In recent years, the SUite of Nonlinear and DIfferential/ALgebraic equation Solvers (SUNDIALS) has been redesigned to better enable the use of application-specific and third-party algebraic solvers and data structures. Throughout this work, we have adhered to specific guiding principles that minimized the impact to current users while providing maximum flexibility for later evolution of solvers and data structures. The redesign was done through creation of new classes for linear and nonlinear solvers, enhancements to the vector class, and the creation of modern Fortran interfaces that leverage interoperability features of the Fortran 2003 standard. The vast majority of this work has been performed "behind-the-scenes," with minimal changes to the user interface and no reduction in solver capabilities or performance. However, these changes now allow advanced users to create highly customized solvers that exploit their problem structure, enabling SUNDIALS use on extreme-scale, heterogeneous computational architectures.

Enabling New Flexibility in the SUNDIALS Suite of Nonlinear and Differential/Algebraic Equation Solvers.

This paper gives a review of integration algorithms for finite dimensional mechanical systems that are based on discrete variational principles. The variational technique gives a unified treatment of many symplectic schemes, including those of higher order, as well as a natural treatment of the discrete Noether theorem. The approach also allows us to include forces, dissipation and constraints in a natural way. Amongst the many specific schemes treated as examples, the Verlet, SHAKE, RATTLE, Newmark, and the symplectic partitioned Runge–Kutta schemes are presented.

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Discrete mechanics and variational integrators

We discuss an Adams-type predictor-corrector method for the numericalsolution of fractional differential equations. The method may be usedboth for linear and for nonlinear problems, and it may be extended tomulti-term equations (involving more than one differential operator)too.

https://ntrs.nasa.gov/api/citations/20020024453/downloads/20020024453.pdf

A Predictor-Corrector Approach for the Numerical Solution of Fractional Differential Equations

ductor, where the conductivity is sufficiently high, the displacement current density can be neglected. In this case, the conduction current density is given by the product of the electric field and the conductance. One of the aspects of with SE is it attenuates the higher frequency components of a signal. The SE was first verified by Kelvin in 1887. Since then many researchers developed work on the subject and presently a comprehensive physical model, based on the Maxwell equations, is well established. The Maxwell formalism plays a fundamental role in the electromagnetic theory. These equations lead to the derivation of mathematical descriptions useful in many applications in physics and engineering. Maxwell is generally The Maxwell equations involve only the integer-order calculus and, therefore, it is natural that the resulting classical models adopted in electrical engineering reflect this perspective. Recently, a closer look of some phenomof precise models, seem to point out the requirement for a fractional calculus approach. Bearing these ideas in mind, in this study we address the SE and we re-evaluate the results demonstrating its fractional-order nature. Department of Electrical Engineering, Institute of Engineering of Porto Rua Dr. Antonio Bernardino de Almeida, 4200-072 Porto, Portugal; E-mail: jtm,isj,amf@isep.ipp.pt Engineering Systems, Vila-Real, Portugal; E-mail: jboavent@utad.pt Institute of Intelligent Engineering Systems, Budapest Tech, John von Neumann Faculty of Informatics, Budapest, Hungary; E-mail: tar@nik.bmf.hu the high-frequency effects is the skin effect (SE ). The fundamental problem regarded as the 19th century scientist who had the greatest influence on 20th century physics, making contributions to the fundamental models of nature. enas present in electrical systems and the motivation towards the development

Advances in Fractional Calculus

We introduce a new family of deep neural network models. Instead of specifying a discrete sequence of hidden layers, we parameterize the derivative of the hidden state using a neural network. The output of the network is computed using a black-box differential equation solver. These continuous-depth models have constant memory cost, adapt their evaluation strategy to each input, and can explicitly trade numerical precision for speed. We demonstrate these properties in continuous-depth residual networks and continuous-time latent variable models. We also construct continuous normalizing flows, a generative model that can train by maximum likelihood, without partitioning or ordering the data dimensions. For training, we show how to scalably backpropagate through any ODE solver, without access to its internal operations. This allows end-to-end training of ODEs within larger models.

/pdf/neural-ordinary-differential-equations-14lrofzsgs.pdf

Neural ordinary differential equations

Solving Ordinary ,Differential Equations II. Stiff and Differential-Algebraic Problems/E. Hairer, G. Wanner, Second Revised Edition with 137 Figures , Vol.: 2

G. Wanner

Papers

Solving Ordinary ,Differential Equations II. Stiff and Differential-Algebraic Problems/E. Hairer, G. Wanner, Second Revised Edition with 137 Figures , Vol.: 2

Solving ordinary differential equations I (2nd revised. ed.): nonstiff problems

Solving Ordinary ,Differential Equations I, Nonstiff problems/E. Hairer, S. P. Norsett, G. Wanner, Second Revised Edition with 135 Figures, Vol.: 1