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We study the diffusion equation with an appropriate change of variables. This equation is in general a partial differential equation (PDE). With the self-similar and related Ansat\"atze we transform the PDE of diffusion to an ordinary…

When the parameters are independently and identically distributed (initialized) neural networks exhibit undesirable properties that emerge as the number of layers increases, e.g. a vanishing dependency on the input and a concentration on…

Solving partial differential equations (PDEs) using neural networks has become a central focus in scientific machine learning. Training neural networks for singularly perturbed problems is particularly challenging due to certain parameters…

In a previous article we have shown how one can employ Artificial Neural Networks (ANNs) in order to solve non-homogeneous ordinary and partial differential equations. In the present work we consider the solution of eigenvalue problems for…

Iterative multiscale methods for electronic structure calculations offer several advantages for large-scale problems. Here we examine a nonlinear full approximation scheme (FAS) multigrid method for solving fixed potential and…

Diffusion models have emerged as a promising class of generative models that map noisy inputs to realistic images. More recently, they have been employed to generate solutions to partial differential equations (PDEs). However, they still…

We propose machine learning methods for solving fully nonlinear partial differential equations (PDEs) with convex Hamiltonian. Our algorithms are conducted in two steps. First the PDE is rewritten in its dual stochastic control…

Deep learning-based partial differential equation(PDE) solvers have received much attention in the past few years. Methods of this category can solve a wide range of PDEs with high accuracy, typically by transforming the problems into…

Machine learning for scientific applications faces the challenge of limited data. We propose a framework that leverages a priori known physics to reduce overfitting when training on relatively small datasets. A deep neural network is…

We present a method to linearize, without approximation, a specific class of eigenvalue problems with eigenvector nonlinearities (NEPv), where the nonlinearities are expressed by scalar functions that are defined by a quotient of linear…

The present study investigates the dynamics of nonlocal beams by establishing a consistent stress-driven integral elastic using the Physics-Informed Neural Network (PINN) approach. Specifically, a PINN is developed to compute the first…

In this paper, we develop a class of interacting particle Langevin algorithms to solve inverse problems for partial differential equations (PDEs). In particular, we leverage the statistical finite elements (statFEM) formulation to obtain a…

In this work, we study physics-informed neural networks (PINNs) constrained by partial differential equations (PDEs) and their application in approximating PDEs with two characteristic scales. From a continuous perspective, our formulation…

Evolutional deep neural networks (EDNN) solve partial differential equations (PDEs) by marching the network representation of the solution fields, using the governing equations. Use of a single network to solve coupled PDEs on large domains…

The choice of parameters, and the design of the network architecture are important factors affecting the performance of deep neural networks. However, there has not been much work on developing an established and systematic way of building…

Diffusion Probabilistic Models (DPMs) are a well-established class of diffusion models for unconditional image generation, while SGMSE+ is a well-established conditional diffusion model for speech enhancement. One of the downsides of…

A method for approximating sixth-order ordinary differential equations is proposed, which utilizes a deep learning feedforward artificial neural network, referred to as a neural solver. The efficacy of this unsupervised machine learning…

The aim of this paper is to discuss potential advances in PET kinetic models and direct reconstruction of kinetic parameters. As a prominent example we focus on a typical task in perfusion imaging and derive a system of…

We consider the problem of making nonparametric inference in a class of multi-dimensional diffusions in divergence form, from low-frequency data. Statistical analysis in this setting is notoriously challenging due to the intractability of…

Collected and annotated datasets, which are obtained through extensive efforts, are effective for training Deep Neural Network (DNN) models. However, these datasets are susceptible to be misused by unauthorized users, resulting in…