Order theory for discrete gradient methods

Abstract The discrete gradient methods are integrators designed to preserve invariants of ordinary differential equations. From a formal series expansion subclass these methods, we derive conditions for arbitrarily high order. We specific results the average vector field gradient, from which get P-series in general case, and B-series canonical Hamiltonian systems. Higher order schemes presented, their applications demonstrated on Hénon–Heiles system Lotka–Volterra system, both training integration pendulum learned data by neural network.