Deep Learning Neural Network Algorithm for Computation of SPICE Transient Simulation of Nonlinear Time Dependent Circuits

In this paper, a special method based on the neural network is presented, which is conveniently used to precompute the steps of numerical integration. This method approximates the behaviour of the numerical integrator with respect to the local truncation error. In other words, it allows the precomputation of the individual steps in such a way that they do not need to be estimated by an algorithm but can be directly estimated by a neural network. Experimental tests were performed on a series of electrical circuits with different component parameters. The method was tested for two integration methods implemented in the simulation program SPICE (Trapez and Gear). For each type of circuit, a custom network was trained. Experimental simulations showed that for well-defined problems with a sufficiently trained network, the method allows in most cases reducing the total number of iteration steps performed by the algorithm during the simulation computation. Applications of this method, drawbacks, and possible further optimizations are also discussed.

A Bayesian neural network predicts the dissolution of compact planetary systems

We introduce a Bayesian neural network model that can accurately predict not only if, but also when a compact planetary system with three or more planets will go unstable. Our model, trained directly from short N-body time series of raw orbital elements, is more than two orders of magnitude more accurate at predicting instability times than analytical estimators, while also reducing the bias of existing machine learning algorithms by nearly a factor of three. Despite being trained on compact resonant and near-resonant three-planet configurations, the model demonstrates robust generalization to both nonresonant and higher multiplicity configurations, in the latter case outperforming models fit to that specific set of integrations. The model computes instability estimates up to 105 times faster than a numerical integrator, and unlike previous efforts provides confidence intervals on its predictions. Our inference model is publicly available in the SPOCK (https://github.com/dtamayo/spock) package, with training code open sourced (https://github.com/MilesCranmer/bnn_chaos_model).

La genèse des neurosciences

À partir des années 1940, plusieurs dynamiques permirent de renforcer le champ des sciences du cerveau et du système nerveux dans un processus interdisciplinaire, favorisé par l’idée que les biochimistes viendraient désormais éclairer les mécanismes physiologiques du système nerveux. Plus globalement, des grands programmes de recherche comme le projet Manhattan ou bien la réalisation de l’ENIAC (electronic numerical integrator and computer), avaient mis en évidence l’interpénétration, sans dépendance unilatérale de l’une à l’égard de l’autre, des sciences et des techniques. L’illusion d’une technique « appliquant » les découvertes scientifiques s’évanouit alors. Le concept de « technosciences »*, en permettant de sortir de cette dichotomie, permet de mieux comprendre comment, entre les années 1940 et 1970, diverses trajectoires convergèrent pour donner naissance aux « neurosciences modernes » [1, 2].

A combination of Lie group-based high order geometric integrator and delta-shaped basis functions for solving Korteweg–de Vries (KdV) equation

In this work, we develop a novel method to obtain numerical solution of well-known Korteweg–de Vries (KdV) equation. In the novel method, we generate differentiation matrices for spatial derivatives of the KdV equation by using delta-shaped basis functions (DBFs). For temporal integration we use a high order geometric numerical integrator based on Lie group methods. This paper is a first attempt to combine DBFs and high order geometric numerical integrator for solving such a nonlinear partial differential equation (PDE) which preserves conservation laws. To demonstrate the performance of the proposed method we consider five test problems. We reckon [Formula: see text], [Formula: see text] and root mean square (RMS) errors and compare them with other results available in the literature. Besides the errors, we also monitor conservation laws of the KDV equation and we show that the method in this paper produces accurate results and preserves the conservation laws quite good. Numerical outcomes show that the present novel method is efficient and reliable for PDEs.

DEVELOPMENT AND IMPLEMENTATION OF A TENTH-ORDER HYBRID BLOCK METHOD FOR SOLVING FIFTH-ORDER BOUNDARY VALUE PROBLEMS

A hybrid convergent method of tenth-order is presented in this work for directly solving fifth-order boundary value problems in ordinary differential equations. A unique direct block approach is obtained by combining multiple Finite Difference Formulas which are derived via the collocation technique. The proposed method is fully analyzed and the existence and uniqueness of the discrete solution is established. Different numerical examples are considered and the results are compared with those provided by existing works in the literature. The comparison shows the good performance of the present method over some cited works in the literature, confirming the competitiveness and superiority of the new numerical integrator.

A Mathematical Method for Predicting the Design Performance of Single and Multi-stage Rockets

Methods for predicting the performance of rockets are not new, however they often exist only within private organizations and in order to ensure competitive advantage, organizations tend to not share any details about their inner performance models. This open-source method gives students, design-teams and hobbyists a method to obtain baseline approximations for the performance of both single and multi-stage tandem rockets and provides a method which can easily be modified to meet the end-user’s requirements. The method solves for the mass, flight-path angle, velocity, altitude, and down-range distance using a numerical integrator to solve a set of nonlinear ordinary differential equations.

A Mathematical Method for Predicting the Design Performance of Single and Multi-stage Rockets

Methods for predicting the performance of rockets are not new, however they often exist only within private organizations and in order to ensure competitive advantage, organizations tend to not share any details about their inner performance models. This open-source method gives students, design-teams and hobbyists a method to obtain baseline approximations for the performance of both single and multi-stage tandem rockets and provides a method which can easily be modified to meet the end-user’s requirements. The method solves for the mass, flight-path angle, velocity, altitude, and down-range distance using a numerical integrator to solve a set of nonlinear ordinary differential equations.

An unconventional robust integrator for dynamical low-rank approximation

We propose and analyse a numerical integrator that computes a low-rank approximation to large time-dependent matrices that are either given explicitly via their increments or are the unknown solution to a matrix differential equation. Furthermore, the integrator is extended to the approximation of time-dependent tensors by Tucker tensors of fixed multilinear rank. The proposed low-rank integrator is different from the known projector-splitting integrator for dynamical low-rank approximation, but it retains the important robustness to small singular values that has so far been known only for the projector-splitting integrator. The new integrator also offers some potential advantages over the projector-splitting integrator: It avoids the backward time integration substep of the projector-splitting integrator, which is a potentially unstable substep for dissipative problems. It offers more parallelism, and it preserves symmetry or anti-symmetry of the matrix or tensor when the differential equation does. Numerical experiments illustrate the behaviour of the proposed integrator.

Adapted block hybrid method for the numerical solution of Duffing equations and related problems

<abstract><p>Problems of non-linear equations to model real-life phenomena have a long history in science and engineering. One of the popular of such non-linear equations is the Duffing equation. An adapted block hybrid numerical integrator that is dependent on a fixed frequency and fixed step length is proposed for the integration of Duffing equations. The stability and convergence of the method are demonstrated; its accuracy and efficiency are also established.</p></abstract>

Solving Quantum Equations with Gauge Fields: How Explicit Integrators Based on a Bipartite Lattice and Affine Transformations Can Help

We proposed an explicit numerical integrator consisting of affine transformation pairs resulting from the checkerboard lattice for spatial discretization. It can efficiently solve time evolution equations that describe dynamical quantum phenomena under gauge fields, e.g., generation, motion, interaction of quantum vortices in superconductors or superfluids.