Modeling Magnetic Transition Temperature of Rare-Earth Transition Metal-Based Double Perovskite Ceramics for Cryogenic Refrigeration Applications Using Intelligent Computational Methods
Abstract
1. Introduction
2. Mathematical Formulation of the Intelligent Algorithms Implemented
2.1. Support Vector Regression Mathematical Background
2.2. Genetic Algorithm Theory and Description
2.3. Extreme Learning Machine Theory and Description
3. Computational Methodology and Approaches
3.1. E2TMO6 Rare-Earth Transition Metal-Based Double Perovskite Ceramic Samples Acquisition and Description
3.2. Computational Development of Hybrid Genetic Algorithm and Support Vector Regression
3.3. Computational Development of ELM-Based Model for Magnetic Transition Temperature Prediction in E2TMO6 System of Ceramics
4. Results and Discussion
4.1. Convergence of GEN-SVR Parameters at Different Chromosome Sizes
4.2. Generalization Strength Comparison Between Developed Models Using Different Metrics
4.3. Computed Predictions of SE-ELM, GEN-SVR, and SM-ELM Models and Comparison with the Measured Magnetic Transition Temperatures
4.4. Dependence of Magnetic Transition Temperature on Applied Magnetic Field Using the SE-ELM Predictive Model
5. Conclusions
Funding
Data Availability Statement
Acknowledgments
Conflicts of Interest
Appendix A
k | Biases | Output Weight | Magnetic Field Weight | Rare-Earth Weight | Transition Metal Weight | Any Metal Weight |
---|---|---|---|---|---|---|
1 | 0.7225130617 | 1.2427827374 | 0.1148810873 | −0.3055849693 | −0.4482516227 | −0.3552237815 |
2 | 0.6849300618 | −1.2434195665 | 0.5617067582 | 0.9311123273 | −0.6659882046 | −0.3743215913 |
3 | 0.1373017009 | −0.9468194910 | −0.7209497397 | 0.6701749553 | 0.1015746494 | 0.5451241719 |
4 | 0.3599920346 | −0.8540009678 | 0.9754589655 | 0.6552374364 | 0.6345787698 | 0.7214028533 |
5 | 0.7907223920 | −1.7162123937 | 0.5423318793 | 0.3483318196 | −0.6136811884 | 0.9418167148 |
6 | 0.5750831796 | −0.1828785703 | −0.0829940562 | −0.9131051465 | −0.9170666451 | −0.1276684793 |
7 | 0.7662664029 | −0.1886235946 | −0.3753636682 | 0.5743482016 | 0.5065171562 | 0.7923093773 |
8 | 0.3746464268 | −2.0139427494 | 0.3628797001 | 0.8929218978 | 0.9170336391 | 0.9389560284 |
9 | 0.3788092433 | 1.5966158208 | 0.5453768478 | −0.9265712958 | 0.6234800595 | 0.5056407731 |
10 | 0.2536340812 | −2.4636479977 | −0.5706482370 | −0.4988611117 | 0.0145844402 | −0.2040967660 |
11 | 0.0851201958 | −1.9694387409 | 0.2371332802 | −0.8610782845 | 0.7489166993 | −0.4903197092 |
12 | 0.5709101967 | 0.9993483906 | −0.0777118592 | 0.2285536523 | 0.3535786575 | −0.2811934590 |
13 | 0.0961277755 | −0.8102393115 | 0.6526073931 | −0.5661174172 | −0.0803850368 | −0.2983802274 |
14 | 0.9619358872 | 0.0098323448 | 0.4677085810 | 0.6354738468 | −0.1657509879 | −0.9582320391 |
15 | 0.4707914879 | −0.9885653735 | 0.9540538818 | 0.6839049860 | 0.8879276198 | 0.7799287663 |
16 | 0.9907834343 | 0.1539170883 | −0.7996574998 | 0.8301807255 | −0.7243864790 | 0.2819266159 |
