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Fourier Feature Physics-Informed Neural Networks for Elasto-Plastic Analysis of Geomaterials with a Non-Associative Mohr-Coulomb Model

This study introduces a Fourier Feature Physics-Informed Neural Network (FF-PINN) that effectively mitigates spectral bias to accurately and efficiently solve two-dimensional elasto-plastic boundary value problems for geomaterials governed by a non-associative Mohr-Coulomb model, achieving significantly higher accuracy and faster convergence than conventional PINNs and Finite Element Method benchmarks.

Original authors: Apisit Robjanghvad, Sompote Youwai

Published 2026-07-29

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Apisit Robjanghvad, Sompote Youwai

Original paper licensed under CC BY 4.0 (http://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

Imagine you are trying to predict how a giant, squishy block of soil will react when you push a heavy building onto it. This isn't just a game of "what happens next"; it's a critical question for engineers designing foundations, dams, and skyscrapers. The soil doesn't just squish like a spring; it has a breaking point. Once you push too hard, it starts to slide and deform permanently, creating sharp, jagged lines where the safe ground meets the failing ground. To figure this out, scientists usually use a method called the Finite Element Method (FEM). Think of FEM like trying to map a mountain by chopping it into millions of tiny Lego bricks and calculating the stress on every single one. It's incredibly accurate, but it's also slow, expensive, and computationally heavy, like trying to solve a puzzle by building every piece from scratch every time.

In recent years, a new kind of "smart guesser" called a Physics-Informed Neural Network (PINN) has entered the scene. Instead of chopping the world into bricks, PINNs are like a super-smart artist who learns the rules of physics (like gravity and pressure) and tries to paint the whole picture at once. They are fast and don't need a massive library of pre-made answers. However, these artists have a weird blind spot: they are terrible at painting sharp edges. They are great at drawing smooth, rolling hills, but when it comes to the jagged cliffs where soil suddenly fails, they tend to blur the lines, smoothing over the very details engineers need to see. This paper tackles that specific problem, asking: Can we teach our digital artist to see the sharp edges clearly, so we can predict soil failure faster and better than the old Lego-brick method?

The authors of this study, working with soil that behaves in a tricky, friction-dependent way (known as a non-associative Mohr-Coulomb model), decided to upgrade the PINN's "eyes." They introduced a technique called Fourier Feature mapping. To understand what this does, imagine the PINN is a musician trying to play a song. The standard version of the network is like a musician who only knows how to play low, deep bass notes. It can hum a smooth tune, but if the song suddenly requires a high-pitched, rapid violin solo (which represents those sharp, sudden changes in soil stress), the musician just can't hear it, let alone play it. This is called "spectral bias."

The researchers fixed this by giving the musician a new set of ears. They added a layer of "Fourier features" right at the beginning of the network. Think of this as handing the musician a sheet of music that already includes all the high-pitched notes and rapid rhythms before they even start playing. Instead of struggling to figure out how to hit those high notes, the network can now instantly access them. This allows the AI to resolve the sharp, jagged boundaries where the soil starts to fail, which is exactly where the standard AI usually fails.

The results of their experiments were quite promising. They tested their new "Fourier Feature PINN" (or FF-PINN) against the old standard PINN and the traditional, slow Lego-brick method (FEM) using three different soil scenarios. The new model didn't just guess; it learned. In terms of accuracy, the FF-PINN reduced errors in predicting how much the ground would move (displacement) by up to 66 percent compared to the old PINN. For the stress forces inside the soil, it cut errors by about 27 percent. Perhaps most importantly, it didn't just get the numbers right; it drew the shape of the failure zone with much sharper fidelity, capturing the exact geometry of where the soil would break, whereas the old model smoothed those dangerous edges into a blur.

The study also looked at how fast this new method was. Even though the new network had slightly more "brain cells" (parameters) to manage, it learned twice as fast. It reached a stable, accurate solution in 17,500 training steps, while the old PINN needed 35,000 steps. This meant the wall-clock training time was cut in half, dropping from 1,366 seconds down to 683 seconds. The researchers also checked if the model could handle "noise," simulating real-world sensor errors up to 2.0 percent. The new model remained stable and didn't get confused, showing it's robust enough for messy, real-world data.

However, the paper is careful to note that this isn't a magic wand for everything. While the new model was a huge improvement, it still struggled with one specific detail: the plastic strain in the direction perpendicular to the soil layer (the out-of-plane strain). This specific component remained difficult to predict, with errors staying high (between 60 and 80 percent) regardless of how much data was fed to the model. The authors suggest this is a fundamental challenge with the current setup, not just a lack of practice.

In short, this paper suggests that by simply changing how the AI "sees" the input data—giving it a better frequency range to work with—we can solve complex soil mechanics problems much faster and more accurately than before. It's a significant step toward making AI a practical tool for geotechnical engineering, offering a way to predict soil failure that is both computationally efficient and physically consistent, provided we can eventually solve that one stubborn blind spot regarding out-of-plane strain.

