{"id":3522,"date":"2023-08-31T11:12:58","date_gmt":"2023-08-31T11:12:58","guid":{"rendered":"https:\/\/neuraldesigner.com\/learning\/superconductivity\/"},"modified":"2026-09-18T16:08:48","modified_gmt":"2026-09-18T14:08:48","slug":"superconductivity","status":"publish","type":"learning","link":"https:\/\/www.neuraldesigner.com\/learning\/examples\/superconductivity\/","title":{"rendered":"Model superconductors&#8217; critical temperature using machine learning"},"content":{"rendered":"<style data-nd-chart-migration=\"1\">\n.nd-chart-bundle{width:100%;max-width:760px;min-width:0;margin:24px auto;box-sizing:border-box;position:relative;overflow-x:auto;text-align:center;background:#fff;color:#30343b}\n.nd-chart-bundle .nd-chart-host{width:100%;height:440px;min-width:0;box-sizing:border-box}\n.nd-chart-bundle:not(.is-ready) .nd-chart-host{position:absolute;visibility:hidden}\n.nd-chart-bundle.is-ready .nd-chart-fallback{display:none!important}\n.nd-chart-bundle 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div:not(:last-child):after{content:\"\u2192\";position:absolute;right:-15px;top:50%;z-index:2;transform:translateY(-50%);color:#56a1c8;font-size:21px}\n.nds-metrics td:nth-child(2){text-align:right;font-variant-numeric:tabular-nums}\n.nds-file-link{display:inline-block;color:#2d799f!important}\n.nds-file-link code{background:#dceef8!important;color:#24789f!important;font-weight:700}\n.nds-file-link:hover code{text-decoration:underline}\n@media(max-width:900px){.nds-flow{grid-template-columns:1fr 1fr}.nds-flow div:after{display:none}}\n@media(max-width:620px){.nds-flow{grid-template-columns:1fr}}\n<\/style>\n<div class=\"nds\" data-science-profile=\"physical-chemical\">\n<div class=\"nds-wrap\">\n<section class=\"nds-executive\">\n<h2>Estimate superconducting critical temperature from composition-derived descriptors<\/h2>\n<p>This regression benchmark maps 81 engineered physicochemical descriptors to critical temperature. On 4,252 held-out rows, the final 81\u201313\u20131 neural network reaches R\u00b2 = 0.817 and RMSE = 14.88 K. It supports comparative screening of represented materials; it does not determine whether an arbitrary compound is superconducting.<\/p>\n<div class=\"nds-kpis\">\n<div class=\"nds-kpi\"><strong>0.817<\/strong><span>testing R\u00b2<\/span><\/div>\n<div class=\"nds-kpi\"><strong>14.88 K<\/strong><span>testing RMSE<\/span><\/div>\n<div class=\"nds-kpi\"><strong>81<\/strong><span>engineered descriptors<\/span><\/div>\n<div class=\"nds-kpi\"><strong>4,252<\/strong><span>held-out testing rows<\/span><\/div>\n<\/div>\n<div class=\"nds-actions\"><a href=\"#6-validation\">Review scientific validation<\/a><a href=\"https:\/\/www.neuraldesigner.com\/wp-content\/uploads\/2023\/10\/superconductor.csv\">Download the data<\/a><\/div>\n<\/section>\n<ul class=\"nds-toc\">\n<li><a href=\"#1-scientific-objective\">Scientific objective<\/a><\/li>\n<li><a href=\"#2-data-provenance\">Data and provenance<\/a><\/li>\n<li><a href=\"#3-model\">Model<\/a><\/li>\n<li><a href=\"#4-training\">Training<\/a><\/li>\n<li><a href=\"#5-selection\">Selection<\/a><\/li>\n<li><a href=\"#6-validation\">Validation<\/a><\/li>\n<li><a href=\"#7-inference\">Inference<\/a><\/li>\n<li><a href=\"#8-validity\">Validity<\/a><\/li>\n<\/ul>\n<section id=\"1-scientific-objective\" class=\"nds-card\">\n<h2>1. Scientific objective<\/h2>\n<p>The objective is to estimate a known superconductor&#8217;s critical temperature (Tc) from descriptors derived from its elemental composition. The output can help materials-informatics teams rank represented candidates, compare modelling approaches and prioritize experiments. Because the training set contains known superconductors, the model is a temperature regressor\u2014not a classifier of superconducting behaviour.