17 | 0.6678873888 | −1.1401651798 | −0.3234151686 | −0.8743317604 | 0.6950044196 | −0.3578771669 |
18 | 0.2376634038 | −2.5425554676 | −0.2767898283 | −0.9726729576 | 0.7707882551 | −0.6746939395 |
19 | 0.8323018721 | 1.3888443024 | −0.4830145061 | 0.3207876845 | 0.2593221792 | 0.4063154274 |
20 | 0.8194160401 | −0.3749088644 | 0.0116857832 | 0.0526458289 | 0.9834831412 | −0.4806278364 |
21 | 0.0971327487 | 2.2993310034 | −0.7055143547 | −0.5006812624 | −0.0324958504 | −0.0889978758 |
22 | 0.7127082668 | −0.6617365338 | −0.1490523076 | −0.7719032061 | −0.3058343469 | 0.9576661155 |
23 | 0.1340897564 | 1.5865820472 | 0.2140103342 | −0.1862000203 | −0.6480876838 | 0.1971435022 |
24 | 0.0908430111 | 0.1061975617 | 0.2004428755 | 0.5649868639 | 0.1986894682 | 0.8913778924 |
25 | 0.1763236663 | 1.4488020633 | 0.1456248639 | 0.6091153115 | −0.9922893515 | 0.7853739345 |
26 | 0.3369806420 | 0.0170290950 | −0.9517564978 | −0.6518093938 | 0.7986093796 | 0.0718938964 |
27 | 0.3993371707 | −1.1988152483 | 0.1083380068 | 0.3303917366 | 0.9758626664 | 0.1973593519 |
28 | 0.8156553780 | −1.2920576406 | −0.9893664948 | 0.4878220974 | 0.5938554601 | −0.0111496625 |
29 | 0.8616999802 | 0.8958653101 | −0.8782046097 | −0.8199701610 | 0.7457298903 | −0.3736714070 |
30 | 0.2959886774 | 1.8906007808 | −0.7107386440 | −0.6714505524 | 0.5855811204 | 0.2868872656 |
31 | 0.8027759212 | −0.3547897695 | −0.1767079645 | −0.0553678761 | 0.0374571552 | −0.6203722170 |
32 | 0.4577495535 | −1.2054657890 | −0.1451153480 | 0.2015119160 | −0.8250790407 | 0.1787523728 |
33 | 0.5550028032 | −0.6662053968 | −0.9912670099 | −0.6222089491 | 0.1131391622 | −0.4404763277 |
34 | 0.6299283784 | −0.8174409202 | −0.3841335379 | 0.8541546066 | −0.3748884618 | 0.4725273257 |
35 | 0.5235554725 | 0.7602517861 | −0.3363700556 | 0.2798180761 | −0.7516126422 | 0.8595361434 |
36 | 0.0424642718 | 1.6497831412 | −0.7371612765 | −0.5793845617 | 0.8957253281 | −0.3219466027 |
37 | 0.0915139834 | 0.3941299563 | 0.9751241704 | −0.8004130391 | −0.8693519293 | −0.0690114476 |
38 | 0.4281027425 | 0.1463530322 | 0.5421960181 | 0.9561890553 | 0.8029879394 | 0.1257953417 |
39 | 0.5275663121 | −1.0685898382 | −0.4911547089 | 0.9272725271 | 0.8448522109 | 0.0587709823 |
40 | 0.1603795738 | 0.9283264592 | 0.1849563719 | −0.3475082642 | 0.3965291135 | 0.3069042295 |
41 | 0.4305470222 | 0.4131597378 | −0.5995995543 | 0.7244229740 | −0.8321645532 | 0.9202493589 |
42 | 0.2068522910 | 0.2696125287 | −0.3370570009 | −0.4464422039 | −0.2413942285 | −0.4854087816 |
43 | 0.4290686318 | 1.1591516256 | 0.7986277631 | 0.3216888986 | 0.2778271091 | −0.4802365031 |
44 | 0.4770785345 | −0.9223261932 | 0.6241152260 | −0.7863318218 | −0.5401958250 | 0.3819927897 |
45 | 0.4159877211 | −0.5150763561 | 0.6444774872 | 0.2565413758 | 0.6034543746 | 0.0052674018 |
46 | 0.1784944762 | −0.8020380120 | −0.2233216232 | 0.1447972871 | 0.8271586672 | −0.6615341834 |
