⚡ electrical engineering

Physics-Informed Neural Network for Stability Analysis of Embankments Supported by GESCs with Bending Deformation

This study proposes a novel physics-informed neural network (PINN) framework that integrates the bending deformation of geosynthetic-encased stone columns (GESCs) as a physical constraint to achieve highly accurate, rapid probabilistic stability assessment and optimized design for embankments on soft soils.

Original authors: Xiaocong Cai, Ling Zhang, Bocheng Peng, Jingpeng Tan, Zijian Yang

Published 2026-09-15

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CC BY 4.0

Original authors: Xiaocong Cai, Ling Zhang, Bocheng Peng, Jingpeng Tan, Zijian Yang

Original paper licensed under CC BY 4.0 (https://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

When engineers build roads or railways across soft, squishy ground, they often face a stubborn problem: the earth simply cannot support the weight. To solve this, they drive thick columns of crushed stone into the soil, creating a reinforced foundation that can hold up the heavy load above. However, in the softest soils, these stone columns can buckle or bend sideways under pressure, much like a dry twig snapping when pushed too hard. To stop this, engineers wrap the stone columns in a strong, woven fabric, similar to a heavy-duty sack, which squeezes the stones together and keeps them from spreading apart. This technique, known as using geosynthetic-encased stone columns, is a standard way to make weak ground strong enough for embankments. Yet, predicting exactly when and how these reinforced columns might fail during an earthquake remains a difficult puzzle, because the ground, the stones, and the fabric all interact in complex, shifting ways that are hard to calculate by hand.

A team of researchers from Hunan University has tackled this complexity by teaching a computer to think like a physicist. Instead of relying solely on massive amounts of data or purely theoretical guesses, they developed a new kind of smart model that combines the learning power of artificial intelligence with the unbreakable rules of mechanics. They started by creating a detailed mathematical description of how these wrapped stone columns bend and break, specifically accounting for the way the fabric wrapping resists the column's tendency to snap. They tested this description against real-world experiments and computer simulations of a three-and-a-half-meter-high embankment built on soft soil. The results confirmed that their new method, which considers both the bending and the shearing of the columns, predicted the point of failure more accurately than older methods that ignored the bending effect. In fact, when they compared their calculations to a physical model test, their method matched the observed failure much more closely than previous approaches, which had tended to overestimate the safety of the structure.

With this reliable physical rule in hand, the researchers built a neural network, a type of computer program designed to find patterns in data. They fed the program thousands of scenarios generated by their physics-based calculations, covering a wide range of soil types, column sizes, and earthquake strengths. The program's job was to learn the relationship between these input conditions and the resulting safety of the embankment. What made this approach special was that the researchers did not just let the computer guess; they forced it to obey the physical law of bending they had derived earlier. Every time the computer made a prediction, it had to check if that prediction made sense according to the laws of physics. If the computer's guess violated the rules of how the column should bend, the system corrected it. This ensured that the final model was not just a statistical trick but a tool grounded in reality.

The result was a highly accurate predictor that could instantly calculate the safety of an embankment without needing to run slow, complex simulations every time. The model learned to predict four key outcomes: the overall safety factor, the maximum bending moment the column could withstand, the strength of the stone material, and the specific angle of the column's compression zone. When tested against data it had never seen before, the model was remarkably precise, with its predictions matching the calculated values almost perfectly. The researchers found that the model could distinguish how different factors influenced the outcome. For instance, they confirmed that stronger wrapping fabric significantly improved stability, while taller embankments or stronger earthquakes reduced it. They also discovered that the diameter of the column and the strength of the fabric were the most critical factors in determining how the column would behave under stress.

This work offers a new way to design safer foundations for roads and railways built on soft ground. By embedding the laws of physics directly into the learning process, the researchers created a tool that is both fast and trustworthy. It avoids the pitfalls of purely data-driven models, which can sometimes produce impossible results, and the slowness of traditional simulations, which take too long for rapid design changes. The study demonstrates that when artificial intelligence is guided by the fundamental rules of how materials bend and break, it can provide engineers with a reliable, instant assessment of stability. This approach allows for the optimization of designs, ensuring that the right amount of material is used to keep structures safe during seismic events, without the need for over-engineering or guesswork. The findings suggest that this method could become a standard tool for designing embankments, offering a clear path to more resilient infrastructure in earthquake-prone regions.

Technical Summary: Physics-Informed Neural Network for Stability Analysis of Embankments Supported by GESCs with Bending Deformation

Problem Statement
The stability assessment of embankments constructed on soft soils improved with Geosynthetic-Encased Stone Columns (GESCs) is complicated by the complex mechanical behavior of the columns under seismic and static loading. While ordinary stone columns (OSCs) typically fail via shear along a sliding surface, GESCs exhibit a more complex failure mode involving combined compression, bending, and shear due to the lateral confinement provided by the geosynthetic encasement. Traditional limit equilibrium methods often neglect the bending deformation and tensile strength contribution of the encasement, potentially leading to inaccurate safety factor predictions. Furthermore, while Finite Element Methods (FEM) can model these interactions, they are computationally prohibitive for parametric studies, reliability analyses, and rapid design iterations required to address geotechnical uncertainties. Purely data-driven machine learning models, conversely, often lack physical consistency and generalizability, functioning as "black boxes" that may violate fundamental mechanical laws.

