Related papers: Hybrid solver methods for ODEs: Machine-Learning c…

A conventional approach to train neural ordinary differential equations (ODEs) is to fix an ODE solver and then learn the neural network's weights to optimize a target loss function. However, such an approach is tailored for a specific…

Machine learning methods have been successful in many areas, like image classification and natural language processing. However, it still needs to be determined how to apply ML to areas with mathematical constraints, like solving PDEs.…

We study gradient-based optimization methods obtained by directly discretizing a second-order ordinary differential equation (ODE) related to the continuous limit of Nesterov's accelerated gradient method. When the function is smooth…

Modern deep learning algorithms use variations of gradient descent as their main learning methods. Gradient descent can be understood as the simplest Ordinary Differential Equation (ODE) solver; namely, the Euler method applied to the…

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…

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…

Recent years have seen a growing interest in understanding acceleration methods through the lens of ordinary differential equations (ODEs). Despite the theoretical advancements, translating the rapid convergence observed in continuous-time…

Hybrid algorithms, which combine black-box machine learning methods with experience from traditional numerical methods and domain expertise from diverse application areas, are progressively gaining importance in scientific machine learning…

A key appeal of the recently proposed Neural Ordinary Differential Equation (ODE) framework is that it seems to provide a continuous-time extension of discrete residual neural networks. As we show herein, though, trained Neural ODE models…

When one wishes to numerically solve an initial value problem, it is customary to rewrite it as an equivalent first-order system to which a method, usually from the class of Runge-Kutta methods, is applied. Directly treating higher-order…

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 connections between ordinary differential equations and optimization algorithms in a non-Euclidean setting. We propose a novel accelerated algorithm for minimising convex functions over a convex constrained set. This algorithm…

Runge-Kutta methods are the classic family of solvers for ordinary differential equations (ODEs), and the basis for the state of the art. Like most numerical methods, they return point estimates. We construct a family of probabilistic…

This paper surveys the recent attempts at leveraging machine learning to solve constrained optimization problems. It focuses on surveying the work on integrating combinatorial solvers and optimization methods with machine learning…

Mixed-precision methods combine low and high precision arithmetics to exploit low precision computational speed and high precision accuracy. Large ODE systems that contain many heterogeneous interactions lead to a high computational cost…

Finding accurate solutions to partial differential equations (PDEs) is a crucial task in all scientific and engineering disciplines. It has recently been shown that machine learning methods can improve the solution accuracy by correcting…

Advances in differentiable numerical integrators have enabled the use of gradient descent techniques to learn ordinary differential equations (ODEs). In the context of machine learning, differentiable solvers are central for Neural ODEs…

This paper contains an error analysis of two randomized explicit Runge-Kutta schemes for ordinary differential equations (ODEs) with time-irregular coefficient functions. In particular, the methods are applicable to ODEs of Carath\'eodory…

Numerically solving ordinary differential equations (ODEs) is a naturally serial process and as a result the vast majority of ODE solver software are serial. In this manuscript we developed a set of parallelized ODE solvers using…

As further progress in the accurate and efficient computation of coupled partial differential equations (PDEs) becomes increasingly difficult, it has become highly desired to develop new methods for such computation. In deviation from…