceedings{chen2018neuralode,
  title     = {Neural Ordinary Differential Equations},
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  year      = {2018},
  url       = {https://arxiv.org/abs/1806.07366}
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@inproceedings{bai2019deep,
  title     = {Deep Equilibrium Models},
  author    = {Bai, Shaojie and Kolter, J. Zico and Koltun, Vladlen},
  booktitle = {Advances in Neural Information Processing Systems},
  year      = {2019},
  url       = {https://arxiv.org/abs/1909.01377}
}

@software{locuslab2019deq,
  title  = {Deep Equilibrium Models Repository},
  author = {Bai, Shaojie and Kolter, J. Zico and Koltun, Vladlen},
  year   = {2019},
  url    = {https://github.com/locuslab/deq}
}

@inproceedings{bai2020multiscale,
  title     = {Multiscale Deep Equilibrium Models},
  author    = {Bai, Shaojie and Koltun, Vladlen and Kolter, J. Zico},
  booktitle = {Advances in Neural Information Processing Systems},
  year      = {2020},
  url       = {https://arxiv.org/abs/2006.08656}
}

@software{locuslab2020mdeq,
  title  = {Multiscale Deep Equilibrium Models Repository},
  author = {Bai, Shaojie and Koltun, Vladlen and Kolter, J. Zico},
  year   = {2020},
  url    = {https://github.com/locuslab/mdeq}
}

@inproceedings{bai2021stabilizing,
  title     = {Stabilizing Equilibrium Models by Jacobian Regularization},
  author    = {Bai, Shaojie and Koltun, Vladlen and Kolter, J. Zico},
  booktitle = {International Conference on Machine Learning},
  year      = {2021},
  url       = {https://arxiv.org/abs/2106.14342}
}

@software{torchdeq2023,
  title  = {TorchDEQ: A Library for Deep Equilibrium Models},
  author = {Geng, Zhengyang and Kolter, J. Zico},
  year   = {2023},
  url    = {https://github.com/locuslab/torchdeq}
}

@inproceedings{teed2020raft,
  title     = {{RAFT}: Recurrent All-Pairs Field Transforms for Optical Flow},
  author    = {Teed, Zachary and Deng, Jia},
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  year      = {2020},
  url       = {https://arxiv.org/abs/2003.12039}
}

@software{princeton2020raft,
  title  = {{RAFT} Repository},
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}

@inproceedings{bai2022deqflow,
  title     = {Deep Equilibrium Optical Flow Estimation},
  author    = {Bai, Shaojie and Geng, Zhengyang and Savani, Yash and Kolter, J. Zico},
  booktitle = {IEEE/CVF Conference on Computer Vision and Pattern Recognition},
  year      = {2022},
  url       = {https://arxiv.org/abs/2204.08442}
}

@software{locuslab2022deqflow,
  title  = {{DEQ-Flow} Repository},
  author = {Bai, Shaojie and Geng, Zhengyang and Savani, Yash and Kolter, J. Zico},
  year   = {2022},
  url    = {https://github.com/locuslab/deq-flow}
}

@inproceedings{gu2020implicit,
  title     = {Implicit Graph Neural Networks},
  author    = {Gu, Fangda and Chang, Heng and Zhu, Wenwu and Sojoudi, Somayeh and El Ghaoui, Laurent},
  booktitle = {Advances in Neural Information Processing Systems},
  year      = {2020},
  url       = {https://proceedings.neurips.cc/paper/2020/hash/8b5c8441a8ff8e151b191c53c1842a38-Abstract.html}
}

@inproceedings{huang2021implicit2,
  title     = {{(Implicit)^2}: Implicit Layers for Implicit Representations},
  author    = {Huang, Zhichun and Bai, Shaojie and Kolter, J. Zico},
  booktitle = {Advances in Neural Information Processing Systems},
  year      = {2021},
  url       = {https://openreview.net/forum?id=AcoMwAU5c0s}
}

@inproceedings{pokle2022deqddim,
  title     = {Deep Equilibrium Approaches to Diffusion Models},
  author    = {Pokle, Ashwini and Geng, Zhengyang and Kolter, J. Zico},
  booktitle = {Advances in Neural Information Processing Systems},
  year      = {2022},
  url       = {https://arxiv.org/abs/2210.12867}
}

