We present a deep learning model for data-driven simulations of random dynamical systems without a distributional assumption. The deep learning model consists of a recurrent neural network, which aims to learn the time marching structure, and a generative adversarial network to learn and sample from the probability distribution of the random dynamical system. Although generative adversarial networks provide a powerful tool to model a complex probability distribution, the training often fails without a proper regularization. Here, we propose a regularization strategy for a generative adversarial network based on consistency conditions for the sequential inference problems. First, the maximum mean discrepancy (MMD) is used to enforce the consistency between conditional and marginal distributions of a stochastic process. Then, the marginal distributions of the multiple-step predictions are regularized by using MMD or from multiple discriminators. The behavior of the proposed model is studied by using three stochastic processes with complex noise structures.

For health prognostic task, ever-increasing efforts have been focused on machine learning-based methods, which are capable of yielding accurate remaining useful life (RUL) estimation for industrial equipment or components without exploring the degradation mechanism. A prerequisite ensuring the success of these methods depends on a wealth of run-to-failure data, however, run-to-failure data may be insufficient in practice. That is, conducting a substantial amount of destructive experiments not only is high costs, but also may cause catastrophic consequences. Out of this consideration, an enhanced RUL framework focusing on data self-generation is put forward for both non-cyclic and cyclic degradation patterns for the first time. It is designed to enrich data from a data-driven way, generating realistic-like time-series to enhance current RUL methods. First, high-quality data generation is ensured through the proposed convolutional recurrent generative adversarial network (CR-GAN), which adopts a two-channel fusion convolutional recurrent neural network. Next, a hierarchical framework is proposed to combine generated data into current RUL estimation methods. Finally, the efficacy of the proposed method is verified through both non-cyclic and cyclic degradation systems. With the enhanced RUL framework, an aero-engine system following non-cyclic degradation has been tested using three typical RUL models. State-of-art RUL estimation results are achieved by enhancing capsule network with generated time-series. Specifically, estimation errors evaluated by the index score function have been reduced by 21.77%, and 32.67% for the two employed operating conditions, respectively. Besides, the estimation error is reduced to zero for the Lithium-ion battery system, which presents cyclic degradation.