Table 3 Test performance metrics

VarvYS pred(MPa) at 298K

RMSE

\(\underline{{\bf{63.079}}\pm {\bf{43.580}}}\)

85.254 ± 60.183

126.506 ± 51.467

–

133.2247 ± 25.0982

–

–

MAE

34.404 ± 21.687

\(\underline{{\bf{32.621}}\pm {\bf{26.955}}}\)

89.187 ± 31.297

–

90.7540 ± 13.8952

–

–

R2

\(\underline{{\bf{0.852}}\pm {\bf{0.122}}}\)

0.712 ± 0.201

0.405 ± 0.164

–

0.4084 ± 0.1114

–

–

Spearman

\(\underline{{\bf{0.944}}\pm {\bf{0.029}}}\)

0.898 ± 0.069

0.667 ± 0.146

–

0.7131 ± 0.0917

–

–

SMAPE

8.6% ± 4.0%

\(\underline{{\bf{6.8}} \% \pm {\bf{5.2}} \% }\)

25.2% ± 6.2%

–

27.75% ± 2.95%

–

–

SFE calc

RMSE

160.015 ± 31.749

154.823 ± 34.594

144.515 ± 29.097

–

\(\underline{{\bf{141.9662}}\pm {\bf{12.8200}}}\)

–

–

MAE

103.249 ± 11.891

101.121 ± 17.958

94.588 ± 11.523

–

\(\underline{{\bf{94.5297}}\pm {\bf{11.6745}}}\)

–

–

R2

0.126 ± 0.126

0.186 ± 0.139

\(\underline{{\bf{0.289}}\pm {\bf{0.100}}}\)

–

0.1921 ± 0.1753

–

–

Spearman

\(\underline{{\bf{0.686}}\pm {\bf{0.066}}}\)

0.567 ± 0.099

0.677 ± 0.117

–

0.6124 ± 0.1351

–

–

SMAPE

\(\underline{{\bf{72.3}} \% \pm {\bf{8.6}} \% }\)

78.0% ± 10.2%

76.3% ± 5.6%

–

82.61% ± 15.56%

–

–

VEC Avg

RMSE

13.345 ± 2.473

\(\underline{{\bf{0.904}}\pm {\bf{0.172}}}\)

62.739 ± 15.508

–

61.4221 ± 23.8710

–

–

MAE

9.900 ± 1.613

\(\underline{{\bf{0.559}}\pm {\bf{0.122}}}\)

45.720 ± 10.894

–

43.5553 ± 15.8566

–

–

R2

0.986 ± 0.005

\(\underline{{\bf{1.000}}\pm {\bf{0.000}}}\)

0.690 ± 0.146

–

0.6758 ± 0.1728

–

–

Spearman

0.992 ± 0.005

\(\underline{{\bf{0.999}}\pm {\bf{0.000}}}\)

0.837 ± 0.073

–

0.7858 ± 0.1334

–

–

SMAPE

1.5% ± 0.3%

\(\underline{{\bf{0.1}} \% \pm {\bf{0.0}} \% }\)

6.8% ± 1.8%

–

6.52% ± 2.55%

–

–

Yield Strength (MPa)

RMSE

89.158 ± 12.990

79.908 ± 10.563

77.999 ± 7.234

63.951 ± 12.507

\(\underline{{\bf{39.4717}}\pm {\bf{13.7846}}}\)

39.8794 ± 14.8138

43.9739 ± 19.9147

MAE

63.459 ± 9.197

57.057 ± 6.396

54.650 ± 6.106

46.235 ± 9.297

28.7471 ± 11.0498

\(\underline{{\bf{28.5152}}\pm {\bf{12.4962}}}\)

31.9583 ± 15.4363

R2

0.585 ± 0.184

0.679 ± 0.110

0.685 ± 0.115

0.732 ± 0.050

\(\underline{{\bf{0.8558}}\pm {\bf{0.1359}}}\)

0.8495 ± 0.1464

0.8038 ± 0.2305

Spearman

0.777 ± 0.093

0.838 ± 0.062

0.874 ± 0.061

0.887 ± 0.024

0.9099 ± 0.0916

\(\underline{{\bf{0.9138}}\pm {\bf{0.0894}}}\)

0.8952 ± 0.1255

SMAPE

15.8% ± 2.7%

14.3% ± 1.6%

13.7% ± 1.9%

11.8% ± 1.4%

\(\underline{{\bf{7.64}} \% \pm {\bf{3.04}} \% }\)

7.64% ± 3.41%

8.68% ± 4.12%

% UTS True (Mpa)

RMSE

182.564 ± 28.431

149.748 ± 25.830

146.172 ± 26.465

180.835 ± 55.784

103.5612 ± 42.8069

\(\underline{{\bf{98.1523}}\pm {\bf{38.1771}}}\)

