Stable and accurate reconstruction of pollutant transport is a crucial and challenging problem, including the inverse problem of identifying pollution sources and physical coefficients, and the forward problem of inferring pollutant transport. Governed by advection, diffusion, and reaction process, this transport phenomenon can be represented by the advection-diffusion-reaction (ADR) equation. In this paper, the physics-informed neural networks (PINN) is applied to solve the forward and inverse ADR problems. To further enhance the stability and accuracy of the original PINN, two improvements are developed. The first adjusts the orthogonal grid (OG) point selection method and the other suggests adding an additional regulation function namely first derivative constraint (FDC). The new method is referred to as OG-PINN with FDC. To verify the effectiveness of the proposed method, five forward and inverse ADR problems are solved, and the results are compared with the analytical and reference solutions. For forward problems, the improved method can solve various ADR problems accurately and stably. For inverse problems, the ability of the OG-PINN for model parameter learning and initial distribution prediction are demonstrated and analyzed. The former gives the missed physical information in the ADR equation from the data, and the latter is used to trace the source of pollutants. The proposed method is quantitatively reliable for investigating various advection-diffusion-reaction processes.

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In this paper, a discretization-free approach based on the physics-informed neural network (PINN) is proposed for solving the forward and inverse problems governed by the nonlinear convection-diffusion-reaction (CDR) systems. By embedding physical information described by the CDR system in the feedforward neural networks, PINN is trained to approximate the solution of the system without the need of labeled data. The good performance of PINN in solving the forward problem of the nonlinear CDR systems is verified by studying the problems of gas-solid adsorption and autocatalytic reacting flow. For CDR systems with different Péclet number, PINN can largely eliminate the numerical diffusion and unphysical oscillations in traditional numerical methods caused by high Péclet number. Meanwhile, the PINN framework is implemented to solve the inverse problem of nonlinear CDR systems and the results show that the unknown parameters can be effectively recognized even with high noisy data. It is concluded that the established PINN algorithm has good accuracy, convergence, and robustness for both the forward and inverse problems of CDR systems.