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arxiv: 2606.00056 · v1 · pith:JYQZOPZMnew · submitted 2026-05-18 · 💻 cs.CE · cs.AI· cs.LG· physics.app-ph

Physics-Informed Neural Networks for Radial Consolidation of Combined Electroosmotic, Vacuum and Surcharge Preloading Considering Smear Effects

Dong Li , Yapeng Cao , Shuai Huang , Yujun Cui , Haiping Fu , Lu Yang , He Wei This is my paper

Pith reviewed 2026-06-30 17:58 UTC · model grok-4.3

classification 💻 cs.CE cs.AIcs.LGphysics.app-ph
keywords physics-informed neural networkselectro-osmotic consolidationradial consolidationsmear effectsvacuum preloadingsurcharge loadingPINNfinite element method
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The pith

A modified gated PINN with hard-constrained boundaries accurately simulates electro-osmotic radial consolidation under combined loadings and smear effects.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a dimensionless multi-domain physics-informed neural network framework to solve the governing equations for radial soil consolidation driven by electro-osmosis, vacuum, and surcharge preloading while including reduced permeability in smear zones around drains. Three variants are tested: a standard soft-constrained PINN, a modified gated version, and a gated version that hard-encodes the cathode boundary and initial conditions into the network output. The hard-constrained model reduces conflicts among competing loss terms for time-dependent loads and produces mean absolute errors of 0.43 kPa, 0.41 kPa, and 0.27 kPa against finite-element references in the exponential-vacuum, ramp-surcharge, and cyclic-surcharge cases. Sensitivity checks confirm the predictions remain stable when network depth, collocation density, and permeability ratios vary within engineering ranges.

Core claim

The study develops a dimensionless multi-domain physics-informed neural network framework for electro-osmotic radial consolidation that incorporates smear effects and combined vacuum and surcharge loading. The modified gated PINN with hard-constraint boundary encoding (Mod-HC-PINN) achieves the lowest mean absolute errors of 0.43 kPa, 0.41 kPa, and 0.27 kPa for exponential vacuum, ramp surcharge, and cyclic haversine surcharge cases by embedding the cathode boundary and initial conditions directly into the output structure, thereby reducing the optimization burden and improving physical consistency compared with soft-constrained models.

What carries the argument

The Mod-HC-PINN architecture, which combines a gated network with hard encoding of the cathode boundary and initial conditions to enforce physical consistency across multiple loading domains.

If this is right

  • The gated architecture resolves steep pressure gradients near the cathode and smear-zone interface under constant vacuum loading.
  • Embedding boundary conditions reduces the simultaneous learning of multiple competing objectives under time-dependent vacuum and surcharge loads.
  • The framework remains robust when network architecture, collocation density, and permeability contrast vary across practical engineering ranges.
  • Mod-HC-PINN predictions match finite-element references for constant, exponential, ramp, and cyclic loading cases with the reported mean absolute errors.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same hard-constraint approach could be adapted to model other coupled transport processes such as contaminant migration under electric fields.
  • Coupling the trained network to real-time sensor data might enable online updating of consolidation forecasts without repeated full simulations.
  • Testing the framework on three-dimensional geometries or anisotropic permeability fields would clarify how far the multi-domain structure generalizes.

Load-bearing premise

The finite element method reference solutions accurately represent the true physical consolidation behavior including smear effects and combined loadings.

What would settle it

Laboratory measurements of pore-pressure evolution during electro-osmotic consolidation with documented smear zones and cyclic surcharge would show whether the Mod-HC-PINN errors stay below 0.5 kPa.

Figures

Figures reproduced from arXiv: 2606.00056 by Dong Li, Haiping Fu, He Wei, Lu Yang, Shuai Huang, Yapeng Cao, Yujun Cui.