118.3553 ± 50.7326

MAE

138.201 ± 22.664

110.308 ± 17.711

106.354 ± 15.304

130.595 ± 42.525

74.4103 ± 32.1561

\(\underline{{\bf{70.2329}}\pm {\bf{28.4956}}}\)

87.2348 ± 41.1554

R2

0.291 ± 0.375

0.574 ± 0.042

0.595 ± 0.046

0.349 ± 0.442

0.7786 ± 0.1402

\(\underline{{\bf{0.7996}}\pm {\bf{0.1224}}}\)

0.6772 ± 0.3124

Spearman

0.630 ± 0.085

0.760 ± 0.039

0.806 ± 0.033

0.687 ± 0.204

0.8741 ± 0.0925

\(\underline{{\bf{0.8761}}\pm {\bf{0.0982}}}\)

0.8085 ± 0.1940

SMAPE

15.6% ± 2.9%

12.4% ± 2.6%

12.1% ± 2.3%

14.9% ± 5.1%

8.91% ± 4.08%

\(\underline{{\bf{8.40}} \% \pm {\bf{3.62}} \% }\)

10.44% ± 5.02%

UTS/YS

RMSE

0.503 ± 0.071

0.417 ± 0.052

0.364 ± 0.066

0.382 ± 0.057

0.3646 ± 0.1254

\(\underline{{\bf{0.3636}}\pm {\bf{0.1405}}}\)

0.525 ± 0.144

MAE

0.383 ± 0.051

0.316 ± 0.043

0.271 ± 0.034

0.294 ± 0.049

0.2458 ± 0.0752

\(\underline{{\bf{0.2447}}\pm {\bf{0.0881}}}\)

0.382 ± 0.108

R2

0.678 ± 0.088

0.785 ± 0.028

\(\underline{{\bf{0.835}}\pm {\bf{0.039}}}\)

0.817 ± 0.065

0.8032 ± 0.1312

0.7988 ± 0.1443

0.556 ± 0.253

Spearman

0.778 ± 0.038

0.851 ± 0.040

0.844 ± 0.055

0.878 ± 0.043

0.8856 ± 0.0748

\(\underline{{\bf{0.8886}}\pm {\bf{0.0797}}}\)

0.809 ± 0.093

SMAPE

15.8% ± 1.6%

12.8% ± 1.5%

11.6% ± 1.0%

11.8% ± 2.1%

\(\underline{{\bf{9.93}} \% \pm {\bf{3.43}} \% }\)

10.06% ± 4.08%

15.301% ± 4.099%

Elong (%)

RMSE

7.782 ± 1.324

7.363 ± 1.222

7.278 ± 1.449

7.776 ± 2.231

6.2017 ± 3.0606

\(\underline{{\bf{6.1778}}\pm {\bf{3.0963}}}\)

6.6826 ± 3.1484

MAE

5.635 ± 1.061

5.212 ± 0.646

5.263 ± 0.757

5.757 ± 1.341

\(\underline{{\bf{3.8575}}\pm {\bf{1.5044}}}\)

3.9204 ± 1.6349

4.2596 ± 1.7478

R2

0.338 ± 0.214

0.367 ± 0.290

0.342 ± 0.457

0.562 ± 0.200

0.6909 ± 0.2418

\(\underline{{\bf{0.6915}}\pm {\bf{0.2434}}}\)

0.6476 ± 0.2678

Spearman

0.740 ± 0.055

0.725 ± 0.117

0.692 ± 0.155

0.743 ± 0.096

\(\underline{{\bf{0.8413}}\pm {\bf{0.1270}}}\)

0.8389 ± 0.1378

0.8044 ± 0.1501

SMAPE

21.3% ± 5.7%

20.4% ± 4.2%

20.7% ± 5.2%

25.2% ± 5.9%

\(\underline{{\bf{17.85}} \% \pm {\bf{4.71}} \% }\)

18.01% ± 4.89%

19.39% ± 6.14%

Hardness (GPa) SRJT

RMSE

0.371 ± 0.096

0.291 ± 0.041

0.300 ± 0.031

0.331 ± 0.064

\(\underline{{\bf{0.1677}}\pm {\bf{0.0465}}}\)

0.1706 ± 0.0514

0.1851 ± 0.0554

MAE

0.280 ± 0.068

0.216 ± 0.027

0.238 ± 0.026

0.230 ± 0.041

\(\underline{{\bf{0.1284}}\pm {\bf{0.0367}}}\)

0.1305 ± 0.0441

0.1366 ± 0.0411

R2

0.514 ± 0.144

0.696 ± 0.050

0.665 ± 0.096

0.561 ± 0.166

\(\underline{{\bf{0.8710}}\pm {\bf{0.1144}}}\)