47 | 0.1457612105 | 0.3408460247 | 0.0216856667 | −0.5782159655 | −0.6303544444 | −0.6247065050 |
48 | 0.4023157185 | 1.6104171411 | −0.7846329653 | −0.4301464485 | 0.5429205965 | −0.5299353098 |
49 | 0.6686333493 | 0.6502204882 | 0.7275401732 | −0.1850086859 | −0.5542099123 | 0.5078068078 |
50 | 0.0332940348 | −0.1509337001 | 0.4192856973 | 0.6203744041 | −0.9066815615 | 0.6009994086 |
51 | 0.5225941328 | 1.1298387384 | 0.6785805165 | 0.5402834724 | −0.4368491298 | −0.3836899471 |
52 | 0.3953559593 | 0.5474267870 | −0.8001374564 | 0.4103617836 | 0.9035865611 | 0.1532144541 |
53 | 0.7039555868 | 0.4711073221 | −0.4430033100 | 0.8014009430 | 0.8652038311 | −0.5053491936 |
k | Biases | Output Weight | Magnetic Field Weight | Rare-Earth Weight | Transition Metal Weight | Any Metal Weight |
---|---|---|---|---|---|---|
1 | 0.495103 | 28.212456 | −0.479931 | 0.101244 | −0.964970 | 0.929804 |
2 | 0.920545 | −10,466,701.398800 | −0.995693 | −0.802332 | −0.937378 | 0.796610 |
3 | 0.550515 | −545,220.272782 | 0.414773 | −0.483924 | −0.059765 | −0.484183 |
4 | 0.328303 | −1,933,091,513.243830 | 0.992638 | 0.115491 | 0.983123 | −0.207775 |
5 | 0.851456 | −1,930,103,987.452320 | −0.472958 | 0.753577 | −0.760417 | 0.475394 |
6 | 0.489914 | −1,929,438,129.932470 | 0.641215 | 0.631100 | −0.459157 | 0.917257 |
7 | 0.193632 | −1,929,820,803.582070 | −0.191521 | 0.532867 | 0.518408 | 0.609534 |
8 | 0.509486 | 10,591.377512 | 0.981898 | −0.650621 | 0.070397 | −0.790181 |
9 | 0.482300 | −41,322,937,065.435100 | −0.933548 | −0.107854 | 0.199715 | −0.307776 |
10 | 0.927984 | −5,547,088,752.814350 | 0.539871 | 0.902726 | −0.790297 | −0.841089 |
11 | 0.923499 | 6.321526 | −0.499757 | −0.480659 | −0.217418 | −0.734662 |
12 | 0.236260 | 11,588.533673 | −0.288171 | −0.041518 | 0.221237 | −0.024587 |
13 | 0.516165 | 11,397,360.963912 | 0.517438 | 0.783815 | −0.822271 | −0.810507 |
14 | 0.862271 | −1,924,164,276.895450 | −0.438467 | 0.981014 | −0.141672 | −0.635907 |
15 | 0.903994 | −697,562.494739 | 0.243786 | −0.856419 | 0.403565 | 0.977871 |
16 | 0.071308 | −1,929,801,307.588630 | 0.522076 | 0.620839 | −0.272625 | 0.421814 |
17 | 0.498830 | −0.000016 | 0.404737 | −0.443972 | −0.184315 | −0.419679 |
18 | 0.954388 | −1,929,801,307.589110 | 0.219814 | 0.590575 | 0.496875 | 0.701237 |
19 | 0.262015 | −1,929,801,307.589120 | −0.852440 | 0.846436 | −0.187352 | 0.430702 |
20 | 0.881432 | −1,408,516.015777 | −0.456174 | 0.606453 | −0.663270 | 0.102670 |
21 | 0.990681 | −1,929,801,307.589120 | −0.334289 | 0.504171 | 0.138129 | 0.352268 |
22 | 0.003085 | −1,929,801,307.589120 | −0.222435 | 0.446963 | 0.927462 | 0.161070 |
23 | 0.546374 | −1,929,801,307.589120 | 0.575198 | 0.573692 | 0.760851 | −0.637897 |
24 | 0.213634 | −1,929,801,307.589120 | −0.563331 | −0.285640 | 0.359468 | 0.799460 |
25 | 0.174239 | 0.000000 | −0.046091 | −0.827885 | −0.630638 | 0.562965 |
26 | 0.641141 | −21,629,467.263229 | 0.053857 | −0.277972 | 0.988591 | −0.321922 |
27 | 0.119745 | −35,206,996.791366 | 0.787812 | −0.627138 | 0.788610 | −0.526778 |
