Estimation and Control of 3D Diffusion-Advection System by Partial Differential Neural Network Combined With Divide-Space Sampling Strategy

In this paper, an optimization framework of partial differential neural network combined with divide-space sampling (PDNN-DSS) is proposed to solve the estimation and control problem of 3D diffusion-advection systems. The PDNN-DSS framework contains a divide-space sampling observer, a model prediction estimator, and a partial differential neural network solver. First, the open-loop state of the diffusion-advection system is obtained by designing the division space observer. The observer provides a deployment scheme for sensors so that mobile robots only need to carry actuators not sensors. Then, the control problem is transformed into an optimization problem by designing a model prediction estimator. Subsequently, a novel partial differential neural network is designed. The neural network is based on partial differential equations containing spatial diffusion features and can solve spatio-temporal optimization problems. Finally, the effectiveness of the proposed method in this paper is demonstrated through simulations from multi-dimensional perspectives of 2D and 3D diffusion-advection systems. Note to Practitioners—Spatio-temporal processes such as marine oil spills and riverwater chemical pollution can be modeled as diffusion-advection systems. This paper utilizes mobile robots and stationary sensors to solve the estimation and control problems of 3D diffusive-advection systems. In the proposed optimization framework, the divide-space sampling observer is used to deploy the stationary sensors, and the partial differential neural network is used to solve the guidance law of the mobile robot. Naturally, if the spatially defined domain of the diffusion-advection system is 2D or 1D, the method of this paper is equally applicable. In addition, if the system does not have an advection term, the design idea of the method in this paper remains the same, but the derivation needs to be simplified.