# Deep Learning for One-dimensional Consolidation

arXiv: https://arxiv.org/abs/2004.11689 · HTML: https://mlsanity.com/paper/2004.11689/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Apr 2020.

## Built on by (most cited first, 14 of 14)

- [Physics-informed neural network simulation of multiphase poroelasticity using stress-split sequential training](https://mlsanity.com/paper/2110.03049/) (2021)
- [Partial Differential Equations Meet Deep Neural Networks: A Survey](https://mlsanity.com/paper/2211.05567/) (2022)
- [Constitutive model characterization and discovery using physics-informed deep learning](https://mlsanity.com/paper/2203.09789/) (2022)
- [Physics-informed neural network solution of thermo-hydro-mechanical (THM) processes in porous media](https://mlsanity.com/paper/2203.01514/) (2022)
- [Inverse modeling of nonisothermal multiphase poromechanics using physics-informed neural networks](https://mlsanity.com/paper/2209.03276/) (2022)
- [A Deep Learning Approach for Predicting Two-dimensional Soil Consolidation Using Physics-Informed Neural Networks (PINN)](https://mlsanity.com/paper/2205.05710/) (2022)
- [Lagged backward-compatible physics-informed neural networks for unsaturated soil consolidation analysis](https://mlsanity.com/paper/2602.07031/) (2026)
- [Physics-informed Deep Learning to Solve Three-dimensional Terzaghi Consolidation Equation: Forward and Inverse Problems](https://mlsanity.com/paper/2401.05439/) (2024)
- [Parsimonious Universal Function Approximator for Elastic and Elasto-Plastic Cavity Expansion Problems](https://mlsanity.com/paper/2407.19074/) (2024)
- [A Critical Assessment of PINNs and Operator Learning for Geotechnical Engineering](https://mlsanity.com/paper/2512.24365/) (2025)
- [CarbonNet: How Computer Vision Plays a Role in Climate Change? Application: Learning Geomechanics from Subsurface Geometry of CCS to Mitigate Global Warming](https://mlsanity.com/paper/2403.06025/) (2024)
- [Operator Learning for Consolidation: An Architectural Comparison for DeepONet Variants](https://mlsanity.com/paper/2507.10368/) (2025)
- [Physics-Informed Neural Networks for Radial Consolidation of Combined Electroosmotic, Vacuum and Surcharge Preloading Considering Smear Effects](https://mlsanity.com/paper/2606.00056/) (2026)
- [Critical evaluation of PINN for FWD inverse analysis and differentiable FEM as an alternative](https://mlsanity.com/paper/2606.03210/) (2026)

## Abstract (arXiv)

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.