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Non-linear convection-reaction-diffusion (CRD) partial differential equations (PDEs) are crucial for modeling complex phenomena in fields such as biology, ecology, population dynamics, physics, and engineering. Numerical approximation of…

While the depth of modern Convolutional Neural Networks (CNNs) surpasses that of the pioneering networks with a significant margin, the traditional way of appending supervision only over the final classifier and progressively propagating…

We consider a nonlinear variational wave equation that models the dynamics of the director field in nematic liquid crystals with high molecular rotational inertia. Being derived from an energy principle, energy stability is an intrinsic…

This paper is concerned with structure-preserving numerical approximations for a class of nonlinear nonlocal Fokker-Planck equations, which admit a gradient flow structure and find application in diverse contexts. The solutions,…

Deep learning is emerging as a new paradigm for solving inverse imaging problems. However, the deep learning methods often lack the assurance of traditional physics-based methods due to the lack of physical information considerations in…

Fully connected multilayer perceptrons are used for obtaining numerical solutions of partial differential equations in various dimensions. Independent variables are fed into the input layer, and the output is considered as solution's value.…

In this paper, an innovative Physical Model-driven Neural Network (PMNN) method is proposed to solve time-fractional differential equations. It establishes a temporal iteration scheme based on physical model-driven neural networks which…

A neural network is essentially a high-dimensional complex mapping model by adjusting network weights for feature fitting. However, the spectral bias in network training leads to unbearable training epochs for fitting the high-frequency…

In deep learning, a central issue is to understand how neural networks efficiently learn high-dimensional features. To this end, we explore the gradient descent learning of a general Gaussian Multi-index model…

Peer-to-peer learning is an increasingly popular framework that enables beyond-5G distributed edge devices to collaboratively train deep neural networks in a privacy-preserving manner without the aid of a central server. Neural network…

The probability density function of stochastic differential equations is governed by the Fokker-Planck (FP) equation. A novel machine learning method is developed to solve the general FP equations based on deep neural networks. The proposed…

This paper investigates efficient deep neural networks (DNNs) to replace dense unstructured weight matrices with structured ones that possess desired properties. The challenge arises because the optimal weight matrix structure in popular…

The numerical solution of time-dependent radiative transfer problems is challenging, both, due to the high dimension as well as the anisotropic structure of the underlying integro-partial differential equation. In this paper we propose a…

Understanding the dynamics of neural networks in different width regimes is crucial for improving their training and performance. We present an exact solution for the learning dynamics of a one-hidden-layer linear network, with…

We consider the initial/boundary value problem for a diffusion equation involving multiple time-fractional derivatives on a bounded convex polyhedral domain. We analyze a space semidiscrete scheme based on the standard Galerkin finite…

We introduce a deep residual recurrent neural network (DR-RNN) as an efficient model reduction technique for nonlinear dynamical systems. The developed DR-RNN is inspired by the iterative steps of line search methods in finding the residual…

This paper investigates the learnability of the nonlinearity property of Boolean functions using neural networks. We train encoder style deep neural networks to learn to predict the nonlinearity of Boolean functions from examples of…

While deep learning has achieved remarkable success in solving partial differential equations (PDEs), it still faces significant challenges, particularly when the PDE solutions have low regularity or singularities. To address these issues,…

We consider a class of time dependent second order partial differential equations governed by a decaying entropy. The solution usually corresponds to a density distribution, hence positivity (non-negativity) is expected. This class of…

We explore a data-driven approach for learning to optimize neural networks. We construct a dataset of neural network checkpoints and train a generative model on the parameters. In particular, our model is a conditional diffusion transformer…