Technical Summary: Fourier Feature Physics-Informed Neural Networks for Elasto-Plastic Analysis of Geomaterials

Problem Statement
Geotechnical engineering relies heavily on the Finite Element Method (FEM) to solve elasto-plastic boundary value problems, particularly those involving the non-associative Mohr-Coulomb model. While FEM is robust, it incurs high computational costs due to the incremental-iterative procedures required for non-linear stress-strain behavior. Physics-Informed Neural Networks (PINNs) offer a mesh-free alternative that embeds governing equations directly into the loss function, bypassing iterative solvers. However, conventional PINNs suffer from "spectral bias," a tendency to preferentially fit low-frequency solution components while failing to resolve sharp local gradients. This limitation is critical for the non-associative Mohr-Coulomb model, where pressure-dependent yield surfaces and non-associative flow rules generate narrow plastic zones and steep stress gradients at elastic-plastic boundaries. Standard PINN architectures often under-resolve these high-gradient regions, leading to inaccuracies in the very areas most critical for geotechnical assessment.

Methodology
This study proposes a Fourier Feature Physics-Informed Neural Network (FF-PINN) to address spectral bias in the context of non-associative plasticity. The methodology involves several key components:

Fourier Feature Mapping: Instead of feeding raw spatial coordinates $(x, y)$ directly into the network, the input layer employs a random Fourier feature mapping. This projects coordinates into a higher-dimensional space using sine and cosine transformations with frequencies sampled from a Gaussian distribution. This modification alters the spectral properties of the network's input representation, enabling the model to access high-frequency basis functions from the outset of training.
Constitutive Modeling: The framework utilizes the non-associative Mohr-Coulomb model, implemented via the Menétrey–Willam yield function to ensure smoothness in the deviatoric plane. The formulation includes a non-associative flow rule with a distinct plastic potential function to accurately capture soil dilatancy.
Multi-Objective Loss Function: The training objective combines physics-based residuals and data-driven terms. The physics loss enforces equilibrium equations, constitutive relations (stress-strain laws), and Karush-Kuhn-Tucker (KKT) conditions (ensuring plastic flow activation only when the stress state is on the yield surface). The data-driven loss penalizes discrepancies against high-fidelity FEM reference solutions for displacement, stress, strain, and plastic strain fields.
Strain-Adaptive Sampling (SAS): To further enhance resolution in critical zones, the study employs a sampling strategy that concentrates collocation points in regions of elevated strain magnitude, rather than using uniform grid sampling.
Benchmarking: The FF-PINN is benchmarked against conventional PINNs and FEM reference solutions across three test cases with varying cohesion, friction angles, and loading conditions.

Key Contributions

First Application to Non-Associative Plasticity: This work represents the first application of random Fourier feature mapping to pressure-dependent, non-associative plasticity within a PINN framework.
Mitigation of Spectral Bias: The study demonstrates that Fourier feature mapping effectively mitigates spectral bias, substantially improving the resolution of high-gradient displacement, stress, and plastic strain fields in localized plastic zones.
Strain-Adaptive Sampling Integration: The framework combines Fourier mapping with a strain-adaptive sampling strategy, showing that the enhanced representational capacity of FF-PINN allows it to effectively exploit concentrated training data in high-gradient regions.
Comprehensive Benchmarking: The framework is rigorously tested against conventional PINNs and FEM, with sensitivity analyses covering training data size, collocation density, loss weighting, and noise levels.

Results
Numerical experiments across three test cases yielded the following findings:

Accuracy Improvements: FF-PINN achieved superior accuracy across most predicted fields compared to conventional PINNs. Error reductions reached approximately 66% for displacement components and 27% for stress components in specific cases.
Plastic Zone Geometry: FF-PINN reproduced the geometry of plastic failure zones with markedly higher fidelity to FEM reference solutions. While conventional PINNs tended to smooth the boundaries of plastic zones, FF-PINN captured the sharper lobes and localized extensions characteristic of the non-associative Mohr-Coulomb model.
Training Efficiency: FF-PINN converged in approximately half the training epochs required by the conventional PINN (17,500 vs. 35,000 epochs), reducing wall-clock training time by 50% despite a modest increase in trainable parameters due to the expanded input dimension.
Sensitivity and Robustness: The model demonstrated robustness to noise levels up to 2.0%. Sensitivity analysis indicated that reliable accuracy is achieved with approximately 1,237 training data points and 5,000 collocation points. The optimal loss weighting ratio (physics-to-data) was identified as 1:15.
Limitations: The study noted a persistent challenge in accurately predicting the out-of-plane plastic strain component ( $\varepsilon_{zz}^p$ ), which exhibited high relative errors (60–80%) across all configurations, suggesting a fundamental approximation difficulty for this specific component under plane strain conditions.

Significance
The paper posits that the proposed FF-PINN framework offers a computationally efficient and physics-consistent alternative to FEM for elasto-plastic geotechnical analysis. By addressing the spectral bias inherent in standard neural networks, the framework enables the accurate resolution of sharp stress and strain gradients that are critical for assessing soil failure. The study concludes that the combination of Fourier feature mapping and strain-adaptive sampling not only improves predictive accuracy but also accelerates convergence, making the approach viable for applications requiring repeated model evaluations, such as parametric studies and reliability analyses. The authors emphasize that while the method shows significant promise, future work is needed to extend the framework to three-dimensional geometries, complex constitutive models, and cyclic loading scenarios.

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