A Universal Concept Based on Cellular Neural Networks for Ultrafast and Flexible Solving of Differential Equations

Abstract

This paper develops and validates a comprehensive and universally applicable computational concept for solving nonlinear differential equations (NDEs) through a neurocomputing concept based on cellular neural networks (CNNs). High-precision, stability, convergence, and lowest-possible memory requirements are ensured by the CNN processor architecture. A significant challenge solved in this paper is that all these cited computing features are ensured in all system-states (regular or chaotic ones) and in all bifurcation conditions that may be experienced by NDEs.One particular quintessence of this paper is to develop and demonstrate a solver concept that shows and ensures that CNN processors (realized either in hardware or in software) are universal solvers of NDE models. The solving logic or algorithm of given NDEs (possible examples are: Duffing, Mathieu, Van der Pol, Jerk, Chua, Rössler, Lorenz, Burgers, and the transport equations) through a CNN processor system is provided by a set of templates that are computed by our comprehensive templates calculation technique that we call nonlinear adaptive optimization. This paper is therefore a significant contribution and represents a cutting-edge real-time computational engineering approach, especially while considering the various scientific and engineering applications of this ultrafast, energy-and-memory-efficient, and high-precise NDE solver concept. For illustration purposes, three NDE models are demonstratively solved, and related CNN templates are derived and used: the periodically excited Duffing equation, the Mathieu equation, and the transport equation.

A Universal Concept for Robust Solving of Shortest Path Problems in Dynamically Reconfigurable Graphs

“…In some of our recent contributions/papers (see [43,44]), the NAOP concept has been successfully used for solving differential equations (ODEs and/or PDEs). We now want to demonstrate that the NAOP concept can be efficiently used as a general and robust framework for modeling shortest path problems (SPP) even in dynamically reconfigurable graphs.…”

“…The NAOP concept in Figure 1 has already been successfully applied for solving both nonlinear ordinary differential equations (ODEs) and partial differential equations (PDEs) [43,44]. The aim of this paper is to demonstrate that the NAOP concept (see Figure 1) can also be efficiently used as a universal framework to realize a robust SPP solver (called NAOP SPP simulator).…”

“…The second step exploits the concept of neuron dynamics as an optimization strategy that maps the optimization problem unto the energy of a neural network (e.g., a Hopfield network) in order to find the optimal solution. In this context, we represent this energy by the Lagrange form and the minimization of the Lagrange function leads to a stable state (due to the already proven convergence of the Basic Differential Multiplier Method (BDMM); see [38,43,44]). Applying the BDMM to the Lagrange function in (4) leads to the derivation oḟ=…”

“…The set of (5a), (5b), and (5c) is the characteristic model of BDMM [38,43,44]. This model (from which the coefficients of the coupled ODEs for a given graph are calculated) reveals the coupling between the dynamics of decision variable with the dynamics of multiplier variables and .…”

“…The values of these parameters are chosen through a stability or convergence analysis (see Section 3.3). A similar study (i.e., stability analysis) is considered (in [43,44]) and it is demonstrated that the sign of the damped mass matrix depends on the parameters and . Specifically, it is demonstrated (see [43,44]) that the convergence is ensured for positive definite damped mass matrix.…”