Figure 3
Figure 3. Figure 3: Schematic diagrams of the three PINN frameworks: (a) standard PINN (Std-PINN), (b) modified MLP PINN (Mod-PINN), and (c) modified MLP PINN with hard constraints (Mod-HC-PINN). 3.6 Collocation and sampling strategy PDE collocation points were sampled in the (R, T) domain using Latin Hypercube Sampling (LHS), which provides a stratified, space-filling distribution and more uniform coverage than purely random… view at source ↗
Figure 4
Figure 4. Figure 4: Comparison of radial excess pore-water pressure profiles at selected times for Case C1: (a) Std￾PINN versus FEM; (b) Mod-PINN versus FEM [PITH_FULL_IMAGE:figures/full_fig_p025_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Spatiotemporal distributions of excess pore-water pressure and corresponding absolute error fields for Case C1: (a) FEM reference solution; (b) Std-PINN prediction; (c) Mod-PINN prediction; (d) absolute error of the Std- PINN relative to the FEM solution. (e) absolute error of the Mod-PINN relative to the FEM solution [PITH_FULL_IMAGE:figures/full_fig_p027_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Convergence histories of the total loss and individual loss components for the soft-constrained PINN models in Case C1: (a) Std-PINN; (b) Mod-PINN. For the standard sequential MLP architecture, each hidden layer depends solely on the output of the previous layer. This sequential structure is more susceptible to spectral bias and therefore tends to learn the smooth, low-frequency component of the solution m… view at source ↗
Figure 7
Figure 7. Figure 7: Radial profiles of excess pore-water pressure at selected times for Case C2: (a) Std- PINN versus FEM; (b) Mod- PINN versus FEM; (c) Mod-HC-PINN versus FEM. The spatiotemporal absolute error maps in [PITH_FULL_IMAGE:figures/full_fig_p030_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: Convergence histories of the total loss and individual loss components for the soft-constrained PINN models in Case C2: (a) Mod-PINN; (b) Mod-HC-PINN [PITH_FULL_IMAGE:figures/full_fig_p032_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Gradient cosine similarities between the PDE, BC, and IC loss terms during Mod-PINN training: (a) Case C1, constant vacuum; (b) Case C2, exponential vacuum. Note: Lines show rolling mean; shaded bands show ±1 rolling standard deviation over a 2500-iteration window. 5.3 Combined electro-osmosis, vacuum, and ramp surcharge loading Case C3 introduces a more complex consolidation response due to the combinati… view at source ↗
Figure 11
Figure 11. Figure 11: Radial profiles of excess pore-water pressure at selected times for Case C3: (a) Mod- PINN versus FEM; (b) Mod-HC-PINN versus FEM. The Mod-PINN [ [PITH_FULL_IMAGE:figures/full_fig_p035_11.png] view at source ↗
Figure 12
Figure 12. Figure 12: Spatiotemporal absolute error distributions of (a) Mod-PINN, and (b) Mod-HC-PINN predictions for Case C3. 5.4 Combined electro-osmosis, vacuum, and cyclic haversine surcharge Case C4 represents the most demanding configuration considered in this study. The surcharge follows a cyclic haversine function (as shown in Eq. (6)) superimposed on the exponential vacuum ramp, producing approximately 13-14 complete… view at source ↗
Figure 13
Figure 13. Figure 13: Radial profiles of excess pore-water pressure at selected times for Case C4: (a) Mod- PINN versus FEM; (b) Mod-HC-PINN versus FEM [PITH_FULL_IMAGE:figures/full_fig_p037_13.png] view at source ↗
Figure 14
Figure 14. Figure 14: Time histories of excess pore-water pressure at selected radial locations for Case C4: (a) Mod￾PINN versus FEM; (b) Mod-HC-PINN versus FEM [PITH_FULL_IMAGE:figures/full_fig_p038_14.png] view at source ↗
Figure 15
Figure 15. Figure 15: Spatiotemporal absolute error distributions of (a) Mod-PINN, and (b) Mod-HC-PINN predictions for Case C4. Case C4 demonstrates that cyclic surcharge loading intensifies the difficulty of the time-dependent boundary-value problem. In the soft-constraint Mod-PINN, the repeated oscillatory loading leads to phase drift and amplitude mismatch in the outer radial zone, resulting in periodic error banding in the… view at source ↗
Figure 16
Figure 16. Figure 16: Heatmap of MAE for different network architectures with varying numbers of hidden layers and neurons per layer. 5.5.2 Effect of training data The influence of collocation point density on prediction accuracy was investigated by varying the total number of sampling points from 5,000 to 60,000 while maintaining the 1:2 ratio between smear zone and undisturbed zone. As shown in [PITH_FULL_IMAGE:figures/full… view at source ↗
Figure 17
Figure 17. Figure 17: Variation of mean absolute error (MAE) with the number of residual collocation points. 5.5.3 Effect of hydraulic permeability coefficients ratio on consolidation behavior The effect of hydraulic permeability coefficients ratio on consolidation behavior was evaluated by varying the hydraulic permeability ratio kr1/kr2 from 0.1 to 1.0, corresponding to permeability reductions of 10× to no reduction, respect… view at source ↗
Figure 18
Figure 18. Figure 18: Effect of conductivity ratios on the mean absolute error (MAE): (a) influence of 𝑘𝑟1/𝑘𝑟2; (b) influence of 𝑘𝑒1/𝑘𝑒2. 6. Summary and Conclusions This study developed a dimensionless multi-domain physics-informed neural network framework for radial electro-osmotic consolidation considering smear effects under combined vacuum pressure and time￾dependent surcharge loading. The smear and undisturbed zones were … view at source ↗
read the original abstract