@inproceedings{geng2021trainingimplicit,
  title     = {On Training Implicit Models},
  author    = {Geng, Zhengyang and Zhang, Xin-Yu and Bai, Shaojie and Wang, Yisen and Lin, Zhouchen},
  booktitle = {Advances in Neural Information Processing Systems},
  year      = {2021},
  url       = {https://arxiv.org/abs/2111.05177}
}

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  url       = {https://arxiv.org/abs/1703.00443}
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@inproceedings{agrawal2019differentiable,
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  booktitle = {Advances in Neural Information Processing Systems},
  year      = {2019},
  url       = {https://arxiv.org/abs/1910.12430}
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@software{agrawal2019cvxpylayers,
  title  = {{CVXPYLayers}},
  author = {Agrawal, Akshay and Amos, Brandon and Barratt, Shane and Boyd, Stephen and Diamond, Steven and Kolter, J. Zico},
  year   = {2019},
  url    = {https://github.com/cvxgrp/cvxpylayers}
}

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@inproceedings{tolstikhin2021mlpmixer,
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  url       = {https://arxiv.org/abs/2105.01601}
}

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  booktitle = {IEEE Conference on Computer Vision and Pattern Recognition},
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  year      = {2023},
  url       = {https://arxiv.org/abs/2301.00808}
}

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}

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  booktitle = {International Conference on Machine Learning},
  year      = {2019},
  url       = {https://arxiv.org/abs/1810.00825}
}

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@article{banach1922operations,
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@misc{dua2019uci,
  title       = {{UCI} Machine Learning Repository},
  author      = {Dua, Dheeru and Graff, Casey},
  institution = {University of California, Irvine, School of Information and Computer Sciences},
  year        = {2019},
  url         = {https://archive.ics.uci.edu/}
}

@inproceedings{marwah2023fnodeq,
  title     = {Deep Equilibrium Based Neural Operators for Steady-State PDEs},
  author    = {Marwah, Tanya and Pokle, Ashwini and Kolter, J. Zico and Lipton, Zachary C. and Lu, Jianfeng and Risteski, Andrej},
  booktitle = {Advances in Neural Information Processing Systems},
  year      = {2023},
  url       = {https://arxiv.org/abs/2312.00234}
}

@inproceedings{rodrigobonet2024pgcndeq,
  title     = {Physics-guided Graph Convolutional Deep Equilibrium Network for Environmental Data},
  author    = {Rodrigo-Bonet, Esther and Deligiannis, Nikos},
  booktitle = {European Signal Processing Conference},
  year      = {2024},
  url       = {https://eurasip.org/Proceedings/Eusipco/Eusipco2024/pdfs/0000987.pdf}
}

@inproceedings{geuter2025ddeq,
  title     = {{DDEQs}: Distributional Deep Equilibrium Models through Wasserstein Gradient Flows},
  author    = {Geuter, Jonathan and Bonet, Clement and Korba, Anna and Alvarez-Melis, David},
  booktitle = {AISTATS},
  series    = {Proceedings of Machine Learning Research},
  volume    = {258},
  pages     = {3988--3996},
  year      = {2025},
  url       = {https://proceedings.mlr.press/v258/geuter25a.html}
}

@inproceedings{ding2023homoode,
  title     = {Two Sides of The Same Coin: Bridging Deep Equilibrium Models and Neural {ODEs} via Homotopy Continuation},
  author    = {Ding, Shutong and Cui, Tianyu and Wang, Jingya and Shi, Ye},
  booktitle = {Advances in Neural Information Processing Systems},
  year      = {2023},
  url       = {https://arxiv.org/abs/2310.09583}
}

@inproceedings{baker2023mignn,
  title     = {Implicit Graph Neural Networks: A Monotone Operator Viewpoint},
  author    = {Baker, Justin and Wang, Qingsong and Hauck, Cory and Wang, Bao},
  booktitle = {International Conference on Machine Learning},
  series    = {Proceedings of Machine Learning Research},
  volume    = {202},
  pages     = {1521--1548},
  year      = {2023},
  url       = {https://proceedings.mlr.press/v202/baker23a.html}
}

@inproceedings{geng2023get,
  title     = {One-Step Diffusion Distillation via Deep Equilibrium Models},
  author    = {Geng, Zhengyang and Pokle, Ashwini and Kolter, J. Zico},
  booktitle = {Advances in Neural Information Processing Systems},
  year      = {2023},
  url       = {https://arxiv.org/abs/2401.08639}
}

