Physics-informed extreme learning machine for Terzaghi consolidation problems and interpretation of coefficient of consolidation based on CPTu data

The gist This paper conducts a preliminary study to investigate the feasibility of a physics-informed extreme learning machine (PIELM) for solving the Terzaghi consolidation equation and interpreting the coefficient of consolidation of soil from piezocone penetration tests (CPTu)

Introduction and Motivation

Understanding soil consolidation behaviour is essential for the design, construction and maintenance of geo-structures such as embankment settlement, slope stability and underground deformation, where settlement and stability are key concerns. Terzaghi (1943) was the first to mathematically describe this process by proposing the wellknown one-dimensional consolidation theory. Biot (1941) further developed a threedimensional consolidation theory that incorporates hydro-mechanical coupling to account for both pore water flow and soil deformation. The recent development of physics-informed machine learning (PIML) has made it possible to solve ordinary differential equations (ODEs) and partial differential equations (PDEs) by implementing physical laws. The most trending branch may be the physics-informed neural networks (PINNs), which enable various ODEs/PDEs to be solved with the universal function approximation capabilities of deep neural networks. Compared to traditional numerical methods, PINNs can more conveniently integrate physical laws with measured data, thereby giving PINNs a significant advantage in physics-data fusion and inverse analyses.

PIELM Framework for Consolidation Analysis

The PIELM framework is introduced as an alternative to conventional PINNs to improve training efficiency by replacing fully connected neural networks with the extreme learning machine (ELM). In the forward problem, the input layer parameters are randomly initialised within the range of [-1,1] and remain fixed throughout training. For forward analysis, when boundary loads are applied suddenly, a time-stepping strategy is incorporated into the PIELM framework to alleviate sharp gradients caused by inconsistent initial and boundary conditions. The TS-PIELM framework divides the consolidation process into a number of time intervals that can either be of equal or varying lengths. The TS-PIELM architecture requires a separate ELM network at each time step, where the output uk from the k-th ELM network will serve as the initial conditions for the (k+1)-th ELM network.

Validation and Performance

The PIELM method for Terzhaghi consolidation problems are first validated by forward problems. In a one-dimensional vertical consolidation problem, both PIELM and PINN predictions are in good agreement with the exact solution, yielding relative L2 error of 5.92e-2 and 3.28e-2, respectively. The training time of PIELM is drastically reduced to 0.86s owing to the adoption of the extreme learning machine (ELM) architecture compared to PINN's training time of approximately 434 s. The TS-PIELM framework demonstrates that the timestepping technique effectively smooths the steep gradients induced by discontinuities and significantly improves the prediction accuracy at early consolidation times. In a two-dimensional radial consolidation with vertical drains, TS-PIELM can accurately predict the pore water pressure evolution in this two-dimensional consolidation problem with discontinuous initial and boundary conditions at two boundaries, giving the relative L2 error of 2.50e-3.

Inverse Analysis for CPTu Interpretation

The PIELM framework is applied to interpret CPTu dissipation tests by treating the coefficient of consolidation as an unknown parameter that must be inferred during the training process. Measured data are necessary and incorporated into the loss vector in the form of data (,0;, 1,2,+, m D, where ND is the total number of measured data. The results demonstrate that PIELM is an effective tool for interpreting CPTu dissipation tests, owing to its ability to fuse data with physical constraints. When initial conditions are assumed known a priori, the predicted coefficient of consolidation 0.986 Predicted / Exact=0.986. Even without prior knowledge of the initial conditions, the coefficient of consolidation can still be inferred provided that sufficient measured data from CPTu dissipation tests are available. When the number of measured data points exceeds approximately 40 and M=200 (or 300), the predicted error for the coefficient of consolidation remains within 30%.

Limitations and Future Work

Certain limitations remain, such as the dissipation rate of pore water pressure generally decreases over time as the hydraulic gradient diminishes. The performance of PIELM depends on factors such as the distribution and quantity of training data, as well as the number of hidden-layer neurons. Future work should include rigorous error estimation, probabilistic analysis, and well-defined assessment metrics to enhance the interpretability and reliability of PIELM. The accuracy of the inferred consolidation coefficient from CPTu tests hinges on data quality. The excess pore water pressure can also dissipate vertically and the ground conditions can be quite complex in practice, which is not considered in this study because accounting for vertical dissipation would require substantially more measured data to train the network.

