Responses of stochastic dynamical systems by the generalized cell mapping method with deep learning

Experimental data is often corrupted by measurement noise in practical engineering and there are multiple observed data under the same experimental condition. The noisy measurement data can be considered as the response of an underlying stochastic dynamical system. To explore the response properties from noisy measurement data with the partial information of the given stochastic dynamical system, a novel method named as the stochastic generalized cell mapping with deep learning (SGCM-DL) is proposed based on the idea of the image processing. Noisy measurement data is reconstructed into a matrix to be expressed by the image, which is used as the training data of the neural network. The obtained neural network agent model is combined to reveal properties of the underlying stochastic dynamical system and employed to predict stochastic responses with given initial values by the way of the image generation. The technique of the image super-resolution (SR) is introduced to generate the image with higher resolution from the output image of the neural network. Based on the image with higher resolution, one-step transition probability matrix with the larger dimension is obtained for stochastic dynamical systems. Then probability density functions (PDFs) of transient and stationary responses can be obtained by the proposed method in this paper. Several examples are presented, and the good effectiveness and time-saving are observed.