Related papers: Stability-Optimized High Order Methods and Stiffne…
This paper studies the robustness of a PDE backstepping delay-compensated boundary controller for a reaction-diffusion partial differential equation (PDE) with respect to a nominal delay subject to stochastic error disturbance. The…
In this paper, we present a deep learning-based numerical method for approximating high dimensional stochastic partial differential equations (SPDEs). At each time step, our method relies on a predictor-corrector procedure. More precisely,…
Numerical resolution of high-dimensional nonlinear PDEs remains a huge challenge due to the curse of dimensionality. Starting from the weak formulation of the Lawson-Euler scheme, this paper proposes a stochastic particle method (SPM) by…
Delayed processes are ubiquitous in biological systems and are often characterized by delay differential equations (DDEs) and their extension to include stochastic effects. DDEs do not explicitly incorporate intermediate states associated…
The problem of solving stochastic differential-algebraic equations (SDAEs) of index one with a scalar driving Brownian motion is considered. Recently, the authors proposed a class of stiffly accurate stochastic Runge-Kutta (SRK) methods…
In this work we analyze the resort to high order exponential solvers for stiff ODEs in the context of cardiac electrophysiology modeling. The exponential Adams-Bashforth and the Rush-Larsen schemes will be considered up to order 4. These…
Review of implicit methods of integrating system of stiff ordinary differential equations is presented. Defines and graphically presents absolute stability region for Gears methods (backward differentiation formula) used to solve system of…
We study the scaling limits of stochastic gradient descent (SGD) with constant step-size in the high-dimensional regime. We prove limit theorems for the trajectories of summary statistics (i.e., finite-dimensional functions) of SGD as the…
We propose a physics-informed consistency modeling framework for solving partial differential equations (PDEs) via fast, few-step generative inference. We identify a key stability challenge in physics-constrained consistency training, where…
In this study, a fast and stable machine-learned hybrid algorithm implemented in TensorFlow for the integration of stiff chemical kinetics is introduced. Numerical solutions to differential equations are at the core of computational fluid…
Classical neural ODEs trained with explicit methods are intrinsically limited by stability, crippling their efficiency and robustness for stiff learning problems that are common in graph learning and scientific machine learning. We present…
We discover restrained numerical instabilities in current training practices of deep networks with stochastic gradient descent (SGD), and its variants. We show numerical error (on the order of the smallest floating point bit and thus the…
The rigorous stability analysis of high-order implicit-explicit multistep (IEMS) methods for nonlinear parabolic equations by using discrete energy arguments is a long standing open issue due to their non-A-stable property. A novel…
We develop and analyze a stabilization term for cut finite element approximations of an elliptic second order partial differential equation on a surface embedded in $\mathbb{R}^d$. The new stabilization term combines properly scaled normal…
Recent applications (e.g. active gels and self-assembly of elastic sheets) motivate the need to efficiently simulate the dynamics of thin elastic sheets. We present semi-implicit time stepping algorithms to improve the time step constraints…
This work introduces a new method to efficiently solve optimization problems constrained by partial differential equations (PDEs) with uncertain coefficients. The method leverages two sources of inexactness that trade accuracy for speed:…
We present a method for approximating solutions of Stochastic Differential Equations (SDEs) with arbitrary rates. This approximation is derived for bounded and measurable test functions. Specifically, we demonstrate that, leveraging the…
We propose a second-order implicit-explicit (IMEX) time-stepping scheme for the isentropic, compressible Cahn-Hilliard-Navier-Stokes equations in the low Mach number regime. The method is based on finite differences on staggered grids and…
We present an adaptive arbitrary-order accurate time-stepping numerical scheme for the flow of vesicles suspended in Stokesian fluids. Our scheme can be summarized as an approximate implicit spectral deferred correction (SDC) method.…
We investigate neural ordinary and stochastic differential equations (neural ODEs and SDEs) to model stochastic dynamics in fully and partially observed environments within a model-based reinforcement learning (RL) framework. Through a…