Abstract The parametric surrogate models for partial differential equations (PDEs) are a necessary component many applications in computational sciences, and the convolutional neural networks (CNNs) have proven to be an excellent tool generate these surrogates when fields present. CNNs commonly trained on labeled data based one-to-one sets of parameter-input PDE-output fields. Recently, residua...

Green’s function plays a significant role in both theoretical analysis and numerical computing of partial differential equations (PDEs). However, most cases, is difficult to compute. The troubles arise the following threefold. Firstly, compared with original PDE, dimension doubled, making it impossible be handled by traditional mesh-based methods. Secondly, usually contains singularities which ...

We study a stochastic spatial epidemic model where the N individuals carry two features: position and an infection state, interact move in $${\mathbb {R}}^d$$ . In this Markovian model, evolution of states are described with help Poisson Point Processes , whereas displacement driven by mean field interactions, (state dependence) diffusion also common noise, so that dynamic is random process. pr...

The aim of this paper is threefold. Firstly, we prove the existence and uniqueness a global strong (in both probabilistic PDE senses) $\mathrm{H}^{1}\_2$-valued solution to 2D stochastic Navier–Stokes equations (SNSEs) driven by multiplicative Lévy noise under natural Lipschitz condition on balls linear growth assumptions jump coefficient. Secondly, Girsanov-type theorem for Poisson random meas...

We consider the spatially inhomogeneous non-cutoff Boltzmann equation with moderately soft potentials and any singularity parameter $$s\in (0,1)$$ , i.e. $$\gamma +2s\in (0,2)$$ on whole space $${\mathbb {R}}^3$$ . prove that if initial data $$f_{{{\,\mathrm{in}\,}}}$$ are close to vacuum solution $$f_{\text {vac}}=0$$ in an appropriate weighted norm then f remains regular globally time approac...

Finite element method is one of powerful numerical methods to solve PDE. Usually, if a finite element solution to a Poisson equation based on a triangulation of the underlying domain is not accurate enough, one will discard the solution and then refine the triangulation uniformly and compute a new finite element solution over the refined triangulation. It is wasteful to discard the original fin...

We present a spectrally accurate embedded boundary method for solving linear, inhomogeneous, elliptic partial differential equations (PDE) in general smooth geometries two dimensions, focusing this manuscript on the Poisson, modified Helmholtz, and Stokes equations. Unlike several recently proposed methods which rely function extension, we propose instead utilizes intension, or truncation of kn...

A method for trimming surfaces generated as solutions to Partial Differential Equations (PDEs) is presented. The work we present here utilises the 2D parameter space on which the trim curves are defined whose projection on the parametrically represented PDE surface is then trimmed out. To do this we define the trim curves to be a set of boundary conditions which enable us to solve a low order e...

This paper presents a method to extend PDE surfaces to high dimensional spaces. We review a common existing analytic solution, and show how it can be used straightforwardly to increase the dimension of the space the surface is embedded within. We then further develop a numerical scheme suitable for increasing the number of variables that parametrise the surface, and investigate some of the prop...