Related papers: A Discrete Resolvent Framework for Operator Splitt…

The solvability of a delay differential equation arising in the construction of quadratic cost functionals, i.e. Lyapunov functionals, for a linear time-delay system with a constant and a distributed delay is investigated. We present a…

Integral deferred correction (IDC) methods have been shown to be an efficient way to achieve arbitrary high order accuracy and possess good stability properties. In this paper, we construct high order operator splitting schemes using the…

Latent ODE models provide flexible descriptions of dynamic systems, but they can struggle with extrapolation and predicting complicated non-linear dynamics. The latent ODE approach implicitly relies on encoders to identify unknown system…

Efficient high order numerical methods for evolving the solution of an ordinary differential equation are widely used. The popular Runge--Kutta methods, linear multi-step methods, and more broadly general linear methods, all have a global…

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…

Modeling sequential patterns from data is at the core of various time series forecasting tasks. Deep learning models have greatly outperformed many traditional models, but these black-box models generally lack explainability in prediction…

Partial differential equations with discrete (concentrated) state-dependent delays are studied. The existence and uniqueness of solutions with initial data from a wider linear space is proven first and then a subset of the space of…

Coupled multi-physics problems are encountered in countless applications and pose significant numerical challenges. Although monolithic approaches offer possibly the best solution strategy, they often require ad-hoc preconditioners and…

We present a framework for recovering/approximating unknown time-dependent partial differential equation (PDE) using its solution data. Instead of identifying the terms in the underlying PDE, we seek to approximate the evolution operator of…

The solvability and stability analysis of linear time invariant systems of delay differential-algebraic equations (DDAEs) is analyzed. The behavior approach is applied to DDAEs in order to establish characterizations of their solvability in…

Stiff ordinary differential equations (ODEs) are common in many science and engineering fields, but standard neural ODE approaches struggle to accurately learn these stiff systems, posing a significant barrier to widespread adoption of…

This work aims to construct an efficient and highly accurate numerical method to address the time singularity at $t=0$ involved in a class of time-fractional parabolic integro-partial differential equations in one and two dimensions. The…

We propose an Exponential DG approach for numerically solving partial differential equations (PDEs). The idea is to decompose the governing PDE operators into linear (fast dynamics extracted by linearization) and nonlinear (the remaining…

This paper focuses on explicit approximations for nonlinear stochastic delay differential equations (SDDEs). Under the weakly local Lipschitz and some suitable conditions, a generic truncated Euler-Maruyama (TEM) scheme for SDDEs is…

The pure time delay operator is considered in continuous and discrete time under the assumption of the input signal being integrable (summable) with square. By making use of a discrete convolution operator with polynomial Markov parameters,…

This paper develops methods for numerically solving stochastic delay-differential equations (SDDEs) with multiple fixed delays that do not align with a uniform time mesh. We focus on numerical schemes of strong convergence orders $1/2$ and…

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 analyze temporal approximation schemes based on overlapping domain decompositions. As such schemes enable computations on parallel and distributed hardware, they are commonly used when integrating large-scale parabolic systems. Our…

In this work, we use the monolithic convex limiting (MCL) methodology to enforce relevant inequality constraints in implicit finite element discretizations of the compressible Euler equations. In this context, preservation of invariant…

Learning how complex dynamical systems evolve over time is a key challenge in system identification. For safety critical systems, it is often crucial that the learned model is guaranteed to converge to some equilibrium point. To this end,…