@inproceedings{cao2024deqir,
  title     = {Deep Equilibrium Diffusion Restoration with Parallel Sampling},
  author    = {Cao, Jiezhang and Shi, Yue and Zhang, Kai and Zhang, Yulun and Timofte, Radu and Van Gool, Luc},
  booktitle = {IEEE/CVF Conference on Computer Vision and Pattern Recognition},
  year      = {2024},
  url       = {https://arxiv.org/abs/2311.11600}
}

@article{daniele2025deqmd,
  title   = {Deep Equilibrium Models for Poisson Imaging Inverse Problems via Mirror Descent},
  author  = {Daniele, Christian and Villa, Silvia and Vaiter, Samuel and Calatroni, Luca},
  journal = {arXiv preprint arXiv:2507.11461},
  year    = {2025},
  url     = {https://arxiv.org/abs/2507.11461}
}

@article{pacheco2024pideq,
  title   = {Solving Differential Equations using Physics-Informed Deep Equilibrium Models},
  author  = {Pacheco, Bruno M. and Camponogara, Eduardo},
  journal = {arXiv preprint arXiv:2406.03472},
  year    = {2024},
  url     = {https://arxiv.org/abs/2406.03472}
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@article{moya2021daepinn,
  title   = {{DAE-PINN}: A Physics-Informed Neural Network Model for Simulating Differential Algebraic Equations with Application to Power Networks},
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  year    = {2021},
  url     = {https://arxiv.org/abs/2109.04304}
}

@inproceedings{bullwinkel2022deqgan,
  title     = {{DEQGAN}: Learning the Loss Function for {PINNs} with Generative Adversarial Networks},
  author    = {Bullwinkel, Blake and Randle, Dylan and Protopapas, Pavlos and Sondak, David},
  booktitle = {AI4Science Workshop at the International Conference on Machine Learning},
  year      = {2022},
  url       = {https://arxiv.org/abs/2209.07081}
}

@misc{pytorch2026sdpa,
  title        = {Scaled Dot Product Attention},
  author       = {{PyTorch Contributors}},
  year         = {2026},
  howpublished = {PyTorch documentation},
  url          = {https://docs.pytorch.org/docs/stable/generated/torch.nn.functional.scaled_dot_product_attention.html}
}

@misc{pytorch2026ddp,
  title        = {DistributedDataParallel},
  author       = {{PyTorch Contributors}},
  year         = {2026},
  howpublished = {PyTorch documentation},
  url          = {https://docs.pytorch.org/docs/stable/generated/torch.nn.parallel.DistributedDataParallel.html}
}

@misc{pytorch2026amp,
  title        = {Automatic Mixed Precision},
  author       = {{PyTorch Contributors}},
  year         = {2026},
  howpublished = {PyTorch documentation},
  url          = {https://docs.pytorch.org/docs/stable/amp.html}
}

@misc{pytorch2026jvp,
  title        = {Jacobian-Vector Product},
  author       = {{PyTorch Contributors}},
  year         = {2026},
  howpublished = {PyTorch documentation},
  url          = {https://docs.pytorch.org/docs/stable/generated/torch.autograd.functional.jvp.html}
}

@inproceedings{pal2023continuousdeq,
  title     = {Continuous Deep Equilibrium Models: Training Neural {ODEs} Faster by Integrating Them to Infinity},
  author    = {Pal, Avik and Edelman, Alan and Rackauckas, Christopher},
  booktitle = {IEEE High Performance Extreme Computing Conference},
  year      = {2023},
  url       = {https://arxiv.org/abs/2201.12240}
}

@article{lin2026cdeq,
  title   = {Consistency Deep Equilibrium Models},
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@inproceedings{bai2022neuraldeqsolvers,
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@inproceedings{gao2026bayesiandeq,
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@inproceedings{wei2022ibpmondeq,
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  year      = {2022},
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@inproceedings{chen2021semimondeq,
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  volume    = {34},
  year      = {2021},
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}

@inproceedings{sato2026lipschitzmdeq,
  title     = {Lipschitz Multiscale Deep Equilibrium Models: A Theoretically Guaranteed and Accelerated Approach},
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  booktitle = {International Conference on Artificial Intelligence and Statistics},
  year      = {2026},
  url       = {https://arxiv.org/abs/2602.03297}
}

