  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.