<\/p>\n<div class=\"nds-value-grid\">\n<article class=\"nds-value\">\n<h3>Candidate ranking<\/h3>\n<p>Compare predicted Tc values before allocating higher-cost synthesis and characterization work.<\/p>\n<\/article>\n<article class=\"nds-value\">\n<h3>Model benchmarking<\/h3>\n<p>Provide a reproducible neural baseline for composition-derived materials descriptors.<\/p>\n<\/article>\n<article class=\"nds-value\">\n<h3>Experimental planning<\/h3>\n<p>Use predictions with domain checks to support\u2014not replace\u2014physics-informed laboratory decisions.<\/p>\n<\/article>\n<\/div>\n<div class=\"nds-audience\"><span>Materials informatics<\/span><span>Superconductivity research<\/span><span>Computational materials<\/span><span>Experimental laboratories<\/span><\/div>\n<div class=\"nds-note\"><strong>Scope.<\/strong> The data-driven model estimates Tc for materials represented by the source collection. Crystal structure, pressure, processing and measurement conditions are not explicit inputs, so predictions require scientific review.<\/div>\n<\/section>\n<section id=\"2-data-provenance\" class=\"nds-card\">\n<h2>2. Data and provenance<\/h2>\n<p>The UCI Superconductivity data set contains 21,263 records with no missing values. The target is <code>critical_temp<\/code> in kelvin. A reproducible 60\/20\/20 row split assigns 12,759 records to training, 4,252 to model selection and 4,252 to testing.<\/p>\n<h3>Why there are two CSV files<\/h3>\n<div class=\"nds-table-scroll\">\n<table>\n<thead>\n<tr>\n<th>File<\/th>\n<th>Role<\/th>\n<th>Contents<\/th>\n<th>Use<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<th><a class=\"nds-file-link\" href=\"https:\/\/www.neuraldesigner.com\/wp-content\/uploads\/2023\/10\/superconductor.csv\" download=\"superconductor.csv\" aria-label=\"Download the primary superconductor modelling data\"><code>superconductor.csv<\/code><\/a><\/th>\n<td>Primary modelling data<\/td>\n<td>81 numeric descriptors plus <code>critical_temp<\/code><\/td>\n<td>Train, validate and run the published model<\/td>\n<\/tr>\n<tr>\n<th><a class=\"nds-file-link\" href=\"https:\/\/www.neuraldesigner.com\/wp-content\/uploads\/2023\/10\/chemical_compounds.csv\" download=\"chemical_compounds.csv\" aria-label=\"Download the auxiliary chemical compounds table\"><code>chemical_compounds.csv<\/code><\/a><\/th>\n<td>Auxiliary composition table<\/td>\n<td>86 elemental stoichiometry columns, <code>critical_temp<\/code> and formula<\/td>\n<td>Identify materials and connect model rows with chemical formulas<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>The files contain the same 21,263 row-aligned target observations but serve different purposes. Keeping both avoids pretending that the exported network accepts a chemical formula directly.