<p>Developing a forecast model of soil liquefaction helps assess the condition of the site and prevent the pertinent losses resulting from an earthquake. In recent years, various types of datasets have been utilized for this purpose, such as field test results (Standard Penetration Test (<InlineEquation ID="IEq1"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10706_2025_3275_Article_IEq1.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="39" /> </InlineMediaObject> <EquationSource Format="TEX">\(SPT\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="italic">SPT</mi> </mrow> </math></EquationSource> </InlineEquation>), Cone Penetration Test (<InlineEquation ID="IEq2"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10706_2025_3275_Article_IEq2.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="41" /> </InlineMediaObject> <EquationSource Format="TEX">\(CPT\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="italic">CPT</mi> </mrow> </math></EquationSource> </InlineEquation>), Shear Wave Velocity (<InlineEquation ID="IEq3"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10706_2025_3275_Article_IEq3.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="25" /> </InlineMediaObject> <EquationSource Format="TEX">\(Vs\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="italic">Vs</mi> </mrow> </math></EquationSource> </InlineEquation>), etc.), laboratory tests (Fine Content (<InlineEquation ID="IEq4"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10706_2025_3275_Article_IEq4.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="30" /> </InlineMediaObject> <EquationSource Format="TEX">\(FC\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="italic">FC</mi> </mrow> </math></EquationSource> </InlineEquation>), Unconfined Compressive Strength (<InlineEquation ID="IEq5"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10706_2025_3275_Article_IEq5.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="22" /> </InlineMediaObject> <EquationSource Format="TEX">\(qu\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="italic">qu</mi> </mrow> </math></EquationSource> </InlineEquation>), etc.), and analytical equations (Cyclic Stress Ratio (<InlineEquation ID="IEq6"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10706_2025_3275_Article_IEq6.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">\(CSR\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="italic">CSR</mi> </mrow> </math></EquationSource> </InlineEquation>), Cyclic Resistance Ratio (<InlineEquation ID="IEq7"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10706_2025_3275_Article_IEq7.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="44" /> </InlineMediaObject> <EquationSource Format="TEX">\(CRR\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="italic">CRR</mi> </mrow> </math></EquationSource> </InlineEquation>), etc.). The present study utilized field test data and group method of data handling (<InlineEquation ID="IEq8"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10706_2025_3275_Article_IEq8.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="65" /> </InlineMediaObject> <EquationSource Format="TEX">\(GMDH\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="italic">GMDH</mi> </mrow> </math></EquationSource> </InlineEquation>)-based metaheuristic models, taking into account the complexity of the liquefaction mechanism. For this aim, Aquila Optimizer (<InlineEquation ID="IEq9"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10706_2025_3275_Article_IEq9.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="30" /> </InlineMediaObject> <EquationSource Format="TEX">\(AO\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="italic">AO</mi> </mrow> </math></EquationSource> </InlineEquation>), Arithmetic Optimization Algorithm (<InlineEquation ID="IEq10"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10706_2025_3275_Article_IEq10.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">\(AOA\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="italic">AOA</mi> </mrow> </math></EquationSource> </InlineEquation>), Genetic Algorithm (<InlineEquation ID="IEq11"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10706_2025_3275_Article_IEq11.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="31" /> </InlineMediaObject> <EquationSource Format="TEX">\(GA\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="italic">GA</mi> </mrow> </math></EquationSource> </InlineEquation>), Gravitational Search Algorithm (<InlineEquation ID="IEq12"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10706_2025_3275_Article_IEq12.