Combination of Physics-Informed Neural Networks and Single-Relaxation-Time Lattice Boltzmann Method for Solving Inverse Problems in Fluid Mechanics

Physics-Informed Neural Networks (PINNs) improve the efficiency of data utilization by combining physical principles with neural network algorithms and thus ensure that their predictions are consistent and stable with the physical laws. PINNs open up a new approach to address inverse problems in fluid mechanics. Based on the single-relaxation-time lattice Boltzmann method (SRT-LBM) with the Bhatnagar–Gross–Krook (BGK) collision operator, the PINN-SRT-LBM model is proposed in this paper for solving the inverse problem in fluid mechanics. The PINN-SRT-LBM model consists of three components. The first component involves a deep neural network that predicts equilibrium control equations in different discrete velocity directions within the SRT-LBM. The second component employs another deep neural network to predict non-equilibrium control equations, enabling the inference of the fluid’s non-equilibrium characteristics. The third component, a physics-informed function, translates the outputs of the first two networks into physical information. By minimizing the residuals of the physical partial differential equations (PDEs), the physics-informed function infers relevant macroscopic quantities of the flow. The model evolves two sub-models that are applicable to different dimensions, named the PINN-SRT-LBM-I and PINN-SRT-LBM-II models according to the construction of the physics-informed function. The innovation of this work is the introduction of SRT-LBM and discrete velocity models as physical drivers into a neural network through the interpretation function. Therefore, the PINN-SRT-LBM allows a given neural network to handle inverse problems of various dimensions and focus on problem-specific solving. Our experimental results confirm the accurate prediction by this model of flow information at different Reynolds numbers within the computational domain. Relying on the PINN-SRT-LBM models, inverse problems in fluid mechanics can be solved efficiently.