Related papers: An adaptive wavelet method for nonlinear partial d…
Can neural networks learn to solve partial differential equations (PDEs)? We investigate this question for two (systems of) PDEs, namely, the Poisson equation and the steady Navier--Stokes equations. The contributions of this paper are…
We develop a new spatial semidiscrete multiscale method based upon the edge multiscale methods to solve semilinear parabolic problems with heterogeneous coefficients and smooth initial data. This method allows for a cheap spatial…
We consider a mesh-based approach for training a neural network to produce field predictions of solutions to parametric partial differential equations (PDEs). This approach contrasts current approaches for "neural PDE solvers" that employ…
Mixed-dimensional partial differential equations (PDEs) are characterized by coupled operators defined on domains of varying dimensions and pose significant computational challenges due to their inherent ill-conditioning. Moreover, the…
Physics-informed deep learning often faces optimization challenges due to the complexity of solving partial differential equations (PDEs), which involve exploring large solution spaces, require numerous iterations, and can lead to unstable…
Solving partial differential equations (PDEs) by numerical methods meet computational cost challenge for getting the accurate solution since fine grids and small time steps are required. Machine learning can accelerate this process, but…
This paper introduces a computationally efficient algorithm in system theory for solving inverse problems governed by linear partial differential equations (PDEs). We model solutions of linear PDEs using Gaussian processes with priors…
Physical models with uncertain inputs are commonly represented as parametric partial differential equations (PDEs). That is, PDEs with inputs that are expressed as functions of parameters with an associated probability distribution.…
Unveiling the underlying governing equations of nonlinear dynamic systems remains a significant challenge. Insufficient prior knowledge hinders the determination of an accurate candidate library, while noisy observations lead to imprecise…
In many scientific fields, the generation and evolution of data are governed by partial differential equations (PDEs) which are typically informed by established physical laws at the macroscopic level to describe general and predictable…
Equations governing physico-chemical processes are usually known at microscopic spatial scales, yet one suspects that there exist equations, e.g. in the form of Partial Differential Equations (PDEs), that can explain the system evolution at…
In this paper we propose a wavelet-based methodology for estimation and variable selection in partially linear models. The inference is conducted in the wavelet domain, which provides a sparse and localized decomposition appropriate for…
In this work we develop a novel domain splitting strategy for the solution of partial differential equations. Focusing on a uniform discretization of the $d$-dimensional advection-diffusion equation, our proposal is a two-level algorithm…
In theory, diffusion curves promise complex color gradations for infinite-resolution vector graphics. In practice, existing realizations suffer from poor scaling, discretization artifacts, or insufficient support for rich boundary…
Partial differential equations have a wide range of applications in modeling multiple physical, biological, or social phenomena. Therefore, we need to approximate the solutions of these equations in computationally feasible terms. Nowadays,…
The ultimate goal of any numerical scheme for partial differential equations (PDEs) is to compute an approximation of user-prescribed accuracy at quasi-minimal computational time. To this end, algorithmically, the standard adaptive finite…
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…
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…
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…
The measured spatiotemporal response of various physical processes is utilized to infer the governing partial differential equations (PDEs). We propose SimultaNeous Basis Function Approximation and Parameter Estimation (SNAPE), a technique…