This study develops a dimensionless multi-domain physics-informed neural network (PINN) framework for electro-osmotic radial consolidation considering smear effects and combined vacuum and surcharge loading. Three PINN-based models are investigated: a standard soft-constrained PINN (Std-PINN), a modified gated PINN (Mod-PINN), and a modified gated PINN with hard-constraint boundary encoding (Mod-HC-PINN). The models are evaluated against FEM reference solutions under four loading cases, including constant vacuum, exponential vacuum, exponential vacuum with ramp surcharge, and exponential vacuum with cyclic haversine surcharge. The results indicate that the gated architecture applied in Mod-PINN improves the resolution of steep pressure gradients near the cathode and smear-zone interface under constant vacuum loading. Under time-dependent loading, the soft-constrained Mod-PINN shows reduced accuracy because it must learn multiple competing objectives simultaneously. The Mod-HC-PINN mitigates this issue by embedding the cathode boundary and initial conditions into the output structure, thereby reducing the optimization burden and improving physical consistency. The Mod-HC-PINN achieves MAE values of 0.43, 0.41, and 0.27 kPa for the exponential vacuum, ramp surcharge, and cyclic surcharge cases, respectively. Sensitivity analyses further demonstrate that the proposed framework remains robust across practical ranges of network architecture, collocation density, and permeability contrast.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The paper develops a dimensionless multi-domain PINN framework (Std-PINN, Mod-PINN, Mod-HC-PINN) for electro-osmotic radial consolidation that incorporates smear-zone permeability reduction and combined time-dependent vacuum/surcharge loadings. The central claim is that the Mod-HC-PINN variant, which hard-encodes the cathode boundary and initial conditions, yields the lowest errors against FEM reference solutions, with reported MAE values of 0.43 kPa (exponential vacuum), 0.41 kPa (ramp surcharge), and 0.27 kPa (cyclic haversine surcharge), while sensitivity studies confirm robustness to network size, collocation density, and permeability contrast.

Significance. If the FEM reference solutions are shown to be accurate, the work would demonstrate a practical advantage of hard-constraint encoding for multi-objective time-dependent consolidation problems that exhibit steep gradients at the smear-zone interface. The explicit comparison of soft vs. hard constraint formulations and the sensitivity analyses constitute reproducible elements that strengthen the contribution to PINN applications in geotechnical modeling.

major comments (2)
  1. [Abstract / Results] Abstract and results section: The headline MAE values (0.43/0.41/0.27 kPa) are defined exclusively relative to FEM solutions; no analytical benchmark for any reduced case (constant vacuum, no surcharge, uniform permeability) or experimental comparison is supplied to anchor the absolute accuracy of either the FEM or the PINN, which is load-bearing for the claim that Mod-HC-PINN solves the physical problem.
  2. [Methodology] Methodology and training description: No information is provided on loss-term weighting coefficients, optimizer hyperparameters, or convergence criteria for the multi-objective optimization under time-dependent loadings; this omission prevents verification that the reported performance gains of Mod-HC-PINN arise from the hard-constraint architecture rather than from favorable hyperparameter choices.
minor comments (1)
  1. [Notation / §2] Notation for the dimensionless groups and the permeability reduction function inside the smear zone should be tabulated for clarity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments and recommendation for major revision. We address each major comment point by point below, with honest indications of where revisions will be incorporated.

read point-by-point responses

  1. Referee: [Abstract / Results] Abstract and results section: The headline MAE values (0.43/0.41/0.27 kPa) are defined exclusively relative to FEM solutions; no analytical benchmark for any reduced case (constant vacuum, no surcharge, uniform permeability) or experimental comparison is supplied to anchor the absolute accuracy of either the FEM or the PINN, which is load-bearing for the claim that Mod-HC-PINN solves the physical problem.

