Physics-informed Deep Learning to Solve Three-dimensional Terzaghi
  Consolidation Equation: Forward and Inverse Problems
Biao Yuan Ana Heitor He Wang Xiaohui Chen
The emergence of neural networks constrained by physical governing equations
has sparked a new trend in deep learning research, which is known as
Physics-Informed Neural Networks (PINNs). However, solving high-dimensional
problems with PINNs is still a substantial challenge, the space complexity
brings difficulty to solving large multidirectional problems. In this paper, a
novel PINN framework to quickly predict several three-dimensional Terzaghi
consolidation cases under different conditions is proposed. Meanwhile, the loss
functions for different cases are introduced, and their differences in
three-dimensional consolidation problems are highlighted. The tuning strategies
for the PINNs framework for three-dimensional consolidation problems are
introduced. Then, the performance of PINNs is tested and compared with
traditional numerical methods adopted in forward problems, and the coefficients
of consolidation and the impact of noisy data in inverse problems are
identified. Finally, the results are summarized and presented from
three-dimensional simulations of PINNs, which show an accuracy rate of over 99%
compared with ground truth for both forward and inverse problems. These results
are desirable with good accuracy and can be used for soil settlement
prediction, which demonstrates that the proposed PINNs framework can learn the
three-dimensional consolidation PDE well.
  Keywords: Three-dimensional Terzaghi consolidation; Physics-informed neural
networks (PINNs); Forward problems; Inverse problems; soil settlement