Tunnel Displacement Field Driven by Physics-Informed Multi-Task Gaussian Process

In deep underground engineering, discrete and sparse monitoring points often fail to accurately characterize the full-field nonlinear deformation features of surrounding rocks. Furthermore, existing purely data-driven methods are highly susceptible to severe overfitting under small sample sizes. To address these limitations, a Physics-Informed Multi-Task Gaussian Process Regression (PI-GPR) method is proposed to reconstruct the displacement field of surrounding rocks. The Intrinsic Coregionalization Model (ICM) is introduced to explicitly construct the spatial cross-covariance structure between horizontal and vertical displacements, thereby achieving synergistic information sharing. Additionally, a hyperparameter constraint strategy, rooted in rock mechanics principles (e.g., continuity and boundary convergence), is designed to restrict the spatial lengthscale within a physically reasonable range. A sampling scheme based on spatial topological features is also formulated to maximize the information entropy derived from limited measurement points. Validations via 3DEC numerical simulations reveal that under extremely sparse monitoring conditions (8 points), traditional purely data-driven models suffer from severe overfitting, yielding spurious striping patterns and boundary divergence with a coefficient of determination (R²) of only 0.186.Conversely, constrained by the physical inductive bias, the PI-GPR model successfully reconstructs a continuous deformation field that complies with Saint-Venant's principle, boosting the R² to 0.399. As the number of points increases to 32, the model's R² converges stably to 0.80, and the inverted lengthscale parameter (l≈0.81) closely aligns with the actual influence radius of the plastic zone. Ultimately, this method significantly enhances inversion robustness under extremely sparse conditions while maintaining high accuracy with sufficient data. It effectively bridges the gap between discrete monitoring data and full-field continuous physical states, providing a reliable methodological foundation for intelligent support design in deep engineering and the development of digital rock mechanics.