    Authors: We agree that absolute accuracy anchoring strengthens the claims. For the complete problem with time-dependent loadings and smear effects, no analytical solutions are available in the literature. The FEM formulation follows established geotechnical models (extensions of radial consolidation theory with electro-osmosis). In revision we will add a validation subsection comparing the FEM to the analytical solution for the reduced case of constant vacuum with uniform permeability (no smear), confirming agreement to within typical tolerances reported in prior work. Experimental comparison is outside the scope of this numerical study, as no laboratory data were generated; the relative gains of Mod-HC-PINN versus the other variants are nevertheless demonstrated consistently against the same reference. revision: partial

  2. Referee: [Methodology] Methodology and training description: No information is provided on loss-term weighting coefficients, optimizer hyperparameters, or convergence criteria for the multi-objective optimization under time-dependent loadings; this omission prevents verification that the reported performance gains of Mod-HC-PINN arise from the hard-constraint architecture rather than from favorable hyperparameter choices.

    Authors: We accept this point and will correct the omission. The revised manuscript will add a dedicated subsection specifying the loss-term weights (PDE residual, boundary, and initial-condition terms), the optimizer (Adam with learning-rate schedule), additional hyperparameters (epochs, collocation batching), and convergence criteria (loss threshold or plateau detection). This documentation will allow readers to confirm that the Mod-HC-PINN improvements stem from the hard-constraint design. revision: yes

Circularity Check

0 steps flagged

No circularity: PINN outputs validated against independent FEM reference solutions

full rationale

The paper trains three PINN variants to minimize PDE residuals for radial consolidation (including smear-zone permeability reduction and time-dependent loadings) and reports MAE against separately computed FEM solutions. No derivation step reduces a claimed prediction to a fitted parameter or self-citation by construction; the FEM benchmark is an external numerical method whose implementation details are independent of the PINN training. This matches the default case of a self-contained numerical method paper with external validation.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The framework rests on standard PINN loss construction and the pre-existing governing equations of electro-osmotic consolidation; no new physical entities are introduced.

axioms (1)
  • domain assumption The governing partial differential equations for electro-osmotic radial consolidation with smear effects are known and can be directly embedded in the neural network loss.
    Invoked implicitly when the models are described as physics-informed.

pith-pipeline@v0.9.1-grok · 5811 in / 1175 out tokens · 33083 ms · 2026-06-30T17:58:15.184032+00:00 · methodology

discussion (0)

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Reference graph

Works this paper leans on

5 extracted references · 2 canonical work pages

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    arXiv preprint arXiv:2602.07031

    Lagged backward-compatible physics-informed neural networks for unsaturated soil consolidation analysis. arXiv preprint arXiv:2602.07031. Liu, Y., Zheng, J.J., Zhao, X., Cao, W. and Huang, Z.,

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    In Dynamics of Earth's Fluid System (pp

    An introduction to the finite element method. In Dynamics of Earth's Fluid System (pp. 199- 226). CRC Press. Rittirong, A. and Shang, J.Q., 2008, July. Numerical analysis for electro -osmotic consolidation in two - dimensional electric field. In ISOPE International Ocean and Polar Engineering Conference (pp. ISOPE- I). ISOPE. Su, J.Q. and Wang, Z.,

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    An Expert’s Guide to Training Physics-informed Neural Networks,

    An expert's guide to training physics-informed neural networks. arXiv preprint arXiv:2308.08468. Wang, S., Li, B., Chen, Y. and Perdikaris, P.,

    work page arXiv

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    Acta Geotechnica, 20(11), 5941-5969

    PINN -based approach to the nonlinear large -strain consolidation under time-dependent drainage boundary. Acta Geotechnica, 20(11), 5941-5969. Zhang, P., Yin, Z.Y. and Sheil, B., 2024 a. A physics -informed data -driven approach for consolidation analysis. Géotechnique, 74(7), 620-631. Zhang, J., Zong, M., Wu, W., Zhang, Y. and Mei, G., 2024 b. Analytical...

    2024