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A new and efficient neural-network and finite-difference hybrid method is developed for solving Poisson equation in a regular domain with jump discontinuities on embedded irregular interfaces. Since the solution has low regularity across…
We introduce a new class of spatially stochastic physics and data informed deep latent models for parametric partial differential equations (PDEs) which operate through scalable variational neural processes. We achieve this by assigning…
A new numerical domain decomposition method is proposed for solving elliptic equations on compact Riemannian manifolds. The advantage of this method is to avoid global triangulations or grids on manifolds. Our method is numerically tested…
Solving high dimensional partial differential equations (PDEs) has historically posed a considerable challenge when utilizing conventional numerical methods, such as those involving domain meshes. Recent advancements in the field have seen…
We are concerned with the numerical solution of a class integro-differential equations, known as Neural Field Equations, which describe the large-scale dynamics of spatially structured networks of neurons. These equations have many…
Partial differential equations (PDEs) are often computationally challenging to solve, and in many settings many related PDEs must be be solved either at every timestep or for a variety of candidate boundary conditions, parameters, or…
In this study, we present a novel computational framework that integrates the finite volume method with graph neural networks to address the challenges in Physics-Informed Neural Networks(PINNs). Our approach leverages the flexibility of…
Fast and accurate solutions of time-dependent partial differential equations (PDEs) are of pivotal interest to many research fields, including physics, engineering, and biology. Generally, implicit/semi-implicit schemes are preferred over…
Developing efficient methods for solving parametric partial differential equations is crucial for addressing inverse problems. This work introduces a Least-Squares-based Neural Network (LS-Net) method for solving linear parametric PDEs. It…
In this paper, we present a new iterative approximate method of solving boundary value problems. The idea is to compute approximate polynomial solutions in the Bernstein form using least squares approximation combined with some properties…
In this paper we show how to augment classical methods for inverse problems with artificial neural networks. The neural network acts as a prior for the coefficient to be estimated from noisy data. Neural networks are global, smooth function…
We deal with probabilistic numerical solutions for linear elliptic equations with Neumann boundary conditions in a Lipschitz domain, by using a probabilistic numerical scheme introduced by Milstein and Tretyakov based on new numerical layer…
Linearized shallow neural networks that are constructed by fixing the hidden-layer parameters have recently shown strong performance in solving partial differential equations (PDEs). Such models, widely used in the random feature method…
Recent work has introduced a simple numerical method for solving partial differential equations (PDEs) with deep neural networks (DNNs). This paper reviews and extends the method while applying it to analyze one of the most fundamental…
Deriving governing equations in Electromagnetic (EM) environment based on first principles can be quite tough when there are some unknown sources of noise and other uncertainties in the system. For nonlinear multiple-physics electromagnetic…
We present a structure-preserving Eulerian algorithm for solving $L^2$-gradient flows and a structure-preserving Lagrangian algorithm for solving generalized diffusions. Both algorithms employ neural networks as tools for spatial…
The $P_1$--nonconforming quadrilateral finite element space with periodic boundary condition is investigated. The dimension and basis for the space are characterized with the concept of minimally essential discrete boundary conditions. We…
The purpose of this article is to study quasi linear parabolic partial differential equations of second order, posed on a bounded network, satisfying a nonlinear and non dynamical Neumann boundary condition at the vertices. We prove the…
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…
Using deep neural networks to solve PDEs has attracted a lot of attentions recently. However, why the deep learning method works is falling far behind its empirical success. In this paper, we provide a rigorous numerical analysis on deep…