The combination of ordinary differential equations and neural networks, ie, neural ordinary
differential equations (Neural ODE), has been widely studied from various angles. However …

This book is designed to be used by mathematicians, engineers, and computer scientists as
a graduate-level introduction to numerical analysis and its methods. Readers are expected …

Some of the most important geometric integrators for both ordinary and partial differential
equations are reviewed and illustrated with examples in mechanics. The class of …

A left pre-Lie algebra over a field k is a k-vector space A with a bilinear binary composition B
that satisfies the left pre-Lie identity. a B b/B ca B. b B c/D. b B a/B cb B. a B c/; for a; b; c 2 A …

Abstract Machine learning methods are widely used in the natural sciences to model and
predict physical systems from observation data. Yet, they are often used as poorly …

Hopf algebra structures on rooted trees are by now a well-studied object, especially in the
context of combinatorics. In this work we consider a Hopf algebra H by introducing a …

B-series are a fundamental tool in practical and theoretical aspects of numerical integrators
for ordinary differential equations. A composition law for B-series permits an elegant …

We propose a novel way to study numerical methods for ordinary differential equations in
one dimension via the notion of multi‐indice. The main idea is to replace rooted trees in …

A new characterization of sufficient conditions for the Lie--Trotter splitting to capture the
numerical invariant measure of nonlinear ergodic Langevin dynamics up to an arbitrary …

Inspired by recent advances in the theory of modified differential equations, we propose a
new methodology for constructing numerical integrators with high weak order for the time …