28 | 0.199852 | −50.706935 | 0.505250 | 0.049929 | −0.671512 | 0.644013 |
29 | 0.410902 | 1.502535 | −0.944841 | −0.306882 | −0.806581 | 0.548474 |
30 | 0.447623 | 0.819332 | 0.631349 | 0.366138 | −0.584833 | −0.657705 |
31 | 0.689286 | 0.000000 | 0.405988 | −0.856346 | 0.352394 | −0.250506 |
32 | 0.772646 | −895,325,993.235744 | 0.053254 | 0.007612 | −0.338845 | 0.799921 |
33 | 0.209038 | −1,929,801,307.589120 | 0.914569 | 0.790592 | −0.339547 | 0.188207 |
34 | 0.595910 | −1,929,801,307.589120 | −0.875421 | 0.703366 | −0.195368 | −0.008984 |
35 | 0.738968 | −0.000009 | −0.991784 | −0.960065 | 0.925270 | −0.468429 |
36 | 0.590238 | 0.000000 | 0.520370 | −0.603955 | −0.341003 | −0.313557 |
37 | 0.950994 | −0.301471 | 0.927915 | −0.438877 | −0.296218 | 0.817231 |
38 | 0.108907 | 7,832,794.611025 | 0.643694 | 0.347742 | 0.006215 | −0.802348 |
39 | 0.967554 | −1,929,801,307.589120 | 0.062504 | 0.991729 | 0.960567 | 0.426048 |
40 | 0.528661 | −1,929,801,307.589120 | −0.827708 | −0.447798 | 0.637542 | 0.932061 |
41 | 0.034482 | 45,519,425,253.511100 | 0.392963 | 0.921239 | −0.814654 | −0.048659 |
42 | 0.066202 | −149,341.524572 | −0.304685 | −0.002606 | −0.273450 | 0.136545 |
43 | 0.041879 | 0.000000 | 0.371048 | 0.190920 | −0.762302 | −0.430662 |
44 | 0.302667 | −12.302782 | 0.560988 | 0.232797 | 0.677680 | −0.872136 |
45 | 0.340062 | −89,457,091.390429 | 0.492860 | 0.795073 | −0.921520 | 0.157988 |
46 | 0.408860 | −1,929,801,307.589120 | 0.431660 | 0.165800 | 0.587793 | 0.419660 |
47 | 0.038721 | −7,847,081,357.644930 | 0.215880 | −0.741172 | 0.871389 | 0.568222 |
48 | 0.883392 | −1,929,801,307.589120 | 0.609993 | 0.432708 | 0.726688 | 0.079436 |
49 | 0.227191 | −55,056,957.589018 | 0.039576 | −0.540297 | 0.901206 | −0.499696 |
50 | 0.621551 | 208.011219 | −0.668210 | −0.060474 | 0.980425 | −0.677664 |
51 | 0.702999 | −277,631,017,251.814000 | 0.194818 | 0.113756 | 0.268420 | −0.863770 |
52 | 0.981236 | −1,929,801,307.589120 | 0.651614 | 0.092626 | 0.152455 | 0.683079 |
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Magnetic Transition Temperature (K) | H (T) | E | T | M | |
---|---|---|---|---|---|
Mean | 6.47 | 5.96 | 104.67 | 77.76 | 77.91 |
Maximum | 12.20 | 7.00 | 107.80 | 94.00 | 94.00 |
Standard deviation | 2.86 | 1.02 | 1.75 | 7.17 | 5.83 |
Minimum | 2.00 | 5.00 | 102.00 | 68.00 | 67.50 |
Correlation coefficient | 1.00 | −0.39 | −0.14 | −0.12 | 0.60 |
Hyperparameter | Optimum Value |
---|---|
Penalty factor | 7970.066 |
Chromosome size | 100 |
Epsilon | 0.154694 |
Option | 1 |
Function | Gaussian |
hyper-plane | E-7 |
Training | Testing Testing Testing | |||
---|---|---|---|---|
CC | CC | RMSE | MAE | |
SE-ELM | 100.00 | 96.92 | 0.92 | 0.74 |
GEN-SVR | 99.95 | 92.84 | 0.98 | 0.87 |
SM-ELM | 94.66 | 71.16 | 1.96 | 1.52 |
Superiority of SE-ELM over GEN-SVR | 0.05 | 4.21 | 6.33 | 15.67 |
Superiority of SE-ELM over SM-ELM | 5.34 | 26.58 | 53.27 | 51.57 |
Superiority of GEN-SVR over SM-ELM | 5.30 | 23.35 | 50.11 | 42.58 |
