多保真 DeepONet 融合仿真与监测数据：隧道施工实时沉降预测

Ground settlement prediction during the process of mechanized tunneling is of paramount importance and remains a challenging research topic.

Typically, two paradigms are existing: a physics-driven approach utilizing process-oriented computational simulation models for the tunnel-soil interaction and the settlement prediction, and a data-driven approach employing machine learning techniques to establish mappings between influencing factors and the ground settlement.

To integrate the advantages of both approaches and to assimilate the data from different sources, we propose a multi-fidelity deep operator network (DeepONet) framework, leveraging the recently developed operator learning methods.

The presented framework comprises of two components: a low-fidelity subnet that captures the fundamental ground settlement patterns obtained from finite element simulations, and a high-fidelity subnet that learns the nonlinear correlation between numerical models and real engineering monitoring data.

A pre-processing strategy for causality is adopted to consider the spatio-temporal characteristics of the settlement during tunnel excavation.

Transfer learning is utilized to reduce the training cost for the low-fidelity subnet.

The results show that the proposed method can effectively capture the physical information provided by the numerical simulations and accurately fit measured data as well.

Remarkably, even with very limited noisy monitoring data, the proposed model can achieve rapid, accurate, and robust predictions of the full-field ground settlement in real-time during mechanized tunnel excavation.

The growing population, urbanization, and demand for high-speed mobility have necessitated the construction of underground transportation systems in the modern era. In recent years, a large number of tunnels has been constructed to improve existing transportation systems. Mechanized tunneling is a highly automated construction process, which has been widely applied to underground engineering projects under various geological and hydrological conditions (Maidl et al.,, 2012). However, the surface settlement induced by the tunnel boring machine (TBM) excavation can pose a threat to the stability of existing buildings above ground, making the real-time prediction of the expected ground settlements ahead of the tunnel face during tunneling a critical engineering issue.

Generally, in a tunnel project, a limited number of monitoring locations are predefined in the design phase, at which the temporal evolution of the ground settlements are recorded during the tunnel construction. However, during the actual construction process, in particular in urban environments, it is desirable to predict the settlements induced by the tunnel advancement expected for the forthcoming excavation steps in order to allow adequately controlling the face and grouting pressure to keep the ground movements within prescribed limits (Cao et al.,, 2016). This information is needed for the complete field in the vicinity of the tunnel boring machine and not only in specific monitoring points. Therefore, to better support the construction process, it is necessary to develop an efficient prediction model, which is capable to predict the complete settlement field at multiple surface points in real-time during the tunneling process.

With the rapid development of deep learning (Lecun et al.,, 2015), scientific machine learning (SciML) has gained increasing attention across diverse disciplines in science and engineering. Supported by sufficient data, deep neural networks can learn functions between arbitrary inputs and outputs, enabling emulation and analysis of various complex physical systems (Jiang et al.,, 2017, 2021; Cabrera et al.,, 2023; Zhang et al.,, 2020; He et al.,, 2022, 2024; Gu et al.,, 2023). However, purely data-driven approaches heavily rely on both the quantity and quality of data. In most cases, available monitoring data in engineering projects are extremely limited, and such a small data set with great noise can lead to poor generalization of neural networks. To address this ubiquitous issue in systems with some physics and some data (Karniadakis et al.,, 2021), a prevalent class of methods are proposed in recent years, known as physics-informed neural networks (PINNs) (Raissi et al.,, 2019; Xu et al.,, 2023; Chew et al.,, 2022; Zobeiry and Humfeld,, 2021; Fernández et al.,, 2023; Haghighat et al.,, 2023; Zhang and Shafieezadeh,, 2023; Ding et al.,, 2023; Sun et al.,, 2023). By leveraging the automatic differentiation technique, physical laws in the form of partial differential equations (PDEs) can be easily embedded into the loss function, thus reducing the requirements for data volume and improving the generalization performance of neural networks. However, the spatio-temporal ground settlement induced by TBM excavation is very complex, which can not be fully described via analytical solutions. In other words, the lack of physics to be incorporated into neural networks greatly limits the practical application of PINNs in this field.