<\/p>\n<h3>Descriptor design<\/h3>\n<p>The 81 inputs comprise <code>number_of_elements<\/code> plus ten statistics for each of eight elemental properties: atomic mass, first ionization energy, atomic radius, density, electron affinity, heat of fusion, thermal conductivity and valence. The statistics are mean, weighted mean, geometric mean, weighted geometric mean, entropy, weighted entropy, range, weighted range, standard deviation and weighted standard deviation.<\/p>\n<div class=\"nds-table-scroll\">\n<table>\n<thead>\n<tr>\n<th>Subset<\/th>\n<th>Rows<\/th>\n<th>Purpose<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<th>Training<\/th>\n<td>12,759<\/td>\n<td>Estimate network parameters<\/td>\n<\/tr>\n<tr>\n<th>Selection<\/th>\n<td>4,252<\/td>\n<td>Select hidden-layer size<\/td>\n<\/tr>\n<tr>\n<th>Testing<\/th>\n<td>4,252<\/td>\n<td>Final held-out evaluation<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"nds-figure-grid nds-data-figures\">\n<figure>\n<div class=\"nd-chart-bundle\" tabindex=\"0\" role=\"group\" aria-label=\"Interactive chart; scroll horizontally on narrow screens\" data-native-chart=\"nd-3522-00\">\n<div class=\"nd-chart-host\" id=\"nd-3522-00\" role=\"img\" aria-label=\"Interactive chart: Distribution of superconducting critical temperature\"><\/div>\n<p><img decoding=\"async\" class=\"nd-chart-fallback\" src=\"https:\/\/www.neuraldesigner.com\/wp-content\/uploads\/2026\/08\/superconductivity-critical-temperature-distribution-2026.png\" alt=\"Distribution of superconducting critical temperature\"><script type=\"application\/json\" id=\"nd-3522-00-data\">{\"config\":{\"displayModeBar\":true,\"displaylogo\":false,\"editable\":false,\"responsive\":true,\"scrollZoom\":true,\"toImageButtonOptions\":{\"filename\":\"neural-designer-chart\",\"format\":\"svg\"}},\"data\":[{\"marker\":{\"color\":\"#209fdf\"},\"name\":\"\",\"opacity\":1,\"showlegend\":false,\"type\":\"bar\",\"x\":[\"9.25\",\"27.8\",\"46.3\",\"64.8\",\"83.3\",\"102\",\"120\",\"139\",\"157\",\"176\"],\"y\":[48.599998474121094,15.800000190734863,7.900000095367432,7.699999809265137,13.800000190734863,4.300000190734863,1.399999976158142,0.5,0,0]}],\"layout\":{\"autosize\":true,\"font\":{\"color\":\"#30343b\",\"family\":\"Arial, sans-serif\",\"size\":14},\"hovermode\":\"closest\",\"margin\":{\"b\":68,\"l\":72,\"pad\":4,\"r\":36,\"t\":112},\"paper_bgcolor\":\"#ffffff\",\"plot_bgcolor\":\"#ffffff\",\"title\":{\"automargin\":true,\"pad\":{\"t\":48},\"text\":\"critical_temp histogram\",\"x\":0.5,\"xanchor\":\"center\",\"y\":1,\"yanchor\":\"top\",\"yref\":\"container\"},\"xaxis\":{\"automargin\":true,\"categoryarray\":[\"9.25\",\"27.8\",\"46.3\",\"64.8\",\"83.3\",\"102\",\"120\",\"139\",\"157\",\"176\"],\"categoryorder\":\"array\",\"gridcolor\":\"#d9dde3\",\"linecolor\":\"#9aa3ad\",\"title\":{\"text\":\"Bin center\"},\"type\":\"category\",\"zerolinecolor\":\"#aeb6c2\"},\"yaxis\":{\"automargin\":true,\"gridcolor\":\"#d9dde3\",\"linecolor\":\"#9aa3ad\",\"range\":[0,60.01],\"title\":{\"text\":\"% frequency\"},\"zerolinecolor\":\"#aeb6c2\"}}}<\/script><\/div><figcaption>The target distribution is broad and concentrated at lower temperatures, with relatively few high-Tc records.