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We explore in detail a method to solve ordinary differential equations using feedforward neural networks. We prove a specific loss function, which does not require knowledge of the exact solution, to be a suitable standard metric to…

Computational Physics · Physics 2020-06-02 Liam L. H. Lau , Denis Werth

Residual neural networks can be viewed as the forward Euler discretization of an Ordinary Differential Equation (ODE) with a unit time step. This has recently motivated researchers to explore other discretization approaches and train ODE…

Machine Learning · Computer Science 2019-07-02 Amir Gholami , Kurt Keutzer , George Biros

A concept of using Neural Ordinary Differential Equations(NODE) for Transfer Learning has been introduced. In this paper we use the EfficientNets to explore transfer learning on CIFAR-10 dataset. We use NODE for fine-tuning our model. Using…

Machine Learning · Computer Science 2020-01-22 Rajath S , Sumukh Aithal K , Natarajan Subramanyam

Transformer is the backbone of modern NLP models. In this paper, we propose RealFormer, a simple and generic technique to create Residual Attention Layer Transformer networks that significantly outperform the canonical Transformer and its…

Machine Learning · Computer Science 2021-09-14 Ruining He , Anirudh Ravula , Bhargav Kanagal , Joshua Ainslie

To understand the fundamental trade-offs between training stability, temporal dynamics and architectural complexity of recurrent neural networks~(RNNs), we directly analyze RNN architectures using numerical methods of ordinary differential…

Machine Learning · Computer Science 2019-05-01 Murphy Yuezhen Niu , Lior Horesh , Isaac Chuang

We introduce a new family of deep neural network models. Instead of specifying a discrete sequence of hidden layers, we parameterize the derivative of the hidden state using a neural network. The output of the network is computed using a…

Machine Learning · Computer Science 2019-12-17 Ricky T. Q. Chen , Yulia Rubanova , Jesse Bettencourt , David Duvenaud

The Transformer architecture is widely used in natural language processing. Despite its success, the design principle of the Transformer remains elusive. In this paper, we provide a novel perspective towards understanding the architecture:…

Machine Learning · Computer Science 2019-06-10 Yiping Lu , Zhuohan Li , Di He , Zhiqing Sun , Bin Dong , Tao Qin , Liwei Wang , Tie-Yan Liu

Deep convolutional neural networks (CNN) have been applied for image dehazing tasks, where the residual network (ResNet) is often adopted as the basic component to avoid the vanishing gradient problem. Recently, many works indicate that the…

Computer Vision and Pattern Recognition · Computer Science 2020-07-14 Jiawei Shen , Zhuoyan Li , Lei Yu , Gui-Song Xia , Wen Yang

A serious problem in image classification is that a trained model might perform well for input data that originates from the same distribution as the data available for model training, but performs much worse for out-of-distribution (OOD)…

Computer Vision and Pattern Recognition · Computer Science 2021-12-07 Rajat Koner , Poulami Sinhamahapatra , Karsten Roscher , Stephan Günnemann , Volker Tresp

Increasing the layer number of on-chip photonic neural networks (PNNs) is essential to improve its model performance. However, the successively cascading of network hidden layers results in larger integrated photonic chip areas. To address…

Machine Learning · Computer Science 2023-02-08 Yun Zhao , Hang Chen , Min Lin , Haiou Zhang , Tao Yan , Xing Lin , Ruqi Huang , Qionghai Dai

We propose a simple extension of residual networks that works simultaneously in multiple resolutions. Our network design is inspired by the iterative back-projection algorithm but seeks the more difficult task of learning how to enhance…

Image and Video Processing · Electrical Eng. & Systems 2021-01-26 Pablo Navarrete Michelini , Hanwen Liu , Yunhua Lu , Xingqun Jiang

Random ordinary differential equations (RODEs), i.e. ODEs with random parameters, are often used to model complex dynamics. Most existing methods to identify unknown governing RODEs from observed data often rely on strong prior knowledge.…

Numerical Analysis · Mathematics 2020-06-04 Junyu Liu , Zichao Long , Ranran Wang , Jie Sun , Bin Dong

Differential equations arise in mathematics, physics,medicine, pharmacology, communications, image processing and animation, etc. An Ordinary Differential Equation (ODE) is a differential equation if it involves derivatives with respect to…

Mathematical Software · Computer Science 2015-12-31 A. O. Anidu , S. A. Arekete , A. O. Adedayo , A. O. Adekoya

The time evolution of dynamical systems is frequently described by ordinary differential equations (ODEs), which must be solved for given initial conditions. Most standard approaches numerically integrate ODEs producing a single solution…

Machine Learning · Computer Science 2020-06-26 Cedric Flamant , Pavlos Protopapas , David Sondak

Ordinary differential equations (ODEs) are the primary means to modelling dynamical systems in many natural and engineering sciences. The number of equations required to describe a system with high heterogeneity limits our capability of…

Mathematical Software · Computer Science 2017-07-17 Andrea Vandin

Transformer networks have become the preferred architecture for many tasks due to their state-of-the-art performance. However, the optimal way to implement residual connections in Transformer, which are essential for effective training, is…

Computation and Language · Computer Science 2023-05-01 Shufang Xie , Huishuai Zhang , Junliang Guo , Xu Tan , Jiang Bian , Hany Hassan Awadalla , Arul Menezes , Tao Qin , Rui Yan

Ordinary differential equation (ODE) is widely used in modeling biological and physical processes in science. In this article, we propose a new reproducing kernel-based approach for estimation and inference of ODE given noisy observations.…

Methodology · Statistics 2021-10-26 Xiaowu Dai , Lexin Li

Statistical regression models whose mean functions are represented by ordinary differential equations (ODEs) can be used to describe phenomenons dynamical in nature, which are abundant in areas such as biology, climatology and genetics. The…

Methodology · Statistics 2017-05-15 Kyoungjae Lee , Jaeyong Lee , Sarat C. Dass

Residual connections are pivotal for deep neural networks, enabling greater depth by mitigating vanishing gradients. However, in standard residual updates, the module's output is directly added to the input stream. This can lead to updates…

Computer Vision and Pattern Recognition · Computer Science 2026-01-13 Giyeong Oh , Woohyun Cho , Siyeol Kim , Suhwan Choi , Youngjae Yu

Large Transformer models routinely achieve state-of-the-art results on a number of tasks but training these models can be prohibitively costly, especially on long sequences. We introduce two techniques to improve the efficiency of…

Machine Learning · Computer Science 2020-02-19 Nikita Kitaev , Łukasz Kaiser , Anselm Levskaya