0.8614 ± 0.1326

0.8371 ± 0.1530

Spearman

0.781 ± 0.041

0.831 ± 0.012

0.794 ± 0.086

0.759 ± 0.078

0.8956 ± 0.0888

\(\underline{{\bf{0.8988}}\pm {\bf{0.0874}}}\)

0.8846 ± 0.0931

SMAPE

10.6% ± 2.3%

8.3% ± 0.8%

9.3% ± 1.2%

8.7% ± 1.3%

\(\underline{{\bf{5.29}} \% \pm {\bf{1.48}} \% }\)

5.35% ± 1.79%

5.57% ± 1.65%

Modulus (GPa) SRJT

RMSE

19.724 ± 3.744

16.760 ± 1.407

16.616 ± 2.231

18.810 ± 2.034

\(\underline{{\bf{12.5840}}\pm {\bf{4.0755}}}\)

12.7843 ± 4.0212

13.1442 ± 2.6331

MAE

14.734 ± 2.604

13.085 ± 0.972

13.036 ± 1.861

14.438 ± 1.283

\(\underline{{\bf{9.5227}}\pm {\bf{3.3418}}}\)

9.5866 ± 3.4949

10.0754 ± 2.3725

R2

-0.031 ± 0.427

0.283 ± 0.096

0.300 ± 0.116

0.124 ± 0.037

0.5064 ± 0.5120

0.4960 ± 0.5135

\(\underline{{\bf{0.5173}}\pm {\bf{0.3643}}}\)

Spearman

0.588 ± 0.118

0.613 ± 0.060

0.658 ± 0.078

0.478 ± 0.048

\(\underline{{\bf{0.7525}}\pm {\bf{0.2478}}}\)

0.7505 ± 0.2624

0.7336 ± 0.2209

SMAPE

7.5% ± 1.3%

6.7% ± 0.4%

6.8% ± 0.9%

7.4% ± 0.6%

\(\underline{{\bf{4.91}} \% \pm {\bf{1.74}} \% }\)

4.93% ± 1.82%

5.20% ± 1.26%

Avg HDYN/HQS

RMSE

0.053 ± 0.004

0.047 ± 0.004

0.042 ± 0.008

0.041 ± 0.005

0.0262 ± 0.0122

\(\underline{{\bf{0.0252}}\pm {\bf{0.0123}}}\)

0.0290 ± 0.0114

MAE

0.045 ± 0.002

0.038 ± 0.004

0.034 ± 0.007

0.033 ± 0.006

0.0206 ± 0.0095

\(\underline{{\bf{0.0199}}\pm {\bf{0.0092}}}\)

0.0232 ± 0.0089

R2

−1.375 ± 1.100

−0.760 ± 0.353

−0.357 ± 0.196

0.043 ± 0.105

0.3371 ± 0.6079

\(\underline{{\bf{0.3772}}\pm {\bf{0.5885}}}\)

0.3023 ± 0.4916

Spearman

0.267 ± 0.260

0.227 ± 0.309

0.227 ± 0.283

0.414 ± 0.145

0.4561 ± 0.5550

\(\underline{{\bf{0.4585}}\pm {\bf{0.5749}}}\)

0.3726 ± 0.5455

SMAPE

3.9% ± 0.2%

3.3% ± 0.4%

2.9% ± 0.6%

2.9% ± 0.5%

1.80% ± 0.82%

\(\underline{{\bf{1.74}} \% \pm {\bf{0.80}} \% }\)

2.03% ± 0.78%

Depth of penetration (mm) FEM

RMSE

0.179 ± 0.034

0.183 ± 0.054

0.154 ± 0.024

0.151 ± 0.039

\(\underline{{\bf{0.1096}}\pm {\bf{0.0367}}}\)

0.1098 ± 0.0399

0.1261 ± 0.0394

MAE

0.126 ± 0.024

0.121 ± 0.026

0.110 ± 0.013

0.113 ± 0.033

0.0800 ± 0.0240

\(\underline{{\bf{0.0788}}\pm {\bf{0.0280}}}\)

0.0954 ± 0.0287

R2

0.460 ± 0.273

0.526 ± 0.188

0.632 ± 0.167

0.587 ± 0.161

\(\underline{{\bf{0.6981}}\pm {\bf{0.2381}}}\)

0.6264 ± 0.2331

0.5967 ± 0.3180

Spearman

0.670 ± 0.167

0.678 ± 0.083

0.6770 ± 0.2993

0.742 ± 0.114

\(\underline{{\bf{0.8649}}\pm {\bf{0.0619}}}\)

0.8563 ± 0.1350

0.8358 ± 0.1374

SMAPE

4.4% ± 0.8%

4.2% ± 1.0%

3.8% ± 0.5%

3.9% ± 1.1%

2.77% ± 0.93%

\(\underline{{\bf{2.73}} \% \pm {\bf{1.03}} \% }\)

3.28% ± 1.05%