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Fractional-order differential equations (FDEs) enhance traditional differential equations by extending the order of differential operators from integers to real numbers, offering greater flexibility in modeling complex dynamical systems…

There is a wave of interest in using unsupervised neural networks for solving differential equations. The existing methods are based on feed-forward networks, {while} recurrent neural network differential equation solvers have not yet been…

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…

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. We first show that a residual block of layers in…

We propose to formulate MRI image reconstruction as an optimization problem and model the optimization trajectory as a dynamic process using ordinary differential equations (ODEs). We model the dynamics in ODE with a neural network and…

Training Artificial Neural Networks poses a challenging and critical problem in machine learning. Despite the effectiveness of gradient-based learning methods, such as Stochastic Gradient Descent (SGD), in training neural networks, they do…

Does the use of auto-differentiation yield reasonable updates for deep neural networks (DNNs)? Specifically, when DNNs are designed to adhere to neural ODE architectures, can we trust the gradients provided by auto-differentiation? Through…

One of the mysteries in the success of neural networks is randomly initialized first order methods like gradient descent can achieve zero training loss even though the objective function is non-convex and non-smooth. This paper demystifies…

The neural ordinary differential equation (ODE) framework has emerged as a powerful tool for developing accelerated surrogate models of complex physical systems governed by partial differential equations (PDEs). A popular approach for PDE…

Deep neural networks (DNNs) have provided brilliant performance across various tasks. However, this success often comes at the cost of unnecessarily large model sizes, high computational demands, and substantial memory footprints.…

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…

Neural Ordinary Differential Equations (NODEs), a framework of continuous-depth neural networks, have been widely applied, showing exceptional efficacy in coping with some representative datasets. Recently, an augmented framework has been…

Designing energy-efficient networks is of critical importance for enabling state-of-the-art deep learning in mobile and edge settings where the computation and energy budgets are highly limited. Recently, Liu et al. (2019) framed the search…

The great success neural networks have achieved is inseparable from the application of gradient-descent (GD) algorithms. Based on GD, many variant algorithms have emerged to improve the GD optimization process. The gradient for…

This paper proposes the Nerual Energy Descent (NED) via neural network evolution equations for a wide class of deep learning problems. We show that deep learning can be reformulated as the evolution of network parameters in an evolution…

Large-scale machine learning models are often trained by parallel stochastic gradient descent algorithms. However, the communication cost of gradient aggregation and model synchronization between the master and worker nodes becomes the…

Finding parameters in a deep neural network (NN) that fit training data is a nonconvex optimization problem, but a basic first-order optimization method (gradient descent) finds a global optimizer with perfect fit (zero-loss) in many…

Neural networks are a popular tool for modeling sequential data but they generally do not treat time as a continuous variable. Neural ODEs represent an important exception: they parameterize the time derivative of a hidden state with a…

In the realm of efficient on-device learning under extreme memory and computation constraints, a significant gap in successful approaches persists. Although considerable effort has been devoted to efficient inference, the main obstacle to…

Scientific machine learning is an emerging field that broadly describes the combination of scientific computing and machine learning to address challenges in science and engineering. Within the context of differential equations, this has…