A family of embedded Runge-Kutta formulae

By J. Dormand, P. Prince. 1980

Abstract A family of embedded Runge-Kutta formulae RK5 (4) are derived. From these are presented formulae which have (a) ‘small’ principal truncation terms in the fifth order and (b) extended regions of absolute stability.

Subfields: Numerical Methods & Scientific Computing

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Citations in the corpus

Score-Based Generative Modeling through Stochastic Differential Equations (2020, positive): “In our experiments, we use the RK45 ODE solver (Dormand & Prince, 1980) provided by scipy.integrate.solve_ivp in all cases.”
Elucidating the Design Space of Diffusion-Based Generative Models (2022, methodology): “As further validation, we show results for the adaptive RK45 method [11] using our schedule as the dashed black curves in Figure 2; the cost of this sophisticated ODE solver outweighs its benefits.”
Equivariant Flows: exact likelihood generative learning for symmetric densities (2020, background): “Thus, Chen et al. propose adaptive-step solvers, such as Dopri5 (Dormand & Prince, 1980), which can require hundreds of vector field evaluations to reach satisfying numerical accuracy.”
DPM-Solver: A Fast ODE Solver for Diffusion Probabilistic Model Sampling in Around 10 Steps (2022, methodology): “Song et al. [3] used the RK45 ODE solver [28] for the diffusion ODEs, which generates samples in ∼ 60 function evaluations to reach comparable quality with a 1000-step SDE solver for Eq.”
Flow Matching for Generative Modeling (2022, positive): “Unless otherwise specified, we evaluate likelihood and samples from the model using dopri5 (Dormand & Prince, 1980) at absolute and relative tolerances of 1e-5.”
Maximum Likelihood Training of Score-Based Diffusion Models (2021, methodology): “In addition, we report FID scores [17] for samples from pODE θ , produced by solving the corresponding ODE with the Dormand–Prince RK45 [14] solver.”
Gotta Go Fast When Generating Data with Score-Based Models (2021, methodology): “We compare our method to the three base solvers used in Song et al. [2020a]: Reverse-Diffusion with Langevin dynamics, Euler-Maruyama (EM), and Probability Flow, where the latter solves an ODE instead of an SDE using Runge-Kutta 45 [Dormand and Prince, 1980].”
Chapter 3 – The Cross-Entropy Method for Optimization (2013, methodology): “For a given ux we solve the nonlinear ODE system in (23) using the Runge–Kutta formulae of Dormand and Prince (1980).”
Speech Enhancement and Dereverberation With Diffusion-Based Generative Models (2022, methodology): “for our task, specifically employing the Runge-Kutta method of fourth order with an error estimator of fifth order [43].”
Matching Normalizing Flows and Probability Paths on Manifolds (2022, methodology): “Furthermore, ODEs have simple higher order solvers like Runga-Kutta methods (Dormand & Prince, 1980) with widely used open-source implementations.”
Score-Based Generative Modeling with Critically-Damped Langevin Diffusion (2021, methodology): “2, we report FID numbers for a Runge–Kutta 4(5) solvers (Dormand & Prince, 1980) as well as the “Gotta Go Fast” solver from Jolicoeur-Martineau et al. (2021a) (see their Table 1).”
Least-Squares Solution of Linear Differential Equations (2017, methodology): “The accuracies obtained using MATLAB’s ode45 [1] integrator and using the least-squares solution (The least-squares problem has been solved”
Computation of Hypergeometric Functions (2009, methodology): “One consistent method that we studied, the Dormand-Prince 5th order method [19], reads as follows:”
A high algebraic order multistage explicit four-step method with vanished phase-lag and its first, second, third, fourth and fifth derivatives for the numerical solution of the Schrödinger equation (2015, methodology): “Using high order Runge–Kutta–Nyström methods(see [124] and [125]) we determine the values qi , i = 1(1)3.”