Embedded Legendre expansion layer of physics-informed neural network with restarting strategy in solving partial differential equation in fluid dynamics

Partial differential equations (PDEs) have long been used to describe and solve problems in fluid dynamics. The advent of Artificial Neural Networks (ANNs) has revolutionized the ways of tackling many problems, including the landscape of scientific computing. The Physics-Informed Neural Network (PINN) is a type of ANN used for unsupervised learning tasks, designed to solve problems by incorporating PDEs. On the other hand, single-layer Functional Link Artificial Neural Network (FLANN) has been a topic of interest in the field of artificial intelligence due to its unique architecture and computational efficiency. Leveraging the optimal structural complexity of FLANN and the effective design of PINN for solving PDEs under Legendre Expansion Layer (LEL) has the potential to tackle complex problems more effectively. This novel approach will be simulated using both the restarting and non-restarting strategies, resulting in LEL-rPINN and LEL-PINN. Three different fluid dynamics PDEs—the Allen-Cahn (AC) equation, the flow mixing (FM) equation, and the Navier-Stokes (NS) system of equations—are employed to demonstrate the improvements and potential drawbacks of incorporating LEL into both PINN and rPINN to solve PDEs. LEL-based methods demonstrate improved agreement with both analytical and ground-truth solutions, outperforming basic PINN and rPINN, as shown in the figures and tables. These results demonstrate that the involvement of LEL shows promising performance across a diverse set of problems, particularly in the field of fluid dynamics.