Deep Learning in Geotechnical Engineering: A Critical Assessment of PINNs and Operator Learning

TL;DR

The study benchmarks ML approaches against traditional solvers for 1D geotechnical problems, revealing that MLPs catastrophically fail in extrapolation ($S(t)=S_\infty(1-e^{-\alpha t})$) and PINNs are thousands of times slower with less accuracy than finite differences for forward problems like $u_{tt}=c^2u_{xx}$; DeepONet demands thousands of training samples and still trails direct solutions in speed. Automatic differentiation through forward solvers yields superior performance for inverse problems, delivering sub-percent parameter recovery efficiently. The work concludes that ML offers limited general forward-modeling value in geotechnics, but can be valuable for digital twins with fixed geometry or pattern recognition within the training envelope, provided rigorous site-based validation and physics constraints are enforced. Practical guidance is provided via break-even analyses and recommended workflow: prioritize AD for inversions, apply site-aware cross-validation, and reserve neural networks for genuinely expensive forward problems or niche data-assimilation tasks.

Abstract

Deep learning methods -- physics-informed neural networks (PINNs), deep operator networks (DeepONet), and graph network simulators (GNS) -- are increasingly proposed for geotechnical problems. This paper tests these methods against traditional solvers on canonical problems: wave propagation and beam-foundation interaction. PINNs run 90,000 times slower than finite difference with larger errors. DeepONet requires thousands of training simulations and breaks even only after millions of evaluations. Multi-layer perceptrons fail catastrophically when extrapolating beyond training data -- the common case in geotechnical prediction. GNS shows promise for geometry-agnostic simulation but faces scaling limits and cannot capture path-dependent soil behavior. For inverse problems, automatic differentiation through traditional solvers recovers material parameters with sub-percent accuracy in seconds. We recommend: use automatic differentiation for inverse problems; apply site-based cross-validation to account for spatial autocorrelation; reserve neural networks for problems where traditional solvers are genuinely expensive and predictions remain within the training envelope. When a method is four orders of magnitude slower with less accuracy, it is not a viable replacement for proven solvers.

Figures (9)

• Figure 1: A single neuron (perceptron) computes a weighted sum of inputs plus bias, then applies a nonlinear activation function $g$. Stacking neurons into layers creates a multi-layer perceptron capable of approximating complex nonlinear relationships.
• Figure 2: Extrapolation failure for multi-layer perceptrons with different activation functions on consolidation settlement. All three networks achieve excellent training accuracy (RMSE $<$ 2 mm on the 0-2 year training data) but fail catastrophically during extrapolation to 10 years. ReLU produces unbounded linear growth (RMSE = 87.6 mm), while tanh and sigmoid saturate prematurely (RMSE = 31 mm). The true solution asymptotically approaches 100 mm.
• Figure 3: Spatial autocorrelation in geotechnical data. (a) Random split creates spatial leakage: test points (red squares) lie within correlation ellipses of training points (circles), inflating reported accuracy. (b) Site-based split ensures test sites (4,5) are geologically independent of training sites (1,2,3), providing honest generalization estimates. Correlation lengths in soil properties are typically 50-100 m horizontally.
• Figure 4: Physics-Informed Neural Network architecture for 1D wave propagation. The network takes spatial ($x$) and temporal ($t$) coordinates as inputs, passes them through hidden layers, and outputs displacement $u(x,t)$. Automatic differentiation computes spatial and temporal derivatives ($\partial u/\partial x$, $\partial^2 u/\partial x^2$, $\partial u/\partial t$, $\partial^2 u/\partial t^2$), which are used to evaluate the wave equation residual. The loss function simultaneously minimizes PDE violations, boundary condition errors, and initial condition errors.
• Figure 5: Wave propagation comparison between PINN and finite difference from notebooks/ad-pinn-fwi.ipynb. The figure shows the 4-panel comparison: (a) constant velocity profile $c = 0.95$ m/s, (b) solution comparison at final time $t = 2.5$ s showing FD (blue) versus PINN (red dashed), (c) pointwise error on logarithmic scale revealing PINN errors of order $10^{-2}$, (d) computational time comparison showing PINN is 96,372$\times$ slower than FD (6.04 ms versus 581.92 s) while achieving RMSE = $3.53 \times 10^{-2}$ compared to FD's machine precision.