Related papers: Towards information optimal simulation of partial …
We consider Galerkin finite element methods for semilinear stochastic partial differential equations (SPDEs) with multiplicative noise and Lipschitz continuous nonlinearities. We analyze the strong error of convergence for spatially…
The aim of this paper is to develop stable and accurate numerical schemes for boundary integral formulations of the heat equation with Dirichlet boundary conditions. The accuracy of Galerkin discretisations for the resulting boundary…
This paper investigates a numerical probabilistic method for the solution of some semilinear stochastic partial differential equations (SPDEs in short). The numerical scheme is based on discrete time approximation for solutions of systems…
A method is introduced for the construction of meshless discretization schemes which preserve Lie symmetries of the differential equations that these schemes approximate. The method exploits the fact that equivariant moving frames provide a…
We address a physics-informed neural network based on the concept of random projections for the numerical solution of IVPs of nonlinear ODEs in linear-implicit form and index-1 DAEs, which may also arise from the spatial discretization of…
Robust physics (e.g., governing equations and laws) discovery is of great interest for many engineering fields and explainable machine learning. A critical challenge compared with general training is that the term and format of governing…
Differential equations in general and neural ODEs in particular are an essential technique in continuous-time system identification. While many deterministic learning algorithms have been designed based on numerical integration via the…
Partial Differential Equations (PDEs) describe phenomena ranging from turbulence and epidemics to quantum mechanics and financial markets. Despite recent advances in computational science, solving such PDEs for real-world applications…
Inferring the parameters of ordinary differential equations (ODEs) from noisy observations is an important problem in many scientific fields. Currently, most parameter estimation methods that bypass numerical integration tend to rely on…
The physics informed neural network (PINN) is evolving as a viable method to solve partial differential equations. In the recent past PINNs have been successfully tested and validated to find solutions to both linear and non-linear partial…
Numerical models based on partial differential equations (PDE), or integro-differential equations, are ubiquitous in engineering and science, making it possible to understand or design systems for which physical experiments would be…
Spatially distributed problems are often approximately modelled in terms of partial differential equations (PDEs) for appropriate coarse-grained quantities (e.g. concentrations). The derivation of accurate such PDEs starting from finer…
The Fisher information matrix (FIM) is a foundational concept in statistical signal processing. The FIM depends on the probability distribution, assumed to belong to a smooth parametric family. Traditional approaches to estimating the FIM…
We present DeepFDM, a differentiable finite-difference framework for learning spatially varying coefficients in time-dependent partial differential equations (PDEs). By embedding a classical forward-Euler discretization into a convolutional…
We perform numerical analysis of a nonlinear gradient flow, which can be regarded as a parabolic minimal surface problem or a regularised total variation flow, using the gradient discretisation method (GDM). GDM is a unified convergence…
We study fully discrete linearized Galerkin finite element approximations to a nonlinear gradient flow, applications of which can be found in many areas. Due to the strong nonlinearity of the equation, existing analyses for implicit schemes…
In this paper we propose an all-in-one statement which includes existence, uniqueness, regularity, and numerical approximations of mild solutions for a class of stochastic partial differential equations (SPDEs) with non-globally monotone…
We propose a methodology that combines generative latent diffusion models with physics-informed machine learning to generate solutions of parametric partial differential equations (PDEs) conditioned on partial observations, which includes,…
Mechanistic knowledge about the physical world is virtually always expressed via partial differential equations (PDEs). Recently, there has been a surge of interest in probabilistic PDE solvers -- Bayesian statistical models mostly based on…
Sparse system identification of nonlinear dynamic systems is still challenging, especially for stiff and high-order differential equations for noisy measurement data. The use of highly correlated functions makes distinguishing between true…