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">\(GSA\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="italic">GSA</mi> </mrow> </math></EquationSource> </InlineEquation>), and Particle Swarm Optimization (<InlineEquation ID="IEq13"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10706_2025_3275_Article_IEq13.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="42" /> </InlineMediaObject> <EquationSource Format="TEX">\(PSO\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="italic">PSO</mi> </mrow> </math></EquationSource> </InlineEquation>) have been employed to tune the hyperparameters of the <InlineEquation ID="IEq14"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10706_2025_3275_Article_IEq8.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="65" /> </InlineMediaObject> <EquationSource Format="TEX">\(GMDH\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="italic">GMDH</mi> </mrow> </math></EquationSource> </InlineEquation> and artificial neural network (<InlineEquation ID="IEq15"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10706_2025_3275_Article_IEq15.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="46" /> </InlineMediaObject> <EquationSource Format="TEX">\(ANN\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="italic">ANN</mi> </mrow> </math></EquationSource> </InlineEquation>) models. The results showed that <InlineEquation ID="IEq16"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10706_2025_3275_Article_IEq16.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="104" /> </InlineMediaObject> <EquationSource Format="TEX">\(GSA-ANN\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mi>S</mi> <mi>A</mi> <mo>-</mo> <mi>A</mi> <mi>N</mi> <mi>N</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq17"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10706_2025_3275_Article_IEq17.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="111" /> </InlineMediaObject> <EquationSource Format="TEX">\(GA-GMDH\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mi>A</mi> <mo>-</mo> <mi>G</mi> <mi>M</mi> <mi>D</mi> <mi>H</mi> </mrow> </math></EquationSource> </InlineEquation> have the best performance in predicting liquefaction triggering potential, while <InlineEquation ID="IEq18"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10706_2025_3275_Article_IEq18.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="124" /> </InlineMediaObject> <EquationSource Format="TEX">\(AOA-GMDH\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mi>O</mi> <mi>A</mi> <mo>-</mo> <mi>G</mi> <mi>M</mi> <mi>D</mi> <mi>H</mi> </mrow> </math></EquationSource> </InlineEquation> has the lowest accuracy. Also, a Shapley Additive Explanations (<InlineEquation ID="IEq19"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10706_2025_3275_Article_IEq19.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="55" /> </InlineMediaObject> <EquationSource Format="TEX">\(SHAP\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="italic">SHAP</mi> </mrow> </math></EquationSource> </InlineEquation>) analysis was done using <InlineEquation ID="IEq20"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10706_2025_3275_Article_IEq20.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="123" /> </InlineMediaObject> <EquationSource Format="TEX">\(GSA-GMDH\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mi>S</mi> <mi>A</mi> <mo>-</mo> <mi>G</mi> <mi>M</mi> <mi>D</mi> <mi>H</mi> </mrow> </math></EquationSource> </InlineEquation>, which showed that the parameters depth (<InlineEquation ID="IEq21"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10706_2025_3275_Article_IEq21.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="16" /> </InlineMediaObject> <EquationSource Format="TEX">\(Z\)</EquationSource> <EquationSource Format="MATHML"><math> <mi>Z</mi> </math></EquationSource> </InlineEquation>), <InlineEquation ID="IEq22"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10706_2025_3275_Article_IEq22.gif" Format="GIF" Height="10" Rendition="HTML" Resolution="72" Type="Linedraw" Width="13" /> </InlineMediaObject> <EquationSource Format="TEX">\(\sigma \)</EquationSource> <EquationSource Format="MATHML"><math> <mi>σ</mi> </math></EquationSource> </InlineEquation>, and <InlineEquation ID="IEq23"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10706_2025_3275_Article_IEq23.gif" Format="GIF" Height="12" Rendition="HTML" Resolution="72" Type="Linedraw" Width="19" /> </InlineMediaObject> <EquationSource Format="TEX">\(\sigma {\prime}\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>σ</mi> <mo>′</mo> </mrow> </math></EquationSource> </InlineEquation> have the most significant impact on soil liquefaction. All models greatly surpass the performance of a random classifier (<InlineEquation ID="IEq24"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10706_2025_3275_Article_IEq24.