### Empirical Models and Soft Computing Techniques
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Method and Technique** & **Model** & **Reference** \\ \hline \multirow{8}{*}{Empirical model} & \(D_{H}=0.75H^{1/2}\theta^{1/3}\) & [PERSON] et al. [11] \\ \cline{2-3}  & \(\log LSI=-3.49-1.86\log R+0.98M_{w}\) & [PERSON] [25] \\ \cline{2-3}  & \(\log(D_{H}+0.01)=\) & \(-17.372+1.248M_{w}-0.923\log R-0.014R\) & \\  & & \(+0.68510R\)\(W+0.3\log T_{5}+4.826\log(100-F_{15})\) & [PERSON] et al. [21] \\  & & \(-1.091\text{D50}_{15}\log\left(D_{H}+0.01\right)\) & \\  & & \(-1.1452+0.988M_{w}-1.049\log R-0.011R\) & \\  & & \(+0.318\log S+0.619\log T_{15}\) & \\  & & \(+4.287\log(100-F_{15})-0.705D50_{15}\) & \\ \cline{2-3}  & \(\log(D_{H})=-16.713+1.532M-1.406\log R^{*}-0.012R+0.592\log W\) & \\  & Analysis & \(+0.540\log T_{5}+3.413\log(100-F_{15})-0.795\log(D50_{15}+0.1\text{mm})\) & \\  & & \(\log(D_{H})=-16.213+1.532M-1.406\log R^{*}-0.012R+0.338\log S\) & [PERSON] et al. [12] \\  & & \(+0.540\log T_{15}+3.413\log(100-F_{15})-0.795\log(D50_{15}+0.1\text{mm})\) & \\  & & \(R^{*}=R_{0}+10^{0.98M_{-5.64}}\) & \\ \cline{2-3}  & \(\log(D_{H})=\) & \(-17.95+1.605M_{w}-1.8673R^{*}-(\log(R+20))^{-3.3836}\) & \\  & & \(+0.540\log N+0.443\log T_{15}+4.1873\log(100-F_{15})\) & \\  & & \(-0.766\log\left(D50_{15}+0.1\text{mm}\right)\log(D_{H})\) & \\  & & \(-19.63+20.37M_{w}-2.6124\log R^{*}\) & \\  & & \(-(-0.68(R+20))^{-2.704}+0.3147\log S\) & \\  & & \(+0.6985\log T_{15}+4.1954\log(100-F_{15})\) & \\  & & \(-0.6772\log(D50_{15}+0.1\text{mm})\) & \\ \hline \multirow{8}{*}{Soft computing methods} & \multirow{2}{*}{ANN} & \(D_{H}=f(M,R,D50_{15},T_{15},F_{15},W,S,N_{160\alpha})\) & [PERSON] and [PERSON] [13] \\ \cline{2-3}  & & \(D_{H}=f(M,R,D50_{15},T_{15},F_{15},W,S)\) & [PERSON] and [PERSON] [33] \\ \cline{2-3}  & & \(D_{H}=-163.1\frac{1}{M^{2}}+57\frac{1}{F_{15}}-0.0035\frac{T_{15}^{2}}{W-D50_{15} }+0.02\frac{T_{15}^{2}}{F_{15}-D50_{15}}\) & \\ \cline{2-3}  & \(\text{GP}\) & \(-0.26\frac{T_{15}^{2}}{F_{15}^{2}}+0.00672\frac{T_{15}^{2}}{F_{15}^{2}}-0.0013 \text{W}^{2}+0.0002M^{2}\cdot W\cdot T_{15}+3.7\) & [PERSON] et al. [31] \\ \cline{2-3}  & & \(D_{H}=-0.8\frac{T_{15}}{M}+0.0014T_{15}^{2}+0.16T_{15}+0.112S+0.04\frac{5T_{15} }{D50_{15}}\) & \\ \cline{2-3}  & & \(-0.026R\cdot D50_{15}+1.14\) & \\ \cline{2-3}  & ANFIS & \(D_{H}=f(M,R,D50_{15},T_{15},F_{15},W,S)\) & [PERSON] [34] \\ \hline \hline \end{tabular} Note: \(N_{1400}\): (\(N1\))\({}_{\theta}\) value corresponds to \(\mathrm{js}\), \(\mathrm{js}\) is the lowest factor of safety below water table using simplified approach; \(\theta\): larger slope of either ground surface or the base of liquefied soil (\(\%\)); \(H\): thickness of liquefied zone (m); \(R^{*}\): modified source distance factor that is a function of earthquake magnitude.  
\end{table} 
Table 1: Empirical and machine learning approaches for liquefaction-induced lateral displacement. face and sloping ground. Adaptive neuro-fuzzy inference system (ANFIS) based approach was suggested by [PERSON] [34] using 426 case histories data and shows an improvement to the [PERSON] et al. [12], [PERSON] [29], [PERSON] et al. [21], and [PERSON] et al. [4] models. Soft computing methodologies are more accurate than analytical formulas, according to all of these studies. The findings revealed that the ML models mentioned above are capable of obtaining the experimental observations with acceptable accuracy. However, this field continues to be further explored. The Gaussian process regression (GPR) approach has been successfully applied in many domains, but its application in geotechnical engineering is limited based on literature surveys. Considering the improved performance of GPR, it is, however, used for the first time in this study to predict the liquefaction-induced lateral spread displacement for free face condition. To demonstrate the efficacy of the proposed GPR-based model, the results are compared with various well-known models for calculating the \(D_{H}\).