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ODE Transformer: An Ordinary Differential Equation-Inspired Model for Sequence Generation [article]

Bei Li, Quan Du, Tao Zhou, Yi Jing, Shuhan Zhou, Xin Zeng, Tong Xiao, JingBo Zhu, Xuebo Liu, Min Zhang
<span title="2022-03-17">2022</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Residual networks are an Euler discretization of solutions to Ordinary Differential Equations (ODE). This paper explores a deeper relationship between Transformer and numerical ODE methods.  ...  Inspired by this, we design a new architecture, ODE Transformer, which is analogous to the Runge-Kutta method that is well motivated in ODE.  ...  The authors would like to thank anonymous reviewers for their valuable comments. And thank Yufan Jiang for his helpful advice to improve the paper.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2203.09176v1">arXiv:2203.09176v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/h272ixys2rekvmebnydvnhvrk4">fatcat:h272ixys2rekvmebnydvnhvrk4</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20220319020141/https://arxiv.org/pdf/2203.09176v1.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/56/a6/56a66d0bb71599b6a01f2b8fd3e0824435b0463c.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2203.09176v1" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

ODE Transformer: An Ordinary Differential Equation-Inspired Model for Sequence Generation

Bei Li, Quan Du, Tao Zhou, Yi Jing, Shuhan Zhou, Xin Zeng, Tong Xiao, JingBo Zhu, Xuebo Liu, Min Zhang
<span title="">2022</span> <i title="Association for Computational Linguistics"> Proceedings of the 60th Annual Meeting of the Association for Computational Linguistics (Volume 1: Long Papers) </i> &nbsp; <span class="release-stage">unpublished</span>
Residual networks are an Euler discretization of solutions to Ordinary Differential Equations (ODE). This paper explores a deeper relationship between Transformer and numerical ODE methods.  ...  Inspired by this, we design a new architecture, ODE Transformer, which is analogous to the Runge-Kutta method that is well motivated in ODE.  ...  The authors would like to thank anonymous reviewers for their valuable comments. And thank Yufan Jiang for his helpful advice to improve the paper.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.18653/v1/2022.acl-long.571">doi:10.18653/v1/2022.acl-long.571</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/wyakht6cfjbdfdrg5lsmhchftm">fatcat:wyakht6cfjbdfdrg5lsmhchftm</a> </span>
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Complementing ODE-Based System Analysis Using Boolean Networks Derived from an Euler-Like Transformation

Claudia Stötzel, Susanna Röblitz, Heike Siebert, Attila Csikász-Nagy
<span title="2015-10-23">2015</span> <i title="Public Library of Science (PLoS)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/s3gm7274mfe6fcs7e3jterqlri" style="color: black;">PLoS ONE</a> </i> &nbsp;
In this paper, we present a systematic transition scheme for a large class of ordinary differential equations (ODEs) into Boolean networks.  ...  It performs an Euler-like step which uses the signs of the right hand sides to obtain the Boolean update functions for every variable of the corresponding discrete model.  ...  Ordinary differential equations (ODEs) are well suited to describe interactions based on concentration changes and allow for detailed predictions.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1371/journal.pone.0140954">doi:10.1371/journal.pone.0140954</a> <a target="_blank" rel="external noopener" href="https://www.ncbi.nlm.nih.gov/pubmed/26496494">pmid:26496494</a> <a target="_blank" rel="external noopener" href="https://pubmed.ncbi.nlm.nih.gov/PMC4619740/">pmcid:PMC4619740</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/7i2jkobd3ngv3bkqd2qdkmcr3e">fatcat:7i2jkobd3ngv3bkqd2qdkmcr3e</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20171011074822/http://journals.plos.org/plosone/article/file?id=10.1371/journal.pone.0140954&amp;type=printable" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/81/38/8138e3d55ce4f93cbf84ffc23fe1df94be852d6d.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1371/journal.pone.0140954"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> plos.org </button> </a> <a target="_blank" rel="external noopener" href="https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4619740" title="pubmed link"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> pubmed.gov </button> </a>

Exchangeable Neural ODE for Set Modeling [article]

Yang Li, Haidong Yi, Christopher M. Bender, Siyuan Shan, Junier B. Oliva
<span title="2020-08-06">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In this work we propose a more general formulation to achieve permutation equivariance through ordinary differential equations (ODE).  ...  Our proposed module, Exchangeable Neural ODE (ExNODE), can be seamlessly applied for both discriminative and generative tasks.  ...  In this work, we propose an exchangeable, invertible flow transformation, ExNODE, based on Neural Ordinary Differential Equation (NODE) [6] .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2008.02676v1">arXiv:2008.02676v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/6spb7auwkvb3jadlx4cd25i6he">fatcat:6spb7auwkvb3jadlx4cd25i6he</a> </span>
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STEER: Simple Temporal Regularization For Neural ODEs [article]