Methodology
This study proposes a novel Physics-Informed Neural Network (PINN) framework designed to bridge the gap between data-driven efficiency and physical fidelity. The methodology proceeds in three main stages:

Theoretical Formulation and Validation:
- A theoretical calculation method for seismic stability is established, explicitly incorporating the bending moment contribution of GESCs. The model treats the geosynthetic encasement similarly to longitudinal steel in reinforced concrete (bearing tensile stress) and the stone aggregates as concrete (bearing compressive stress).
- Equations for the critical bending moment ( $M_u$ ), uniaxial compressive strength ( $f_c$ ), and the compression central angle ( $\alpha$ ) are derived based on limit equilibrium principles and geometric constraints.
- The theoretical method is validated against model tests, numerical simulations (FLAC3D), and existing analytical studies. Results indicate that considering bending failure yields a Factor of Safety (FS) more consistent with experimental observations than methods ignoring bending.
Dataset Generation:
- A high-fidelity dataset of 10,000 samples is generated using a physics-embedded Monte Carlo (MC) simulation.
- Fifteen input parameters are modeled as random variables (truncated normal or uniform distributions), including geotechnical properties (cohesion, friction angle, unit weight of soil, fill, and column), geometric parameters (column diameter, embankment height, slope ratio, area replacement ratio), and loading conditions (seismic coefficient, geosynthetic tensile strength).
- For each input vector, the theoretical model computes the corresponding outputs: Factor of Safety (FS), critical bending moment ( $M_u$ ), compressive strength ( $f_c$ ), and compression central angle ( $\alpha$ ).
PINN Architecture and Training:
- A fully connected feedforward neural network (15-20-15-10-3 architecture) is employed.
- Key Innovation: The algebraic physical constraint governing the bending equilibrium of the GESC (derived from the theoretical equations) is "hard-coded" into the loss function as a penalty term.
- The total loss function is a weighted sum of the data-driven loss (Mean Squared Error between predictions and MC simulation targets) and the physics-informed loss (residual of the physical constraint equation).
- The model is trained using the Levenberg-Marquardt algorithm to minimize this composite loss, ensuring predictions are statistically accurate and mechanically admissible.

Key Contributions

Novel PINN Framework: Development of a specific PINN architecture for GESC-supported embankments that integrates column bending mechanics directly into the learning process.
Physical Constraint Integration: The derivation and direct integration of algebraic physical constraints (bending equilibrium) into the loss function, ensuring the model adheres to fundamental mechanical laws rather than relying solely on statistical correlations.
Comprehensive Theoretical Model: Establishment and validation of a seismic stability analysis method that accounts for combined compression, bending, and shear failure modes, addressing a gap in current design practices.
High-Fidelity Surrogate Model: Creation of a rapid, differentiable surrogate model capable of predicting complex stability metrics without the computational cost of repeated FEM simulations.

Results

Predictive Accuracy: The trained PINN demonstrated exceptional performance across training, validation, and testing sets. The coefficient of determination ( $R^2$ ) for the Factor of Safety (FS) and key physical parameters ( $M_u$ , $f_c$ , $\alpha$ ) exceeded 99% (specifically reaching up to 99.99% for $M_u$ ).
Generalization: The near-perfect overlap of loss curves for training, validation, and test sets indicates the model has learned the underlying physical mapping rather than memorizing noise, confirming strong generalization capabilities.
Sensitivity Analysis: Global sensitivity analysis revealed that the geosynthetic tensile strength ( $T$ ) has a strong positive correlation with FS, while embankment height ( $H$ ), column diameter ( $D_c$ ), and seismic coefficient ( $k_h$ ) exhibit negative correlations. The model successfully captured these non-linear relationships.
Validation: The theoretical method used to generate data was validated against FLAC3D simulations and model tests, showing that neglecting bending deformation leads to overestimation of stability, whereas the proposed method aligns closely with observed failure modes.

Significance and Claims
The paper claims that this work represents a transformative direction for geotechnical design tools by offering a method that is both computationally efficient and physically trustworthy. By enforcing physical laws within the neural network, the model avoids the "black box" limitations of pure machine learning and the computational bottlenecks of traditional numerical methods. The authors assert that the developed PINN framework enables rapid probabilistic stability assessment and optimized design of GESC-supported embankments. The study concludes that while the framework is promising, future work should focus on incorporating more complex loading conditions (dynamic/cyclic), advanced constitutive models, and further validation against broader field monitoring data.

Technical Summary: Physics-Informed Neural Network for Stability Analysis of Embankments Supported by GESCs with Bending Deformation

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