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We study the numerical approximation of a class of degenerate parabolic stochastic partial differential equations on non-compact metric graphs, which naturally arise in the asymptotic analysis of Hamiltonian flows under small noise…
A direct method for the computation of polynomial conservation laws of polynomial systems of nonlinear partial differential equations (PDEs) in multi-dimensions is presented. The method avoids advanced differential-geometric tools. Instead,…
This paper aims to investigate the numerical approximation of semilinear non-autonomous stochastic partial differential equations (SPDEs) driven by multiplicative or additive noise. Such equations are more realistic than autonomous SPDEs…
Many stochastic differential equations (SDEs) in the literature have a superlinearly growing nonlinearity in their drift or diffusion coefficient. Unfortunately, moments of the computationally efficient Euler-Maruyama approximation method…
The behavior of many dynamical systems follow complex, yet still unknown partial differential equations (PDEs). While several machine learning methods have been proposed to learn PDEs directly from data, previous methods are limited to…
This work proposes stochastic partial differential equations (SPDEs) as a practical tool to replicate clustering effects of more detailed particle-based dynamics. Inspired by membrane-mediated receptor dynamics on cell surfaces, we…
This paper considers the filtering problem which consists in reconstructing the state of a dynamical system with partial observations coming from sensor measurements, and the knowledge that the dynamics are governed by a physical PDE model…
Stochastic differential equations (SDEs) are established tools to model physical phenomena whose dynamics are affected by random noise. By estimating parameters of an SDE intrinsic randomness of a system around its drift can be identified…
Models incorporating uncertain inputs, such as random forces or material parameters, have been of increasing interest in PDE-constrained optimization. In this paper, we focus on the efficient numerical minimization of a convex and smooth…
Traditional nonparametric estimation methods often lead to a slow convergence rate in large dimensions and require unrealistically enormous sizes of datasets for reliable conclusions. We develop an approach based on partial derivatives,…
We present a new scientific machine learning method that learns from data a computationally inexpensive surrogate model for predicting the evolution of a system governed by a time-dependent nonlinear partial differential equation (PDE), an…
We consider the problem of estimating parameters of stochastic differential equations (SDEs) with discrete-time observations that are either completely or partially observed. The transition density between two observations is generally…
Physical laws governing population dynamics are generally expressed as differential equations. Research in recent decades has incorporated fractional-order (non-integer) derivatives into differential models of natural phenomena, such as…
In this paper we propose a new method to stabilise non-symmetric indefinite problems. The idea is to solve a forward and an adjoint problem simultaneously using a suitable stabilised finite element method. Both stabilisation of the element…
In this paper, we present the Partial Integral Equation (PIE) representation of linear Partial Differential Equations (PDEs) in one spatial dimension, where the PDE has spatial integral terms appearing in the dynamics and the boundary…
In this paper we present the theoretical framework needed to justify the use of a kernel-based collocation method (meshfree approximation method) to estimate the solution of high-dimensional stochastic partial differential equations…
The present work introduces and investigates an explicit time discretization scheme, called the projected Euler method,to numerically approximate random periodic solutions of semi-linear SDEs under non-globally Lipschitz conditions. The…
Scientific research and engineering practice often require the modeling and decomposition of nonlinear systems. The Dynamic Mode Decomposition (DMD) is a novel Koopman-based technique that effectively dissects high-dimensional nonlinear…
This article deals with the approximation of a stochastic partial differential equation (SPDE) via amplitude equations. We consider an SPDE with a cubic nonlinearity perturbed by a general multiplicative noise that preserves the constant…
One of the pivotal tasks in scientific machine learning is to represent underlying dynamical systems from time series data. Many methods for such dynamics learning explicitly require the derivatives of state data, which are not directly…