Related papers: Learning Runge-Kutta Integration Schemes for ODE S…
Many problems in science and engineering require an efficient numerical approximation of integrals or solutions to differential equations. For systems with rapidly changing dynamics, an equidistant discretization is often inadvisable as it…
We study the learning of numerical algorithms for scientific computing, which combines mathematically driven, handcrafted design of general algorithm structure with a data-driven adaptation to specific classes of tasks. This represents a…
A new approach for the construction of high order A-stable explicit integrators for ordinary differential equations (ODEs) is theoretically studied. Basically, the integrators are obtained by splitting, at each time step, the solution of…
Numerous applications necessitate the computation of numerical solutions to differential equations across a wide range of initial conditions and system parameters, which feeds the demand for efficient yet accurate numerical integration…
A general purpose, modular program package for the integration of large number of independent ordinary differential equation systems capable of using professional graphics cards is presented. The available numerical schemes are the explicit…
The combination of ordinary differential equations and neural networks, i.e., neural ordinary differential equations (Neural ODE), has been widely studied from various angles. However, deciphering the numerical integration in Neural ODE is…
System identification through learning approaches is emerging as a promising strategy for understanding and simulating dynamical systems, which nevertheless faces considerable difficulty when confronted with power systems modeled by…
A semi-implicit-explicit (semi-IMEX) Runge-Kutta (RK) method is proposed for the numerical integration of ordinary differential equations (ODEs) of the form $\mathbf{u}' = \mathbf{f}(t,\mathbf{u}) + G(t,\mathbf{u}) \mathbf{u}$, where…
Differential equations arise in mathematics, physics,medicine, pharmacology, communications, image processing and animation, etc. An Ordinary Differential Equation (ODE) is a differential equation if it involves derivatives with respect to…
Statistical regression models whose mean functions are represented by ordinary differential equations (ODEs) can be used to describe phenomenons dynamical in nature, which are abundant in areas such as biology, climatology and genetics. The…
Applied to the master equation, the usual numerical integration methods, such as Runge-Kutta, become inefficient when the rates associated with various transitions differ by several orders of magnitude. We introduce an integration scheme…
Classical and new numerical schemes are generated using evolutionary computing. Differential Evolution is used to find the coefficients of finite difference approximations of function derivatives, and of single and multi-step integration…
Complex dynamical networks appear in a wide range of physical, biological, and engineering systems. The coupling of subsystems with varying time scales often results in multirate behavior. During the simulation of highly integrated…
Numerical integrators could be used to form interpolation conditions when training neural networks to approximate the vector field of an ordinary differential equation (ODE) from data. When numerical one-step schemes such as the Runge-Kutta…
In this technical note a general procedure is described to construct internally consistent splitting methods for the numerical solution of differential equations, starting from matching pairs of explicit and diagonally implicit Runge-Kutta…
We propose a new probabilistic scheme which combines deep learning techniques with high order schemes for backward stochastic differential equations belonging to the class of Runge-Kutta methods to solve high-dimensional semi-linear…
Implicit Runge--Kutta (IRK) methods are highly effective for solving stiff ordinary differential equations (ODEs) but can be computationally expensive for large-scale problems due to the need of solving coupled algebraic equations at each…
Learning large scale nonlinear ordinary differential equation (ODE) systems from data is known to be computationally and statistically challenging. We present a framework together with the adaptive integral matching (AIM) algorithm for…
We consider the efficient numerical solution of coupled dynamical systems, consisting of a small nonlinear part and a large linear time invariant part, possibly stemming from spatial discretization of an underlying partial differential…
Differential equations in general and neural ODEs in particular are an essential technique in continuous-time system identification. While many deterministic learning algorithms have been designed based on numerical integration via the…