More recently, the neural operator method has emerged as a new and promising research field in the SciML community, with notable methods such as the Deep Operator Network (DeepONet) proposed by Lu et al. (Lu et al.,, 2021) and the Fourier neural operator (FNO) proposed by Li et al. (Li et al.,, 2021). Unlike previous approaches, the neural operator can learn mappings between infinite-dimensional Banach spaces (Lanthaler et al.,, 2022; Kovachki et al.,, 2021). Through various methodological improvements (Goswami et al., 2022a, ; Wang et al.,, 2021; Zhu et al.,, 2023; Lu et al., 2022a, ; Garg and Chakraborty,, 2023), it has demonstrated remarkable effectiveness in a wide range of scientific problems (Lin et al.,, 2021; Cai et al.,, 2021; Goswami et al., 2022b, ; Yin et al.,, 2022; Pickering et al.,, 2022; Di Leoni et al.,, 2021; Wang and Perdikaris,, 2023; Lin et al.,, 2023).

In this paper, we propose a framework for Multi-Fidelity DeepONet to enable the seamless fusion of information from the two types of data mentioned above. Since the tunneling process is time-dependent, the Causality-DeepONet (Liu et al.,, 2022) is adopted for data pre-processing. This framework consists of two parts: the low-fidelity DeepONet trained on simulation-based data to learn the underlying physical mechanism during tunnel excavations, and the residual DeepONet trained on very limited monitoring data to bridge the gap between numerical models and measurements. The final reconstructed settlement field can exhibit physical patterns similar to numerical simulation results and fit well with the monitoring data. To the best of our knowledge, no study has ever managed to use the operator learning technique for settlement prediction in tunnel construction.

1. In this paper, we propose a framework for Multi-Fidelity DeepONet to enable the seamless fusion of information from the two types of data mentioned above. Since the tunneling process is time-dependent, the Causality-DeepONet (Liu et al.,, 2022) is adopted for data pre-processing.

2. It is an efficient architecture that combines the characteristics of two recent advances in DeepONet (Lu et al., 2022b, ; Howard et al.,, 2023). Transfer learning is employed at the online stage to reduce the computational cost.

3. The incorporation of transfer learning significantly reduces training costs, leading to efficient field reconstruction for each excavation step in TBM operations. The computing time for every time step is less than one minute, showcasing great potential of our approach to be applied in tunnel engineering projects. The presented method exhibits robustness against noise, yielding R² scores above 0.8 even for small high-fidelity data sets with a 50% error level.

A multi-fidelity DeepONet framework has been adapted to perform real-time reconstruction of the settlement field induced by machine driven tunnel construction based on a priori trained data from computational simulation and data from continuous monitoring acquired during tunnel advancement at limited monitoring locations. The proposed DeepONet framework provides an efficient fusion of model based data, representing the physics of the problem with unavoidable simplifications and limited knowledge of parameters (low-fidelity data), and measurement data (high-fidelity data).

The main findings from this study can be summarized below: (a) This reconstruction not only captures the anticipated trends derived from the physical model but also achieves satisfactory agreement with the observed data. (b) The successful implementation of causality pre-processing can take into the account the time-dependent characteristics of both inputs and outputs. (c) The incorporation of transfer learning significantly reduces training costs, leading to efficient field reconstruction for each excavation step in TBM operations. The computing time for every time step is less than one minute, showcasing great potential of our approach to be applied in tunnel engineering projects. (d) The presented method exhibits robustness against noise, yielding R² scores above 0.8 even for small high-fidelity data sets with a 50% error level.

Moreover, some potential improvements can be accomplished in future work: (a) the arrangement of monitoring points remains fixed throughout all analyses, an upcoming study will extend the proposed multi-fidelity DeepONet framework to examine the influence of monitoring point layouts and suggest the optimal number and positions for monitoring sensors; (b) only two TBM process parameters, the tail void grouting pressure 𝑷𝑮 and the face support pressure 𝑷𝑺, are considered in this work; (c) the pre-processing procedure for causality can effectively deal with causal time series, but it significantly inflates the size of the data set, which in turn influences training efficiency. It is emphasized, that although the technology proposed in this paper has specifically been applied to tunneling engineering, it can be used for the process control in a variety of engineering applications.