Conclusion

This paper investigates the feasibility of physics-informed machine learning (PIML) for solving Terzaghi consolidation equation using a more efficient framework termed PIELM. The deep neural network employed in conventional physics-informed neural network (PINN) frameworks is replaced by a single-hidden-layer extreme learning machine (ELM). Both physical laws and measured data are incorporated into a loss vector, making it convenient for the fusion of data and physics. For inverse analysis, this paper presents a preliminary study on applying PIELM to CPTu dissipation tests, from which the coefficient of consolidation of soil can be inferred. Since initial conditions are typically unavailable in such field tests, PIELM is particularly well-suited due to its ability to seamlessly integrate partial physical laws with incomplete measurement data. As long as sufficient high-quality data are available, it shows that PIELM can still accurately estimate the coefficient of consolidation from CPTu dissipation curves, even in the absence of explicit initial conditions.

Appendix

To improve the accuracy of PIML, the network architecture can be modified by applying hard constraints, which ensure that initial/boundary conditions are strictly satisfied. The constrained expression represents the family of all possible functions that satisfy the constraints, transforming constrained problems into unconstrained problems. In Case 1, the constrained expression is z H u z t g z t g t z g z = − −   (30). In Case 2, the constrained expression is (31). Finally, in Case 3, the constrained expression is (32). Data will be available from the corresponding author upon reasonable request. The authors declare that there is no known conflict of interest. We used large language models (LLMs) to help refine the writing and presentation of this paper (e.g., clarifying explanations, improving readability, and rephrasing drafts). All technical content—including problem formulation, method design, theoretical results, and experiments—was conceived, implemented, and validated by the authors.

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The gist This paper conducts a preliminary study to investigate the feasibility of a physics-informed extreme learning machine (PIELM) for solving the Terzaghi consolidation equation and interpreting the coefficient of consolidation of soil from piezocone penetration tests (CPTu)

How it works

1. The PIELM framework replaces fully connected neural networks with a single-hidden-layer extreme learning machine (ELM). The input layer parameters are randomly initialised within the range of [-1,1] and remain fixed throughout training.

2. For forward analysis, the output layer parameters are directly determined by minimising the loss vector using the Moore–Penrose generalized inverse.

3. For inverse analysis, both the output layer weights θ and the coefficient of consolidation c are treated as trainable parameters, and an iterative least squares method is adopted to training the ELM network.

4. A time-stepping strategy is incorporated into the PIELM framework to alleviate sharp gradients caused by inconsistent initial and boundary conditions, leading to the TS-PIELM framework.

Key Findings

**/The PIELM framework can directly calculate the network parameters to reduce the training cost compared to PINN which employs gradient descent methods. 2. The TS-PIELM technique helps mitigate the issue of excessively large gradients and thereby improves prediction accuracy at early consolidation times. 3. For inverse analysis, PIELM can still accurately estimate the coefficient of consolidation from CPTu dissipation curves even in the absence of explicit initial conditions. 4.

1. The PIELM framework can be implemented as an efficient alternative to PINNs by replacing fully connected neural networks with a single-hidden-layer ELM network, which can directly calculate the network parameters to reduce the training cost. This allows for significantly improved training efficiency compared to PINNs, potentially saving up to 98% of training time in certain cases.

2. TS-PIELM can be incorporated into the PIELM framework to alleviate sharp gradients caused by inconsistent initial and boundary conditions, which significantly improves the prediction accuracy at early consolidation times by dividing the process into time intervals.

3. The PIELM framework is suitable for interpreting CPTu data because it allows for the inference of the coefficient of consolidation without requiring the initial distribution of excess water pressure, fusing physical laws with measured data.

4. The PIELM system can be applied to infer soil parameters from CPTu dissipation tests by treating the coefficient of consolidation as an unknown parameter and must be inferred during the training process when incorporating measured data into the loss vector.

This paper conducts a preliminary study to investigate the feasibility of a physics-informed extreme learning machine (PIELM) for solving the Terzaghi consolidation equation and interpreting the coefficient of consolidation of soil from piezocone penetration tests (CPTu). In the PIELM framework, the target solution is approximated by a single-layer feed-forward extreme learning machine (ELM) network, instead of the deep neural networks typically employed in physics-informed neural networks (PINNs). Physical laws and measured data are integrated into a loss vector, which is minimized via least squares methods during ELM training. As a result, training efficiency is significantly improved by avoiding the gradient-descent optimisation commonly used in PINNs. The performance of PIELM is evaluated using three forward-problem case studies. Notably, a time-stepping strategy is incorporated into the PIELM framework to alleviate sharp gradients caused by inconsistent initial and boundary conditions. This paper further applies PIELM to estimate the soil consolidation coefficient, given that initial distributions of excess water pressure are often unavailable in CPTu dissipation tests (conducted following the pauses of penetration). By combining physical laws (excluding initial conditions) with measured data (i.e., excess pore-water pressure at the probe surface), the results demonstrate that PIELM is an effective tool for interpreting CPTu dissipation tests, owing to its ability to fuse data with physical constraints. This study contributes to the interpretation of consolidation coefficients from CPTu dissipation tests, particularly in scenarios where initial distributions of excess water pressure are not prior-known.