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

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.

💡 Research Summary

This paper introduces a novel physics‑informed extreme learning machine (PIELM) for solving the one‑dimensional Terzaghi consolidation equation and for estimating the coefficient of consolidation (c v) from piezocone penetration test (CPTu) dissipation data. Unlike conventional physics‑informed neural networks (PINNs) that rely on deep multilayer architectures and iterative gradient‑descent optimization, PIELM employs a single‑layer feed‑forward extreme learning machine (ELM). In an ELM, hidden‑layer weights are randomly assigned and fixed, while only the output weights are determined analytically by solving a least‑squares problem. This eliminates the need for back‑propagation, dramatically reducing training time.

The governing PDE, ∂u/∂t = c v ∂²u/∂z², together with boundary conditions, is embedded into a loss vector that also contains measured data. The loss function is constructed as the concatenation of (i) the residual of the PDE evaluated at collocation points and (ii) the discrepancy between the network prediction and observed excess pore‑water pressure at the cone tip. A regularization parameter balances the physics term against the data term. Because the loss is a linear function of the output weights, the optimal weights are obtained by a closed‑form solution of the normal equations, i.e., a simple matrix inversion or pseudo‑inverse operation.

A key difficulty in consolidation problems is the presence of sharp gradients caused by inconsistent initial and boundary conditions. To mitigate this, the authors adopt a time‑stepping strategy: the total simulation time is divided into several sub‑intervals, and a separate PIELM is trained on each interval. The solution at the end of one interval serves as the initial condition for the next, ensuring continuity and stabilizing the training process.

The performance of PIELM is evaluated on three forward‑problem case studies: (1) a standard consolidation scenario with uniform initial excess pressure, (2) a case with a sudden load increase that generates steep pressure gradients, and (3) a problem with a non‑uniform initial pressure distribution. In all three cases, the PIELM predictions match high‑resolution finite‑difference solutions with mean absolute errors on the order of 10⁻³ and maximum errors below 5 × 10⁻³. Compared with a typical PINN implementation (thousands of training epochs), PIELM converges in a few hundred milliseconds, achieving a 5–10× speed‑up while maintaining comparable accuracy.

The most innovative application presented is the inverse estimation of c v from CPTu dissipation tests. In practice, the initial excess pressure field generated during cone penetration is unknown, making conventional inverse methods ill‑posed. The authors therefore omit the initial condition from the physics term and rely solely on the measured surface pressure time series as data. By simultaneously enforcing the consolidation PDE (without the initial condition) and fitting the observed pressure, the network identifies the value of c v that best reconciles the two. Validation against laboratory triaxial tests and empirical correlations shows that the PIELM‑derived c v values deviate by less than 5 % on average, even when the initial pressure distribution is completely unknown. This demonstrates that PIELM can fuse sparse field measurements with governing equations to retrieve hidden soil parameters.

The paper discusses several advantages of the proposed framework: (i) extreme computational efficiency due to analytical weight determination, (ii) natural integration of physics and data without extensive hyper‑parameter tuning, (iii) robustness to sharp gradients via the time‑stepping scheme, and (iv) potential for real‑time field deployment. Limitations are also acknowledged: the choice of hidden‑layer size and activation function can affect solution quality; extending to higher‑dimensional or strongly nonlinear problems may increase memory demands; and when no boundary or initial information is available, the physics alone may not guarantee a stable solution.

Future work suggested includes (a) extending PIELM to multi‑physics consolidation models (e.g., anisotropic permeability, nonlinear soil stiffness), (b) applying the method to three‑dimensional consolidation problems, and (c) incorporating adaptive collocation point selection and automated regularization strategies to further improve accuracy and robustness.

In summary, the study provides compelling evidence that a physics‑informed extreme learning machine can solve the Terzaghi consolidation equation with high accuracy and dramatically reduced computational cost, and that it can be successfully employed to interpret CPTu dissipation tests for the estimation of the consolidation coefficient, even in the absence of prior knowledge of the initial excess pressure field. This contribution opens a pathway toward fast, data‑augmented geotechnical analysis tools suitable for on‑site engineering practice.