Deep Learning for One-dimensional Consolidation

TL;DR

This work addresses solving one-dimensional consolidation problems by enforcing the governing pore-pressure PDE as a constraint within a neural network. The authors propose a fully-connected network that predicts pore pressure p(z,t) from (z,t) and uses automatic differentiation to impose the PDE constraint, combining data loss with a collocation-based physics loss. They demonstrate forward solutions under drained-top and drained-top-and-bottom conditions, achieving high accuracy in p(z,t) predictions, and perform inverse analyses to recover the coefficient of consolidation c_v with small training samples. The approach promises faster real-time predictions for digital twins and improved constitutive-parameter identification, with broad applicability to PDE-governed systems in science and engineering.

Abstract

Neural networks with physical governing equations as constraints have recently created a new trend in machine learning research. In line with such efforts, a deep learning model for one-dimensional consolidation where the governing equation is applied as a constraint in the neural network is presented here. A review of related research is first presented and discussed. The deep learning model relies on automatic differentiation for applying the governing equation as a constraint. The total loss is measured as a combination of the training loss (based on analytical and model predicted solutions) and the constraint loss (a requirement to satisfy the governing equation). Two classes of problems are considered: forward and inverse problems. The forward problems demonstrate the performance of a physically constrained neural network model in predicting solutions for one-dimensional consolidation problems. Inverse problems show prediction of the coefficient of consolidation. Terzaghi's problem with varying boundary conditions are used as example and the deep learning model shows a remarkable performance in both the forward and inverse problems. While the application demonstrated here is a simple one-dimensional consolidation problem, such a deep learning model integrated with a physical law has huge implications for use in, such as, faster real-time numerical prediction for digital twins, numerical model reproducibility and constitutive model parameter optimization.

Figures (13)

• Figure 1: One-dimensional consolidation.
• Figure 2: Illustration of the neural network architecture with input, hidden and output layers. The activation function used at the hidden units is $\sigma(x) = \tanh(x)$. Automatic differentiation is used to determined the partial derivatives in the governing equation and is used as a physical constraint to optimize together with the prediction error based on training data. The number of hidden layers and hidden units in this figure is for illustration only; the actual number of layers and hidden units used for different cases are discussed in a later section.
• Figure 3: Example for one-dimensional consolidation.
• Figure 4: Initial and boundary condition training data for forward problems. The total number $N$ of training data points $(z,t)$ depends on the spatial and temporal discretization used to obtain the exact solution. The training data is shuffled and divided into batches according a specified batch size during training.
• Figure 5: Results from analytical solution and model prediction in terms of color plots on $(z,t)$ grid for a drained top boundary. The color plots are obtained using interpolation with the nearest available values. The batch size used for training the model is 100.