Compound | MTT (K) | SE-ELM | SE-ELM Residual | SM-ELM | SM-ELM Residual | GEN-SVR | GEN-SVR Residual |
---|---|---|---|---|---|---|---|
Ho2CrMnO6 | 6.1 [61] | 6.1 | 0.0 | 6.1 | 0.0 | 6.1 | 0.0 |
Gd2ZnMnO6 | 6.4 [62] | 6.4 | 0.0 | 6.4 | 0.0 | 6.2 | 0.2 |
Er2FeCrO6 | 11.7 [63] | 11.7 | 0.0 | 11.7 | 0.0 | 11.5 | 0.2 |
Er2CuMnO6 | 3.6 [64] | 3.6 | 0.0 | 5.3 | 1.7 | 3.8 | 0.2 |
Er2CoMnO6 | 7.5 [29] | 7.5 | 0.0 | 6.5 | 1.0 | 7.3 | 0.2 |
Ho2NiMnO6 | 6.0 [19] | 6.0 | 0.0 | 5.5 | 0.5 | 5.4 | 0.6 |
Dy2FeAlO6 | 7.8 [65] | 7.8 | 0.0 | 7.8 | 0.0 | 7.6 | 0.2 |
Dy2NiMnO6 | 6.0 [19] | 6.0 | 0.0 | 5.6 | 0.4 | 5.9 | 0.1 |
Dy2CoMnO6 | 5.0 [29] | 5.0 | 0.0 | 5.4 | 0.4 | 5.2 | 0.2 |
Gd2FeAlO6 | 2.0 [65] | 2.0 | 0.0 | 2.0 | 0.0 | 2.2 | 0.2 |
Ho2ZnMnO6 | 6.8 [62] | 6.8 | 0.0 | 6.8 | 0.0 | 8.4 | 1.6 |
Er2NiMnO6 | 5.0 [19] | 5.0 | 0.0 | 6.3 | 1.3 | 5.2 | 0.2 |
Gd2CuMnO6 | 7.5 [64] | 7.5 | 0.0 | 7.4 | 0.1 | 7.3 | 0.2 |
Er2NiMnO6 | 5.0 [19] | 5.0 | 0.0 | 5.7 | 0.7 | 5.2 | 0.2 |
Ho2FeCoO6 | 4.0 [66] | 4.0 | 0.0 | 3.9 | 0.1 | 4.2 | 0.2 |
Er2FeCoO6 | 2.7 [67] | 2.7 | 0.0 | 2.7 | 0.0 | 2.9 | 0.2 |
Er2CrMnO6 | 5.2 [61] | 5.2 | 0.0 | 5.2 | 0.0 | 6.1 | 0.9 |
Ho2FeAlO6 | 2.0 [65] | 2.0 | 0.0 | 2.0 | 0.0 | 2.2 | 0.2 |
Dy2CuMnO6 | 12.1 [64] | 12.1 | 0.0 | 13.5 | 1.4 | 11.1 | 1.0 |
Dy2NiMnO6 | 6.0 [19] | 6.0 | 0.0 | 5.5 | 0.5 | 6.2 | 0.2 |
Gd2FeCoO6 | 4.9 [67] | 4.9 | 0.0 | 5.0 | 0.1 | 5.1 | 0.2 |
Ho2CuMnO6 | 12.2 [64] | 12.2 | 0.0 | 9.2 | 3.0 | 12.0 | 0.2 |
Tm2FeCrO6 | 10.5 [63] | 10.6 | 0.1 | 12.2 | 1.7 | 10.3 | 0.2 |
Ho2CoMnO6 | 8.0 [29] | 6.8 | 1.2 | 6.4 | 1.6 | 7.8 | 0.2 |
Gd2FeCoO6 | 4.9 [67] | 4.3 | 0.6 | 5.5 | 0.6 | 5.1 | 0.2 |
Dy2ZnMnO6 | 10.4 [62] | 11.9 | 1.5 | 6.7 | 3.7 | 10.2 | 0.2 |
Ho2NiMnO6 | 5.5 [19] | 5.3 | 0.2 | 5.4 | 0.1 | 5.7 | 0.2 |
Mean absolute error (K) | 0.1 | 0.7 | 0.3 |
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Ibn Shamsah, S.M. Modeling Magnetic Transition Temperature of Rare-Earth Transition Metal-Based Double Perovskite Ceramics for Cryogenic Refrigeration Applications Using Intelligent Computational Methods. Materials 2025, 18, 4594. https://doi.org/10.3390/ma18194594
Ibn Shamsah SM. Modeling Magnetic Transition Temperature of Rare-Earth Transition Metal-Based Double Perovskite Ceramics for Cryogenic Refrigeration Applications Using Intelligent Computational Methods. Materials. 2025; 18(19):4594. https://doi.org/10.3390/ma18194594
Chicago/Turabian StyleIbn Shamsah, Sami M. 2025. "Modeling Magnetic Transition Temperature of Rare-Earth Transition Metal-Based Double Perovskite Ceramics for Cryogenic Refrigeration Applications Using Intelligent Computational Methods" Materials 18, no. 19: 4594. https://doi.org/10.3390/ma18194594
APA StyleIbn Shamsah, S. M. (2025). Modeling Magnetic Transition Temperature of Rare-Earth Transition Metal-Based Double Perovskite Ceramics for Cryogenic Refrigeration Applications Using Intelligent Computational Methods. Materials, 18(19), 4594. https://doi.org/10.3390/ma18194594