An efficient thermo-mechanical coupling solution for soil surrounding energy pile based on physics-informed neural networks

To investigate the thermo-mechanical coupling for soil surrounding energy pile, an efficient mesh-free numerical model based on the physics-informed neural networks (PINNs) method is proposed. The model incorporates the heat diffusion equation and Navier stress equilibrium equation as physical constraints by embedding the residuals of the governing equations into the loss function, thereby ensuring that the training process strictly adheres to physical conservation laws. Validation against finite element simulation results indicates that the optimal configuration of the proposed PINNs model consists of four hidden layers with 50 neurons per layer and the Tanh activation function, achieving maximum relative errors of only 0.53% and 5.72% for the temperature and displacement fields, respectively. In terms of optimization strategy, the hybrid Adam+L-BFGS optimizer reduces the total loss to 4.42×10⁻⁴, improving performance by 51.52% and 15.33% compared with using Adam and L-BFGS individually. Moreover, the Latin hypercube sampling strategy significantly enhances model accuracy, reducing the average relative temperature prediction error by 77.03% and 76.81% compared with uniform and random sampling, respectively. With the introduction of a transfer learning framework, the model achieves comparable accuracy to full retraining under new working conditions at only 9.35% of the computational cost, improving overall computational efficiency by 16.55% compared with the traditional finite element method. The proposed approach provides a new solution for studying thermo-mechanical coupling problems in underground engineering, combining physical rigor with high computational efficiency.