@article{sittoni2024subhomogeneous,
  title   = {Subhomogeneous Deep Equilibrium Models},
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  year    = {2024},
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@inproceedings{georgiev2024dear,
  title     = {Deep Equilibrium Algorithmic Reasoning},
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  booktitle = {Advances in Neural Information Processing Systems},
  volume    = {37},
  year      = {2024},
  url       = {https://arxiv.org/abs/2410.15059}
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@inproceedings{wang2024deqh,
  title     = {Infusing Self-Consistency into Density Functional Theory Hamiltonian Prediction via Deep Equilibrium Models},
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  booktitle = {Advances in Neural Information Processing Systems},
  volume    = {37},
  year      = {2024},
  url       = {https://arxiv.org/abs/2406.03794}
}

@article{gilton2021inverse,
  title   = {Deep Equilibrium Architectures for Inverse Problems in Imaging},
  author  = {Gilton, Davis and Ongie, Gregory and Willett, Rebecca},
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  year    = {2021},
  url     = {https://arxiv.org/abs/2102.07944}
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@article{zhao2023deqsci,
  title   = {Deep Equilibrium Models for Snapshot Compressive Imaging},
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  volume  = {37},
  number  = {3},
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  year    = {2023},
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@article{gungor2023deqmpi,
  title   = {{DEQ-MPI}: A Deep Equilibrium Reconstruction with Learned Consistency for Magnetic Particle Imaging},
  author  = {G{\"u}ng{\"o}r, Alper and Askin, Baris and Soydan, Damla Alptekin and Top, Can Bar{\i}{\c s} and Saritas, Emine Ulku and {\c C}ukur, Tolga},
  journal = {IEEE Transactions on Medical Imaging},
  year    = {2023},
  doi     = {10.1109/TMI.2023.3300704},
  url     = {https://arxiv.org/abs/2212.13233}
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@article{gkillas2023hyperspectral,
  title   = {Connections between Deep Equilibrium and Sparse Representation Models with Application to Hyperspectral Image Denoising},
  author  = {Gkillas, Alexandros and Ampeliotis, Dimitris and Berberidis, Kostas},
  journal = {IEEE Transactions on Image Processing},
  volume  = {32},
  pages   = {1513--1528},
  year    = {2023},
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@inproceedings{gao2024serialized,
  title     = {Certified Robustness for Deep Equilibrium Models via Serialized Random Smoothing},
  author    = {Gao, Weizhi and Hou, Zhichao and Xu, Han and Liu, Xiaorui},
  booktitle = {Advances in Neural Information Processing Systems},
  volume    = {37},
  year      = {2024},
  url       = {https://arxiv.org/abs/2411.00899}
}

@inproceedings{cao2024deqir,
  title     = {Deep Equilibrium Diffusion Restoration with Parallel Sampling},
  author    = {Cao, Jiezhang and Shi, Yue and Zhang, Kai and Zhang, Yulun and Timofte, Radu and Van Gool, Luc},
  booktitle = {IEEE/CVF Conference on Computer Vision and Pattern Recognition},
  year      = {2024},
  url       = {https://arxiv.org/abs/2311.11600}
}

@article{revay2023ren,
  title   = {Recurrent Equilibrium Networks: Flexible Dynamic Models with Guaranteed Stability and Robustness},
  author  = {Revay, Max and Wang, Ruigang and Manchester, Ian R.},
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  year    = {2023},
  url     = {https://arxiv.org/abs/2104.05942}
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@inproceedings{havens2023lipschitz,
  title     = {Exploiting Connections between Lipschitz Structures for Certifiably Robust Deep Equilibrium Models},
  author    = {Havens, Aaron and Araujo, Alexandre and Garg, Siddharth and Khorrami, Farshad and Hu, Bin},
  booktitle = {Advances in Neural Information Processing Systems},
  volume    = {36},
  year      = {2023},
  url       = {https://proceedings.neurips.cc/paper_files/paper/2023/hash/4462db5eee6823b2abad0d1f955e187a-Abstract-Conference.html}
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@inproceedings{liu2024deqmatt,
  title     = {Image Matting Based on Deep Equilibrium Models},
  author    = {Liu, Xinshuang and Zhao, Yue},
  booktitle = {Artificial Neural Networks and Machine Learning -- ICANN 2024},
  pages     = {379--391},
  year      = {2024},
  publisher = {Springer Nature Switzerland},
  doi       = {10.1007/978-3-031-72335-3_26}
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@article{azinovic2022deep,
  title   = {Deep Equilibrium Nets},
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  number  = {4},
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  year    = {2022},
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