Arnab Ghosh, Harkirat Singh Behl, Emilien Dupont, Philip H. S. Torr, Vinay Namboodiri
<span title="2020-11-02">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Training Neural Ordinary Differential Equations (ODEs) is often computationally expensive.  ...  Indeed, computing the forward pass of such models involves solving an ODE which can become arbitrarily complex during training.  ...  Neural Ordinary Differential Equations (ODE) are an elegant approach for building continuous depth neural networks [4] .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2006.10711v3">arXiv:2006.10711v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/grlhctsvxvh6zanoql4bl3n2mq">fatcat:grlhctsvxvh6zanoql4bl3n2mq</a> </span>
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The ODE/IM correspondence

Patrick Dorey, Clare Dunning, Roberto Tateo
<span title="2007-07-24">2007</span> <i title="IOP Publishing"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/nzcpmvl7nzgojf4q47bpg2qrdu" style="color: black;">Journal of Physics A: Mathematical and Theoretical</a> </i> &nbsp;
This article reviews a recently-discovered link between integrable quantum field theories and certain ordinary differential equations in the complex domain.  ...  Along the way, aspects of PT-symmetric quantum mechanics are discussed, and some elementary features of the six-vertex model and the Bethe ansatz are explained.  ...  B.2 The values of C(0, l) and D(0, l) For general values of M , the ordinary differential equation cannot be solved in closed form.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1088/1751-8113/40/32/r01">doi:10.1088/1751-8113/40/32/r01</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/3j4p3hhr5vbunm5trye657o5y4">fatcat:3j4p3hhr5vbunm5trye657o5y4</a> </span>
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Stateful ODE-Nets using Basis Function Expansions [article]

Alejandro Queiruga, N. Benjamin Erichson, Liam Hodgkinson, Michael W. Mahoney
<span title="2021-11-06">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
The recently-introduced class of ordinary differential equation networks (ODE-Nets) establishes a fruitful connection between deep learning and dynamical systems.  ...  Performance is demonstrated by applying our stateful ODE-Block to (a) image classification tasks using convolutional units and (b) sentence-tagging tasks using transformer encoder units.  ...  Acknowledgments We are grateful for the generous support from Amazon AWS and Google Cloud.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2106.10820v2">arXiv:2106.10820v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/qdqsmwmsyjfypmybc4aidt67py">fatcat:qdqsmwmsyjfypmybc4aidt67py</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20211117135021/https://arxiv.org/pdf/2106.10820v2.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/6e/c4/6ec408a30831ab535d3962224187c5873c4ab689.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2106.10820v2" title="arxiv.org access"> <button class="ui compact blue labeled icon button serp-button"> <i class="file alternate outline icon"></i> arxiv.org </button> </a>

LTI ODE-valued neural networks

Manel Velasco, Enric X. Martín, Cecilio Angulo, Pau Martí
<span title="2014-05-31">2014</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/khkbaqss45bgnjenoehwasx35u" style="color: black;">Applied intelligence (Boston)</a> </i> &nbsp;
In this new approach, artificial neurons are characterized by: i) inputs in the form of differentiable continuous-time signals, ii) linear time-invariant ordinary differential equations (LTI ODE) for connection  ...  It will be shown that this new characterization of the constitutive nodes in an artificial neural network, namely LTI ODE-valued neural network (LTI ODEVNN), allows solving multiple problems at the same  ...  Acknowledgments The authors would like to thank the editor and the anonymous reviewers for their valuable comments and suggestions to improve the quality of the paper.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s10489-014-0548-7">doi:10.1007/s10489-014-0548-7</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/gsp5fn36rva5fb5fv4v6bhghpy">fatcat:gsp5fn36rva5fb5fv4v6bhghpy</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200710064758/http://people-esaii.upc.edu/people/pmarti/14AI.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/a6/f9/a6f90c28f1f6ae345ab3ba4c67d1757177556bd4.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s10489-014-0548-7"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

NeurInt : Learning to Interpolate through Neural ODEs [article]

Avinandan Bose, Aniket Das, Yatin Dandi, Piyush Rai
<span title="2021-11-07">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
Differential Equations.  ...  However, most generative models mapping a fixed prior to the generated images lead to interpolation trajectories lacking smoothness and containing images of reduced quality.  ...  This framework is known as Neural Ordinary Differential Equations (Chen et al. 2018) .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2111.04123v1">arXiv:2111.04123v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/4zfiqe2odbfnnd4vybhtarzc64">fatcat:4zfiqe2odbfnnd4vybhtarzc64</a> </span>
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Toward the Fully Physics-Informed Echo State Network – an ODE Approximator Based on Recurrent Artificial Neurons [article]

