Related papers: A Dynamical Model for Spatio-Temporal Processes Mo…
Dynamical processes can be classified in various ways as deterministic or stochastic, and continuous or discrete time. All these types can be studied by the path-spaces they generate, and stationary measures on that path-space. Such…
We introduce a novel grid-independent model for learning partial differential equations (PDEs) from noisy and partial observations on irregular spatiotemporal grids. We propose a space-time continuous latent neural PDE model with an…
This manuscript proposes a class of fractional stochastic integro-differential equation (FSIDE) with non-instantaneous impulses in an arbitrary separable Hilbert space. We use a projection scheme of increasing sequence of finite dimensional…
Turbulent dynamical systems are characterized by nonlinear interactions and stochastic effects that generate coupled statistical quantities, such as non-zero higher-order moments, which are difficult to capture from data with accuracy. We…
We introduce a guided stochastic sampling method that augments sampling from diffusion models with physics-based guidance derived from partial differential equation (PDE) residuals and observational constraints, ensuring generated samples…
This chapter reviews and compares discontinuous Galerkin time-stepping methods for the numerical approximation of second-order ordinary differential equations, particularly those stemming from space finite element discretization of wave…
Stochastic reduced-order models are widely used to represent the effective dynamics of complex systems, but estimating their drift and diffusion coefficients from data remains challenging. Standard approaches often rely on short-time…
The last decades have seen an unprecedented increase in the availability of data sets that are inherently global and temporally evolving, from remotely sensed networks to climate model ensembles. This paper provides a view of statistical…
We consider the numerical approximation of stochastic partial differential equations (SPDEs) based models for a quasi-periodic climate pattern in the tropical Pacific Ocean known as El Ni\~no phenomenon. We show that for these models the…
We introduce a discrete time microscopic single particle model for kinetic transport. The kinetics is modeled by a two-state Markov chain, the transport by deterministic advection plus a random space step. The position of the particle after…
In this paper, we introduce a class of stochastic partial differential equations (SPDEs) with fractional time-derivatives, and study the $L_2$-theory of the equations. This class of SPDEs can be used to describe random effects on transport…
For spatiotemporal chaos described by partial differential equations, there are generally locations where the dynamical variable achieves its local extremum or where the time partial derivative of the variable vanishes instantaneously. To a…
We consider a stochastic partial differential equation (SPDE) which describes the velocity field of a viscous, incompressible non-Newtonian fluid subject to a random force. Here, the extra stress tensor of the fluid is given by a polynomial…
This thesis is mainly concerned with state-space approaches for solving deep (temporal) Gaussian process (DGP) regression problems. More specifically, we represent DGPs as hierarchically composed systems of stochastic differential equations…
We consider linear dynamical systems of ordinary differential equations or differential algebraic equations. Physical parameters are substituted by random variables for an uncertainty quantification. We expand the state variables as well as…
We introduce a general framework for approximating parabolic Stochastic Partial Differential Equations (SPDEs) based on fluctuation-dissipation balance. Using this approach we formulate Stochastic Discontinuous Galerkin Methods (SDGM). We…
This manuscript is devoted to the study of a class of nonlinear non-instantaneous impulsive first order abstract retarded type functional differential equations in an arbitrary separable Hilbert space H. A new set of sufficient conditions…
The Latent Stochastic Differential Equation (SDE) is a powerful tool for time series and sequence modeling. However, training Latent SDEs typically relies on adjoint sensitivity methods, which depend on simulation and backpropagation…
We analyze a system of nonlinear stochastic partial differential equations (SPDEs) of mixed elliptic-parabolic type that models the propagation of electric signals and their effect on the deformation of cardiac tissue. The system governs…
We study parametric estimation for second order linear parabolic stochastic partial differential equations (SPDEs) in two space dimensions driven by two types of $Q$-Wiener processes based on high frequency spatio-temporal data. First, we…