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">\(AUC\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="italic">AUC</mi> </mrow> </math></EquationSource> </InlineEquation>= 0.5), therefore validating their predictive efficacy. The elevated <InlineEquation ID="IEq25"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10706_2025_3275_Article_IEq24.gif" Format="GIF" Height="14" Rendition="HTML" Resolution="72" Type="Linedraw" Width="43" /> </InlineMediaObject> <EquationSource Format="TEX">\(AUC\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi mathvariant="italic">AUC</mi> </mrow> </math></EquationSource> </InlineEquation> values of <InlineEquation ID="IEq26"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10706_2025_3275_Article_IEq18.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="124" /> </InlineMediaObject> <EquationSource Format="TEX">\(AOA-GMDH\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>A</mi> <mi>O</mi> <mi>A</mi> <mo>-</mo> <mi>G</mi> <mi>M</mi> <mi>D</mi> <mi>H</mi> </mrow> </math></EquationSource> </InlineEquation> and <InlineEquation ID="IEq27"> <InlineMediaObject> <ImageObject Color="BlackWhite" FileRef="10706_2025_3275_Article_IEq17.gif" Format="GIF" Height="15" Rendition="HTML" Resolution="72" Type="Linedraw" Width="111" /> </InlineMediaObject> <EquationSource Format="TEX">\(GA-GMDH\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>G</mi> <mi>A</mi> <mo>-</mo> <mi>G</mi> <mi>M</mi> <mi>D</mi> <mi>H</mi> </mrow> </math></EquationSource> </InlineEquation> indicate that these hybrid methodologies are especially efficacious for the job, presumably owing to their refined parameter optimization and resilient feature selection processes.</p>

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Predicting Liquefaction Triggering Potential Using Metaheuristic GMDH Approaches

  • Nazli Khankeshi Oghli,
  • Mahzad Esmaeili-Falak

摘要

Developing a forecast model of soil liquefaction helps assess the condition of the site and prevent the pertinent losses resulting from an earthquake. In recent years, various types of datasets have been utilized for this purpose, such as field test results (Standard Penetration Test ( \(SPT\) SPT ), Cone Penetration Test ( \(CPT\) CPT ), Shear Wave Velocity ( \(Vs\) Vs ), etc.), laboratory tests (Fine Content ( \(FC\) FC ), Unconfined Compressive Strength ( \(qu\) qu ), etc.), and analytical equations (Cyclic Stress Ratio ( \(CSR\) CSR ), Cyclic Resistance Ratio ( \(CRR\) CRR ), etc.). The present study utilized field test data and group method of data handling ( \(GMDH\) GMDH )-based metaheuristic models, taking into account the complexity of the liquefaction mechanism. For this aim, Aquila Optimizer ( \(AO\) AO ), Arithmetic Optimization Algorithm ( \(AOA\) AOA ), Genetic Algorithm ( \(GA\) GA ), Gravitational Search Algorithm ( \(GSA\) GSA ), and Particle Swarm Optimization ( \(PSO\) PSO ) have been employed to tune the hyperparameters of the \(GMDH\) GMDH and artificial neural network ( \(ANN\) ANN ) models. The results showed that \(GSA-ANN\) G S A - A N N and \(GA-GMDH\) G A - G M D H have the best performance in predicting liquefaction triggering potential, while \(AOA-GMDH\) A O A - G M D H has the lowest accuracy. Also, a Shapley Additive Explanations ( \(SHAP\) SHAP ) analysis was done using \(GSA-GMDH\) G S A - G M D H , which showed that the parameters depth ( \(Z\) Z ), \(\sigma \) σ , and \(\sigma {\prime}\) σ have the most significant impact on soil liquefaction. All models greatly surpass the performance of a random classifier ( \(AUC\) AUC = 0.5), therefore validating their predictive efficacy. The elevated \(AUC\) AUC values of \(AOA-GMDH\) A O A - G M D H and \(GA-GMDH\) G A - G M D H indicate that these hybrid methodologies are especially efficacious for the job, presumably owing to their refined parameter optimization and resilient feature selection processes.