Physics informed neural networks for analyzing m-method in pile foundations

The m-method is widely used to analyze laterally loaded piles, with extensive data available for recommended values of m. However, it currently lacks an analytical solution and typically requires numerical methods. Numerical methods necessitate meshing the model and iterative computations, and they can only compute one set of input parameters at a time, resulting in inefficiencies for tasks that demand extensive computations. Recently, Physics informed neural networks (PINN) have been proposed, which do not require meshing and can solve high-dimensional partial differential equations (PDEs). Based on this, a PINN model that considers all input parameters of the m-method, including loads, pile parameters, and soil parameters, is proposed to analyze arbitrary input parameters without the need for re-modeling or re-training. The model normalizes the input parameters and employs a hard constraint method, significantly enhancing training efficiency. The dimensions of the loss terms are unified, addressing the issue of large differences among them. The accuracy of the PINN model is validated by comparison with the dimensionless coefficients obtained from the tabular method. The advantages of the PINN model include: it is trained directly on differential equations without requiring labeled data; it provides higher accuracy than the tabular method and greater computational efficiency than FEM; it enables efficient parallel computations for multiple parameter sets without re-modeling, making it highly advantageous for reliability analysis; it efficiently computes input-output derivatives, offering significant benefits for optimization design; and it demonstrates excellent scalability, allowing application to other pile analysis models, such as the c-method.