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The solution to partial differential equations using deep learning approaches has shown promising results for several classes of initial and boundary-value problems. However, their ability to surpass, particularly in terms of accuracy,…

By working with the periodic resolvent kernel and Bloch-decomposition, we establish pointwise bounds for the Green function of the linearized equation associated with spatially periodic traveling waves of a system of reaction diffusion…

In this paper, based on the combination of finite element mesh and neural network, a novel type of neural network element space and corresponding machine learning method are designed for solving partial differential equations. The…

Predicting time-dependent dynamics of complex systems governed by non-linear partial differential equations (PDEs) with varying parameters and domains is a challenging task motivated by applications across various fields. We introduce a…

The aim of this paper is to give a systematic mathematical interpretation of the diffusion problem on which Graph Neural Networks (GNNs) models are based. The starting point of our approach is a dissipative functional leading to dynamical…

Physics-informed neural networks solve partial differential equations by training neural networks. Since this method approximates infinite-dimensional PDE solutions with finite collocation points, minimizing discretization errors by…

A numerical explicit method to evaluates transient solutions of linear partial differential non-homogeneous equation with constant coefficients is proposed.

Partial Differential Equations (PDE) are fundamental to model different phenomena in science and engineering mathematically. Solving them is a crucial step towards a precise knowledge of the behaviour of natural and engineered systems. In…

In this paper, we propose a novel machine learning method based on adaptive tensor neural network subspace to solve linear time-fractional diffusion-wave equations and nonlinear time-fractional partial integro-differential equations. In…

A growing body of literature has been leveraging techniques of machine learning (ML) to build novel approaches to approximating the solutions to partial differential equations. Noticeably absent from the literature is a systematic…

Diffusion models have emerged as powerful generative tools with applications in computer vision and scientific machine learning (SciML), where they have been used to solve large-scale probabilistic inverse problems. Traditionally, these…

Strongly correlated materials exhibit complex electronic phenomena that are challenging to capture with traditional theoretical methods, yet understanding these systems is crucial for discovering new quantum materials. Addressing the…

In this paper, we introduce a shallow (one-hidden-layer) physics-informed neural network for solving partial differential equations on static and evolving surfaces. For the static surface case, with the aid of level set function, the…

This short note describes the concept of guided training of deep neural networks (DNNs) to learn physically reasonable solutions. DNNs are being widely used to predict phenomena in physics and mechanics. One of the issues of DNNs is that…

This paper introduces a meta-learning approach for parameterized pseudo-differential operators with deep neural networks. With the help of the nonstandard wavelet form, the pseudo-differential operators can be approximated in a compressed…

Neural operators, which learn mappings between the function spaces, have been applied to solve boundary value problems in various ways, including learning mappings from the space of the forcing terms to the space of the solutions with the…

This paper presents the potential of applying physics-informed neural networks for solving nonlinear multiphysics problems, which are essential to many fields such as biomedical engineering, earthquake prediction, and underground energy…

It is well known that second order linear ordinary differential equations with slowly varying coefficients admit slowly varying phase functions. This observation is the basis of the Liouville-Green method and many other techniques for the…

We proposed a framework for solving inverse problems in differential equations based on neural networks and automatic differentiation. Neural networks are used to approximate hidden fields. We analyze the source of errors in the framework…

Diffusion, a fundamental internal mechanism emerging in many physical processes, describes the interaction among different objects. In many learning tasks with limited training samples, the diffusion connects the labeled and unlabeled data…