# Physics-informed Deep Learning to Solve Three-dimensional Terzaghi Consolidation Equation: Forward and Inverse Problems

arXiv: https://arxiv.org/abs/2401.05439 · HTML: https://mlsanity.com/paper/2401.05439/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 80 references that are in the map.
- First posted on arXiv 8 Jan 2024.

## Builds on (most cited first, 30 of 30)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [DeepXDE: A deep learning library for solving differential equations](https://mlsanity.com/paper/1907.04502/) (2019)
- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Physics-informed neural networks (PINNs) for fluid mechanics: A review](https://mlsanity.com/paper/2105.09506/) (2021)
- [On the eigenvector bias of Fourier feature networks: From regression to solving multi-scale PDEs with physics-informed neural networks](https://mlsanity.com/paper/2012.10047/) (2020)
- [Physics-Constrained Deep Learning for High-dimensional Surrogate Modeling and Uncertainty Quantification without Labeled Data](https://mlsanity.com/paper/1901.06314/) (2019)
- [A comprehensive study of non-adaptive and residual-based adaptive sampling for physics-informed neural networks](https://mlsanity.com/paper/2207.10289/) (2022)
- [Self-Adaptive Physics-Informed Neural Networks using a Soft Attention Mechanism](https://mlsanity.com/paper/2009.04544/) (2020)
- [A disciplined approach to neural network hyper-parameters: Part 1 -- learning rate, batch size, momentum, and weight decay](https://mlsanity.com/paper/1803.09820/) (2018)
- [PDE-Net 2.0: Learning PDEs from Data with A Numeric-Symbolic Hybrid Deep Network](https://mlsanity.com/paper/1812.04426/) (2018)
- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [PhyGeoNet: Physics-Informed Geometry-Adaptive Convolutional Neural Networks for Solving Parameterized Steady-State PDEs on Irregular Domain](https://mlsanity.com/paper/2004.13145/) (2020)
- [Physics-informed learning of governing equations from scarce data](https://mlsanity.com/paper/2005.03448/) (2020)
- [Activation Functions: Comparison of trends in Practice and Research for Deep Learning](https://mlsanity.com/paper/1811.03378/) (2018)
- [PPINN: Parareal Physics-Informed Neural Network for time-dependent PDEs](https://mlsanity.com/paper/1909.10145/) (2019)
- [Parallel Physics-Informed Neural Networks via Domain Decomposition](https://mlsanity.com/paper/2104.10013/) (2021)
- [Multi-Objective Loss Balancing for Physics-Informed Deep Learning](https://mlsanity.com/paper/2110.09813/) (2021)
- [Residual-based attention in physics-informed neural networks](https://mlsanity.com/paper/2307.00379/) (2023)
- [DAS-PINNs: A deep adaptive sampling method for solving high-dimensional partial differential equations](https://mlsanity.com/paper/2112.14038/) (2021)
- [Learning Activation Functions to Improve Deep Neural Networks](https://mlsanity.com/paper/1412.6830/) (2014)
- [Multi-stage Neural Networks: Function Approximator of Machine Precision](https://mlsanity.com/paper/2307.08934/) (2023)
- [Hidden Fluid Mechanics: A Navier-Stokes Informed Deep Learning Framework for Assimilating Flow Visualization Data](https://mlsanity.com/paper/1808.04327/) (2018)
- [Physics-informed neural network simulation of multiphase poroelasticity using stress-split sequential training](https://mlsanity.com/paper/2110.03049/) (2021)
- [Fully Connected Deep Structured Networks](https://mlsanity.com/paper/1503.02351/) (2015)
- [A practical PINN framework for multi-scale problems with multi-magnitude loss terms](https://mlsanity.com/paper/2308.06672/) (2023)
- [Deep Learning for One-dimensional Consolidation](https://mlsanity.com/paper/2004.11689/) (2020)
- [Solving multiphysics-based inverse problems with learned surrogates and constraints](https://mlsanity.com/paper/2307.11099/) (2023)
- [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)

## Built on by (most cited first, 1 of 1)

- [Physics-Informed Extreme Learning Machine (PIELM) for Tunnelling-Induced Soil-Pile Interactions](https://mlsanity.com/paper/2510.00698/) (2025)

## Abstract (arXiv)

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