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A series of recent articles introduced a method to construct stochastic partial differential equations (SPDEs) which are invariant with respect to the distribution of a given conditioned diffusion. These works are restricted to the case of…
The shallow water flow model is widely used to describe water flows in rivers, lakes, and coastal areas. Accounting for uncertainty in the corresponding transport-dominated nonlinear PDE models presents theoretical and numerical challenges…
Irregular sampling intervals and missing values in real-world time series data present challenges for conventional methods that assume consistent intervals and complete data. Neural Ordinary Differential Equations (Neural ODEs) offer an…
Numerical discretisations of partial differential equations (PDEs) can be written as discrete convolutions, which, themselves, are a key tool in AI libraries and used in convolutional neural networks (CNNs). We therefore propose to…
Explicit, unconditionally stable, high-order schemes for the approximation of some first- andsecond-order linear, time-dependent partial differential equations (PDEs) are proposed.The schemes are based on a weak formulation of a…
Many successful methods to learn dynamical systems from data have recently been introduced. However, ensuring that the inferred dynamics preserve known constraints, such as conservation laws or restrictions on the allowed system states,…
Motivated by the modeling of three-dimensional fluid turbulence, we define and study a class of stochastic partial differential equations (SPDEs) that are randomly stirred by a spatially smooth and uncorrelated in time forcing term. To…
Recent work in deep learning focuses on solving physical systems in the Ordinary Differential Equation or Partial Differential Equation. This current work proposed a variant of Convolutional Neural Networks (CNNs) that can learn the hidden…
This article is focused on two related topics within the study of partial differential equations (PDEs) that illustrate a beautiful connection between dynamics, topology, and analysis: stability and spatial dynamics. The first is a property…
The goal of this thesis is the development and implementation of a non-perturbative solution method for Wegner's flow equations. We show that a parameterization of the flowing Hamiltonian in terms of a scalar function allows the flow…
Since its introduction in 2017, physics-informed deep learning (PIDL) has garnered growing popularity in understanding the evolution of systems governed by physical laws in terms of partial differential equations (PDEs). However, empirical…
The static dielectric constant of liquid water is computed using classical force field based molecular dynamics simulation at fixed electric displacement D. The method to constrain the electric displacement is the finite temperature…
Every linear system of partial differential equations (PDEs) admits a scaling symmetry in its dependent variables. In conjunction with other admitted symmetries of linear type, the associated invariant solution condition poses a linear…
We develop an unsupervised machine learning algorithm for the automated discovery and identification of traveling waves in spatio-temporal systems governed by partial differential equations (PDEs). Our method uses sparse regression and…
Over the centuries mathematicians have been challenged by the partial differential equations (PDEs) that describe the motion of fluids in many physical contexts. Important and beautiful results were obtained in the past one hundred years,…
When nonconforming discontinuous Galerkin methods are implemented for hyperbolic equations using quadrature, exponential energy growth can result even when the underlying scheme with exact integration does not support such growth. Using…
This work focuses on the accuracy and stability of high-order nodal discontinuous Galerkin (DG) methods for under-resolved turbulence computations. In particular we consider the inviscid Taylor-Green vortex (TGV) flow to analyse the…
Learning the full family of solutions to parameterized partial differential equations (PDEs) is a central challenge to our ability to model the behavior of heterogeneous systems, with a variety of fundamental and application-oriented…
Partial differential equations (PDEs) are among the most universal and parsimonious descriptions of natural physical laws, capturing a rich variety of phenomenology and multi-scale physics in a compact and symbolic representation. This…
Neural differential equations offer a powerful approach for learning dynamics from data. However, they do not impose known constraints that should be obeyed by the learned model. It is well-known that enforcing constraints in surrogate…