Dong Keun Oh
<span title="2020-11-13">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
As the plainest work on such a purpose, an ODE (ordinary differential equation) approximator is designed to replicate the solution in sequence with respect to the recurrent evaluations.  ...  On the principal invariance of differential equations, the constraint in recurrence just takes shape to secure a proper regression method for the ESN-based ODE approximator.  ...  The invariance of differential equations just establishes an essential idea to constrain the recurrent models in terms of the given ODE.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2011.06769v1">arXiv:2011.06769v1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/743qq4bx45c3lf4oe2b7dqym7u">fatcat:743qq4bx45c3lf4oe2b7dqym7u</a> </span>
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Time-Reversal Symmetric ODE Network [article]

In Huh, Eunho Yang, Sung Ju Hwang, Jinwoo Shin
<span title="2021-01-07">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In this paper, we propose a novel loss function that measures how well our ordinary differential equation (ODE) networks comply with this time-reversal symmetry; it is formally defined by the discrepancy  ...  We also show that, even for systems that do not possess the full time-reversal symmetry, TRS-ODENs can achieve better predictive performances over baselines.  ...  Acknowledgments and Disclosure of Funding The authors received no third party funding for this work.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2007.11362v3">arXiv:2007.11362v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/ytfun5zqxresth2h6f2ektoa6u">fatcat:ytfun5zqxresth2h6f2ektoa6u</a> </span>
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Second-Order Neural ODE Optimizer [article]

Guan-Horng Liu, Tianrong Chen, Evangelos A. Theodorou
<span title="2021-11-06">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
We propose a novel second-order optimization framework for training the emerging deep continuous-time models, specifically the Neural Ordinary Differential Equations (Neural ODEs).  ...  to derive backward ODEs for higher-order derivatives at the same O(1) memory cost.  ...  Acknowledgments and Disclosure of Funding The authors would like to thank Chia-Wen Kuo and Chen-Hsuan Lin for the meticulous proofreading, and Keuntaek Lee for providing additional computational resources  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2109.14158v2">arXiv:2109.14158v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/rxflzdyh6zg57d2a3pkbhrdaky">fatcat:rxflzdyh6zg57d2a3pkbhrdaky</a> </span>
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Temporal Knowledge Graph Forecasting with Neural ODE [article]

Zhen Han, Zifeng Ding, Yunpu Ma, Yujia Gu, Volker Tresp
<span title="2022-06-02">2022</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
To this end, we propose a novel continuum model by extending the idea of neural ordinary differential equations (ODEs) to multi-relational graph convolutional networks.  ...  There has been an increasing interest in inferring future links on temporal knowledge graphs (KG).  ...  neural ordinary differential equation.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2101.05151v3">arXiv:2101.05151v3</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/6hcybiyjsbcgjavif7x3v4ihqq">fatcat:6hcybiyjsbcgjavif7x3v4ihqq</a> </span>
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Learning Compositional Representation for 4D Captures with Neural ODE [article]

Boyan Jiang, Yinda Zhang, Xingkui Wei, Xiangyang Xue, Yanwei Fu
<span title="2021-04-20">2021</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
To model the motion, a neural Ordinary Differential Equation (ODE) is trained to update the initial state conditioned on the learned motion code, and a decoder takes the shape code and the updated state  ...  code to reconstruct the 3D model at each time stamp.  ...  We train a model replacing ODE to an MLP that directly produces the pose code for a specified time, and get IoU=80.4% and Chamfer Distance=0.073 for re- construction on D-FAUST, while our ODE model achieves  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/2103.08271v2">arXiv:2103.08271v2</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/4k7o32ffnvachjcppbxyxwul2q">fatcat:4k7o32ffnvachjcppbxyxwul2q</a> </span>
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Symplectic ODE-Net: Learning Hamiltonian Dynamics with Control [article]

Yaofeng Desmond Zhong, Biswadip Dey, Amit Chakraborty
<span title="2020-04-30">2020</span> <i > arXiv </i> &nbsp; <span class="release-stage" >pre-print</span>
In this paper, we introduce Symplectic ODE-Net (SymODEN), a deep learning framework which can infer the dynamics of a physical system, given by an ordinary differential equation (ODE), from observed state  ...  This framework, by offering interpretable, physically-consistent models for physical systems, opens up new possibilities for synthesizing model-based control strategies.  ...  ACKNOWLEDGMENTS This research was inspired by the ideas and plans articulated by N. E. Leonard and A. Majumdar, Princeton University, in their ONR grant #N00014-18-1-2873.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener" href="https://arxiv.org/abs/1909.12077v4">arXiv:1909.12077v4</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/73asaqs6h5bmxgjqdaxqtqbsae">fatcat:73asaqs6h5bmxgjqdaxqtqbsae</a> </span>
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