Potential contribution of CNN-based solving of stiff ODEs & PDEs to enabling real-time Computational Engineering

One of the most common approaches to avoid complexity while numerically solving stiff ordinary differential equations (ODEs) is approximating them by ignoring the nonlinear terms. While facing stiff partial differential equations (PDEs) the same is done by avoiding/suppressing the nonlinear terms from the Taylor's series expansion. By so doing, the traditional methods for solving stiff PDEs and ODEs do compromise on both efficiency and precision of the resulting computations. This does inevitably lead to less accurate results that consequently cannot provide the full insight that may be needed in diverse cutting-edge situations in the 'real' nonlinear dynamical behavior experienced by the various engineering and natural systems (generally modeled by nonlinear differential equations of the types ODE or PDE), which are analyzed in the frame of the novel discipline called Computational Engineering. For many of these systems, even a real-time simulation and/or control of the behavior is wished or needed; this sets evidently extremely high challenging requirements to the computing capability with regard to both computing speed and precision. This paper develops/proposes and validate through a series of presentable examples a comprehensive high-precision and ultra-fast computing concept for solving stiff ODEs and PDEs with Cellular Neural Networks (CNN). The core of this concept is a straight-forward scheme that we call 'Nonlinear Adaptive Optimization (NAOP)', which is used for a precise template calculation for solving any (stiff) nonlinear ODE through CNN processors. One of the key contributions of this work, this is a real breakthrough, is to demonstrate the possibility of mapping/transforming different types of nonlinearities displayed by various classical and well-known oscillators (e.g. van der Pol-, Rayleigh-, Duffing-, Ro?ssler-, Lorenz-, and Jerk- oscillators, just to name a few) unto first-order CNN elementary cells, and thereby enabling the easy derivation of corresponding CNN templates. Furthermore, in case of PDE solving, the same concept also allows a mapping unto first-order CNN cells while considering one or even more nonlinear terms of the Taylor's series expansion generally used in the transformation of a PDE in a set of coupled nonlinear ODEs. Therefore, the concept of this paper does significantly contribute to the consolidation of CNN as a universal and ultra-fast solver of stiff differential equations (both ODEs and ODEs). This clearly enables a CNN-based, realtime, ultra-precise, and low-cost Computational Engineering. As proof of concept some well-known prototypes of stiff equations (van der Pol, Lorenz, and Ro?ssler oscillators) have been considered; the corresponding precise CNN templates are derived to obtain precise solutions of corresponding equations. An implantation of the concept developed is possible even on embedded digital platforms (e.g. FPGA, DSP, GPU, etc.); this opens a broad range of applications. On-going works (as outlook) are using NAOP for deriving precise templates for a selected set of practically interesting PDE models such as Navier Stokes, Schro?dinger, Maxwell, etc.