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The ocean wave distribution in a specific region of space and time is described by its sea state. Knowledge about the sea states a ship encounters on a journey can be used to assess various parameters of risk and wear associated with the…
Animal groups exhibit emergent properties that are a consequence of local interactions. Linking individual-level behaviour to coarse-grained descriptions of animal groups has been a question of fundamental interest. Here, we present two…
Stochastic difference equations and a stochastic partial differential equation (SPDE) are simultaneously derived for the time-dependent neutron angular density in a general three-dimensional medium where the neutron angular density is a…
Are there hidden dynamical common patterns in the evolution of social and cultural history? While the growing availability of digitized social data invites us to answer this question, prevailing quantitative methods often rely on…
We propose a new type of SPDEs, singular or with regularized noises, motivated by a study of the fluctuation of the density field in a microscopic interacting particle system. They include a large scaling parameter $N$, which is the ratio…
A new method for constructing of composite coherent states of the hydrogen atom, based on the dynamical group approach and various schemes of reduction to subgroups, is presented. A wide class of well-localized (Gaussian) hydrogenic wave…
We consider a stochastic partial differential equation (SPDE) on a lattice \partial_t X=(\Delta-m^2)X-\lambda X^p+\eta where $\eta$ is a space-time L\'evy noise. A perturbative (in the sense of formal power series) strong solution is given…
A fully discrete approximation of the semi-linear stochastic wave equation driven by multiplicative noise is presented. A standard linear finite element approximation is used in space and a stochastic trigonometric method for the temporal…
We propose a predictor-corrector adaptive method for the study of hyperbolic partial differential equations (PDEs) under uncertainty. Constructed around the framework of stochastic finite volume (SFV) methods, our approach circumvents…
Modeling real-world problems with partial differential equations (PDEs) is a prominent topic in scientific machine learning. Classic solvers for this task continue to play a central role, e.g. to generate training data for deep learning…
Uncertainty quantification appears today as a crucial point in numerous branches of science and engineering. In the past two decades, a growing interest has been devoted to stochastic finite element method (SFEM) for the propagation of…
Data-driven discovery of partial differential equations (PDEs) has attracted increasing attention in recent years. Although significant progress has been made, certain unresolved issues remain. For example, for PDEs with high-order…
We consider statistics for stochastic evolution equations in Hilbert space with emphasis on stochastic partial differential equations (SPDEs). We observe a solution process under additional measurement errors and want to estimate a real or…
We investigate numerical methods for wave equations in $n+2$ spacetime dimensions, written in spherical coordinates, decomposed in spherical harmonics on $S^n$, and finite-differenced in the remaining coordinates $r$ and $t$. Such an…
This paper aims to investigate numerical approximation of a general second order non-autonomous semilinear parabolic stochastic partial differential equation (SPDE) driven by multiplicative noise. Numerical approximations of autonomous…
In this paper, we study large and moderate deviation principles for stochastic partial differential equations (SPDEs) on metric graphs and their associated multiscale models via the weak convergence approach, providing a refined…
We present a nonlinear dynamical approximation method for time-dependent Partial Differential Equations (PDEs). The approach makes use of parametrized decoder functions, and provides a general, and principled way of understanding and…
In this paper, we address the question of the discretization of Stochastic Partial Differential Equations (SPDE's) for excitable media. Working with SPDE's driven by colored noise, we consider a numerical scheme based on finite differences…
The stochastic interpolant framework offers a powerful approach for constructing generative models based on ordinary differential equations (ODEs) or stochastic differential equations (SDEs) to transform arbitrary data distributions.…
The multiscale complexity of modern problems in computational science and engineering can prohibit the use of traditional numerical methods in multi-dimensional simulations. Therefore, novel algorithms are required in these situations to…