<\/figcaption><\/figure>\n<figure>\n<div class=\"nd-chart-bundle\" tabindex=\"0\" role=\"group\" aria-label=\"Interactive chart; scroll horizontally on narrow screens\" data-native-chart=\"nd-3522-01\">\n<div class=\"nd-chart-host\" id=\"nd-3522-01\" role=\"img\" aria-label=\"Interactive chart: Pearson correlations between descriptors and critical temperature\"><\/div>\n<p><img decoding=\"async\" class=\"nd-chart-fallback\" src=\"https:\/\/www.neuraldesigner.com\/wp-content\/uploads\/2026\/08\/superconductivity-correlations-2026.png\" alt=\"Pearson correlations between descriptors and critical temperature\"><script type=\"application\/json\" id=\"nd-3522-01-data\">{\"config\":{\"displayModeBar\":true,\"displaylogo\":false,\"editable\":false,\"responsive\":true,\"scrollZoom\":true,\"toImageButtonOptions\":{\"filename\":\"neural-designer-chart\",\"format\":\"svg\"}},\"data\":[{\"marker\":{\"color\":\"#209fdf\"},\"name\":\"\",\"opacity\":1,\"orientation\":\"h\",\"showlegend\":false,\"type\":\"bar\",\"x\":[-0.6690000295639038,-0.6589999794960022,-0.6589999794960022,0.6539999842643738,0.6539999842643738,0.6549999713897705,0.6710000038146973,0.6880000233650208,0.7070000171661377,0.7210000157356262],\"y\":[\"wtd_mean_Valence\",\"wtd_gmean_Density\",\"wtd_gmean_Valence\",\"std_ThermalConductivity\",\"entropy_Valence\",\"wtd_entropy_atomic_mass\",\"wtd_std_atomic_radius\",\"range_ThermalConductivity\",\"range_atomic_radius\",\"wtd_std_ThermalConductivity\"]}],\"layout\":{\"autosize\":true,\"font\":{\"color\":\"#30343b\",\"family\":\"Arial, sans-serif\",\"size\":14},\"hovermode\":\"closest\",\"margin\":{\"b\":68,\"l\":72,\"pad\":4,\"r\":36,\"t\":112},\"paper_bgcolor\":\"#ffffff\",\"plot_bgcolor\":\"#ffffff\",\"title\":{\"automargin\":true,\"pad\":{\"t\":48},\"text\":\"critical_temp Pearson correlations chart\",\"x\":0.5,\"xanchor\":\"center\",\"y\":1,\"yanchor\":\"top\",\"yref\":\"container\"},\"xaxis\":{\"automargin\":true,\"gridcolor\":\"#d9dde3\",\"linecolor\":\"#9aa3ad\",\"range\":[-1,1],\"title\":{\"text\":\"Correlation\"},\"zerolinecolor\":\"#aeb6c2\"},\"yaxis\":{\"automargin\":true,\"autorange\":\"reversed\",\"categoryarray\":[\"wtd_mean_Valence\",\"wtd_gmean_Density\",\"wtd_gmean_Valence\",\"std_ThermalConductivity\",\"entropy_Valence\",\"wtd_entropy_atomic_mass\",\"wtd_std_atomic_radius\",\"range_ThermalConductivity\",\"range_atomic_radius\",\"wtd_std_ThermalConductivity\"],\"categoryorder\":\"array\",\"gridcolor\":\"#d9dde3\",\"linecolor\":\"#9aa3ad\",\"type\":\"category\",\"zerolinecolor\":\"#aeb6c2\"}}}<\/script><\/div><figcaption>Univariate correlations provide orientation only; correlated engineered descriptors and nonlinear effects prevent causal interpretation.<\/figcaption><\/figure>\n<figure class=\"nds-figure-full\">\n<div class=\"nd-chart-bundle\" tabindex=\"0\" role=\"group\" aria-label=\"Interactive chart; scroll horizontally on narrow screens\" data-native-chart=\"nd-3522-02\">\n<div class=\"nd-chart-host\" id=\"nd-3522-02\" role=\"img\" aria-label=\"Interactive chart: Critical temperature against weighted standard deviation of thermal conductivity\"><\/div>\n<p><img decoding=\"async\" class=\"nd-chart-fallback\" src=\"https:\/\/www.neuraldesigner.com\/wp-content\/uploads\/2026\/08\/superconductivity-thermal-conductivity-scatter-2026.png\" alt=\"Critical temperature against weighted standard deviation of thermal conductivity\"><script type=\"application\/json\" 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nds-note--provenance\"><strong>Provenance.