NDAWL-PINN: a new non-dimensionalization and multi-task learning approach for efficient training of physics-informed neural networks to solve the shallow water equations

The exploration of deep learning methodologies has recently generated significant interest in the use of Physics-Informed Neural Networks (PINNs) to address complex physical problems governed by partial differential equations (PDEs). The PINN is trained using information from physical laws, including governing PDEs, boundary conditions, and initial conditions. However, achieving a well-trained PINN typically necessitates an appropriate balance between the weights of each loss function, which can considerably increase manual effort. This paper introduces a novel training approach that integrates non-dimensionalization with a multi-task learning technique, termed Automatic Weighted Loss (AWL), to autonomously achieve an optimal balance for each loss function. In the baseline PINN training for solving time-dependent PDEs, multiple weights (usually more than six) must be manually tuned for the model, whereas this method can reduce the number of scaling weights to only one. The proposed approach, referred to as the Non-dimensionalization Automatic Weighted Loss (NDAWL), is evaluated through six free-surface flow problems modelled by the Shallow Water Equations (SWEs). Furthermore, a comparative analysis is conducted between the solutions obtained using NDAWL-PINN and those from the original PINN, which relies on manual fine-tuning of loss functions. The numerical results indicate that the NDAWL-PINN method achieves comparable or superior accuracy to the original PINN, demonstrating its effectiveness in automating the balancing of loss functions.