Abstract
Using machine-learning models as surrogate models is a popular technique to increase the computational efficiency of stochastic analysis. In this technique, a smaller number of numerical simulations are conducted for a case, and obtained results are used to train machine-learning surrogate models specific for this case. This study presents a new framework using deep learning, where models are trained with a big dataset covering any soil properties, spatial variabilities, or load conditions encountered in practice. These models are very accurate for new data without re-training. So, the small number of numerical simulations and training process are not needed anymore, which further increases efficiency. The prediction of bearing capacity of shallow strip footings is taken as an example. We start with a simple scenario, and progressively consider more complex scenarios until the full problem is considered. More than 12,000 data are used in training. It is shown that one-hidden-layer fully connected networks can give reasonable results for simple problems, but they are ineffective for complex problems, where deep neural networks show a competitive edge, and a deep-learning model achieves a very high accuracy (the root-mean-square relative error is 3.1% for unseen data). In testing examples, this model is proven very accurate if the parameters of specific cases are well in the defined limits. Otherwise, the capability of deep-learning models can be extended by simply generating more data outside the current limits and re-training the models.
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Abbreviations
- AvgPool:
-
Average pooling
- CNN:
-
Convolutional neural networks
- Conv:
-
Convolutional
- COV:
-
Coefficient of variation
- FC:
-
Fully connected
- FDQS:
-
Finite-difference quasi-static
- FEQS:
-
Finite-element quasi-static
- FES:
-
Finite-element static
- LC:
-
Locally connected
- MaxPool:
-
Max pooling
- PDF:
-
Probability density function
- RMSRE:
-
Root-mean-square relative error
- \(A\) :
-
Extent of slip lines
- \(B\) :
-
Width of strip footing
- \(c\) :
-
Cohesion
- \(c^{\prime}\) :
-
Dimensionless cohesion, \(c^{\prime} = \frac{c}{{c_{r} }}\)
- \({\mathbf{C}}^{\prime}\) :
-
Finite field of the dimensionless cohesion, represented by a matrix
- \(c_{r}\) :
-
A reference strength
- \(f_{R} \left( R \right)\) :
-
Probability density function of a random variable/field \(R\)
- \(H_{i}\) :
-
A random field
- \({\varvec{H}}\) :
-
Vector of random fields
- \(l_{x}\) :
-
Horizontal scale of fluctuation for a random field
- \(l_{y}\) :
-
Vertical scale of fluctuation for a random field
- \(N_{MC}\) :
-
Number of Monte Carlo simulations
- \(q_{0}\) :
-
Overburden load
- \(q_{0}^{^{\prime}}\) :
-
Dimensionless overburden load \(\frac{{q_{0} }}{{c_{r} }}\)
- \(q_{u}\) :
-
Bearing capacity
- \(q_{u}^{^{\prime}}\) :
-
Dimensionless bearing capacity \(\frac{{q_{u} }}{{c_{r} }}\)
- \(R\) :
-
Structural response
- \(s_{u}\) :
-
Undrained strength
- \(v\) :
-
Coefficient of variation for a random field/variable
- \({\varvec{x}}\) :
-
Position vector
- \(\user2{x^{\prime}}\) :
-
Dimensionless position vector, \(\user2{x^{\prime}} = \frac{{\varvec{x}}}{B}\)
- \(\gamma\) :
-
Unit weight
- \(\gamma ^{\prime}\) :
-
Dimensionless unit weight, \(\gamma ^{\prime} = \frac{\gamma B}{{c_{r} }}\)
- \({{\varvec{\Gamma}}}^{\prime}\) :
-
Finite field of the dimensionless unit weight, represented by a matrix
- \(\phi\) :
-
Friction angle
- \({{\varvec{\Phi}}}\) :
-
Finite field of the friction angle, represented by a matrix
- \(\mu\) :
-
Mean of a random field/variable
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He, X., Wang, F., Li, W. et al. Deep learning for efficient stochastic analysis with spatial variability. Acta Geotech. 17, 1031–1051 (2022). https://doi.org/10.1007/s11440-021-01335-1
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DOI: https://doi.org/10.1007/s11440-021-01335-1