<\/strong> The benchmark is the UCI Superconductivity data set contributed with Kam Hamidieh&#8217;s 2018 study and derived from the SuperCon materials collection. The published downloads preserve UCI&#8217;s two complementary tables and are provided under CC BY 4.0.<\/div>\n<\/section>\n<section id=\"3-model\" class=\"nds-card\">\n<h2>3. Model<\/h2>\n<p>The modelling process starts with an 81\u20133\u20131 baseline. All 81 inputs use mean-and-standard-deviation scaling; a dense hidden layer with three tanh neurons learns nonlinear descriptor interactions, and an identity output neuron returns Tc before output unscaling. This initial network contains 250 trainable parameters and provides the reference architecture for training and neuron selection.<\/p>\n<div class=\"nds-note\"><strong>Input contract.<\/strong> The model requires all 81 descriptors in the exact order listed in the Python package. Formula parsing and descriptor generation are separate upstream responsibilities.<\/div>\n<figure class=\"nds-architecture-figure\"><img decoding=\"async\" class=\"nds-architecture\" src=\"https:\/\/www.neuraldesigner.com\/wp-content\/uploads\/2026\/08\/superconductivity-initial-network-architecture-2026.png\" alt=\"Initial neural network with 81 inputs, three hidden neurons and one critical-temperature output\"><figcaption>Initial 81\u20133\u20131 approximation network used for the baseline training run.<\/figcaption><\/figure>\n<\/section>\n<section id=\"4-training\" class=\"nds-card\">\n<h2>4. Training strategy<\/h2>\n<p>The network minimizes normalized squared error with L2 regularization of 0.01 using the Quasi-Newton method. The initial three-neuron network finishes with training error 0.192 and selection error 0.0668 after 99 iterations.<\/p>\n<figure class=\"nds-centered-figure nds-figure-medium\">\n<div class=\"nd-chart-bundle\" tabindex=\"0\" role=\"group\" aria-label=\"Interactive chart; scroll horizontally on narrow screens\" data-native-chart=\"nd-3522-04\">\n<div class=\"nd-chart-host\" id=\"nd-3522-04\" role=\"img\" aria-label=\"Interactive chart: Quasi-Newton training and selection error history\"><\/div>\n<p><img decoding=\"async\" class=\"nd-chart-fallback\" src=\"https:\/\/www.neuraldesigner.com\/wp-content\/uploads\/2026\/08\/superconductivity-training-history-2026.png\" alt=\"Quasi-Newton training and selection error history\"><script type=\"application\/json\" 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error\",\"opacity\":1,\"showlegend\":true,\"type\":\"scatter\",\"x\":[0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99],\"y\":[1.0700000524520874,1.059999942779541,0.7720000147819519,0.48399999737739563,0.4740000069141388,0.45899999141693115,0.4519999921321869,0.4269999861717224,0.4189999997615814,0.39800000190734863,0.375,0.3610000014305115,0.34200000762939453,0.33000001311302185,0.31200000643730164,0.29499998688697815,0.2849999964237213,0.27300000190734863,0.2639999985694885,0.2529999911785126,0.25,0.24799999594688416,0.2460000067949295,0.24199999868869781,0.23800000548362732,0.23499999940395355,0.23199999332427979,0.2290000021457672,0.22699999809265137,0.2240000069141388,0.2199999988079071,0.21799999475479126,0.2160000056028366,0.2150000035762787,0.21299999952316284,0.21199999749660492,0.20900000631809235,0.20800000429153442,0.2070000022649765,0.20600000023841858,0.20399999618530273,0.2029999941587448,0.2029999941587448,0.20200000703334808,0.20100000500679016