V.M. Zhuravlev, A.V. Zhuravlev
This work presents a new approach to generation of nonlinear stochastic models from experimental data in the form of fields of physical parameters or images. The method is based on the modified generalized Cole-Hopf substitution method applied to discrete systems. In this work the generalized Cole-Hopf substitution method (GCHS) was modified by replacing general partial derivative with their difference analogue. It was shown that all results of GCHS for persistent nonlinear models are correct for proposed approach. This result can be propagated to wider class of models. It is essential to show that the model use Leibniz rule of differentiation or its generalization that does not increase an order of derivatives. In the work it was found a system of discrete analogs of Burgers-type equations (integrable with Cole-Hopf substitution) for two types of difference derivatives. On that basis it was shown that such method can be used for identification of nonlinear empiric models from experimental data. It was developed a procedure for restoration of model’s parameters in 3 steps. The first step is the generation of auxiliary field controlled with multidimensional autoregression model from the experimental data. On the second step parameters of multidimensional autoregression model of auxiliary field were estimated. The third step consists in transformation of auxiliary linear models to nonlinear models of initial process using discrete Cole-Hopf substitutions. In the work the general methodology of such approach was developed and several formal examples were given.