Active learning with physics-informed neural networks for optimal sensor placement in deep tunneling through transversely isotropic elastic rocks

TL;DR

This work presents a framework that couples Physics-Informed Neural Networks with an active-learning strategy to reconstruct the displacement field around a deep circular tunnel in transversely isotropic elastic rocks and to back-calculate constitutive and stress parameters from sparse, noisy measurements. By employing Monte Carlo dropout to quantify epistemic uncertainty, the method greedily selects the most informative sensors (extensometers or convergence points) and iteratively augments the training set, achieving accurate inverse estimates of $E_h$, $E_v$, $G_{vh}$, $K$, and $\beta$ from limited data. The approach leverages non-dimensionalization for training stability and uses a high-fidelity synthetic dataset from MOOSE to validate performance, showing strong displacement-field reconstruction and parameter recovery even with few, scattered observations. Practically, this framework supports optimal subsurface monitoring and adaptive tunnel design by enabling data-efficient sensor placement and rapid inverse analysis on modest hardware (roughly 2–5 minutes per active-learning step on an Apple M4 chip).

Abstract

This paper presents a deep learning strategy to simultaneously solve Partial Differential Equations (PDEs) and back-calculate their parameters in the context of deep tunnel excavation. A Physics-Informed Neural Network (PINN) model is trained with synthetic data that emulates in situ displacement measurements in the host rock and at the cavity wall, obtained from extensometers and convergence monitoring. As acquiring field observations can be costly, a sequential training approach based on active learning is implemented to determine the most informative locations for new sensors. In particular, Monte Carlo dropout is used to quantify epistemic uncertainty and query measurements in regions where the model is least confident. This approach reduces the amount of required field data and optimizes sensor placement. The PINN is tested to reconstruct the displacement field around a deep tunnel of circular section excavated in transversely isotropic elastic rock and to determine rock constitutive and stress-field parameters. Results demonstrate excellent performance on small, scattered, and noisy datasets, achieving high precision for the Young's moduli, shear modulus, horizontal-to-vertical far-field stress ratio, and the orientation of the bedding planes. The proposed framework shall ultimately support decision-making for optimal subsurface monitoring and for adaptive tunnel design and control.

Figures (13)

• Figure 1: Tunnel with circular cross-section or radius $R$ excavated in transversely isotropic elastic rocks.
• Figure 2: Training set. Left: pool of measurements $\mathcal{P}$ from which the model can query data, including both extensometers and convergence points. Right: grid of collocation and boundary points $\mathcal{G}$ used to solve the PDE and BC in the domain.
• Figure 3: PINN architecture used in this study. Independent neural networks, $\mathcal{N}_{u_x}(\boldsymbol{x}; \boldsymbol{\theta})$ and $\mathcal{N}_{u_y}(\boldsymbol{x}; \boldsymbol{\theta})$, are defined to predict the displacement components $u_x$ and $u_y$, respectively. Each network takes the spatial coordinates $(x, y)$ as input features.
• Figure 4: Sequential active learning process where sensors are actively selected by the model and added to the training set. The color scale represents the total epistemic uncertainty $\sigma^2 = \sigma_{u_x}^2 + \sigma_{u_y}^2$. Current sensors are shown in white, and the next sensor to be queried is highlighted in black. (a)-(b) Active learning when only extensometer observation points are added sequentially. (c)-(d) Active learning when only convergence observation points are added sequentially.
• Figure 5: Comparison of relative test errors obtained using active learning versus random selection for different types of sensors. The solid lines correspond to the mean values of the errors, and the shaded regions represent the associated standard deviations.