,0.20000000298023224,0.1979999989271164,0.1979999989271164,0.19699999690055847,0.1979999989271164,0.1979999989271164,0.19699999690055847,0.19699999690055847,0.19699999690055847,0.19599999487400055,0.19599999487400055,0.19499999284744263,0.19499999284744263,0.19499999284744263,0.19499999284744263,0.1940000057220459,0.1940000057220459,0.1940000057220459,0.19300000369548798,0.19300000369548798,0.19300000369548798,0.19300000369548798,0.19300000369548798,0.19300000369548798,0.19300000369548798,0.19300000369548798,0.19300000369548798,0.19300000369548798,0.19300000369548798,0.19200000166893005,0.19200000166893005,0.19200000166893005,0.19200000166893005,0.19200000166893005,0.19200000166893005,0.19200000166893005,0.19200000166893005,0.19200000166893005,0.19200000166893005,0.19200000166893005,0.19200000166893005,0.19200000166893005,0.19200000166893005,0.19200000166893005,0.19200000166893005,0.19200000166893005,0.19200000166893005,0.19200000166893005,0.19200000166893005,0.19200000166893005,0.19200000166893005,0.19200000166893005,0.19200000166893005,0.19200000166893005,0.19200000166893005]},{\"connectgaps\":false,\"line\":{\"color\":\"#ed982a\",\"dash\":\"solid\",\"width\":2},\"mode\":\"lines\",\"name\":\"Selection 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sans-serif\",\"size\":14},\"hovermode\":\"closest\",\"legend\":{\"orientation\":\"h\",\"x\":0.5,\"xanchor\":\"center\",\"y\":-0.24,\"yanchor\":\"top\"},\"margin\":{\"b\":120,\"l\":72,\"pad\":4,\"r\":36,\"t\":112},\"paper_bgcolor\":\"#ffffff\",\"plot_bgcolor\":\"#ffffff\",\"title\":{\"automargin\":true,\"pad\":{\"t\":48},\"text\":\"Quasi-Newton method error history\",\"x\":0.5,\"xanchor\":\"center\",\"y\":1,\"yanchor\":\"top\",\"yref\":\"container\"},\"xaxis\":{\"automargin\":true,\"gridcolor\":\"#d9dde3\",\"linecolor\":\"#9aa3ad\",\"range\":[0,99],\"title\":{\"text\":\"Epoch\"},\"zerolinecolor\":\"#aeb6c2\"},\"yaxis\":{\"automargin\":true,\"gridcolor\":\"#d9dde3\",\"linecolor\":\"#9aa3ad\",\"range\":[0,1.2002000000000002],\"title\":{\"text\":\"NormalizedSquaredError\"},\"zerolinecolor\":\"#aeb6c2\"}}}<\/script><\/div><figcaption>The stored training history belongs to the three-neuron baseline used before hidden-layer selection.<\/figcaption><\/figure>\n<\/section>\n<section id=\"5-selection\" class=\"nds-card\">\n<h2>5. Model selection and baseline<\/h2>\n<p>Neuron selection evaluates one to 15 hidden neurons while keeping the data partition and training strategy fixed. The minimum selection error is 0.0616 at 13 neurons, approximately 7.4% below the three-neuron result; the final model therefore uses 13 hidden neurons.<\/p>\n<figure class=\"nds-centered-figure nds-figure-medium\">\n<div class=\"nd-chart-bundle\" tabindex=\"0\" role=\"group\" aria-label=\"Interactive chart; scroll horizontally on narrow screens\" data-native-chart=\"nd-3522-05\">\n<div class=\"nd-chart-host\" id=\"nd-3522-05\" role=\"img\" aria-label=\"Interactive chart: Neuron selection errors for one to 15 hidden neurons\"><\/div>\n<p><img decoding=\"async\" class=\"nd-chart-fallback\" src=\"https:\/\/www.neuraldesigner.com\/wp-content\/uploads\/2026\/08\/superconductivity-neuron-selection-2026.png\" alt=\"Neuron selection errors for one to 15 hidden neurons\"><script type=\"application\/json\" id=\"nd-3522-05-data\">{\"config\":{\"displayModeBar\":true,\"displaylogo\":false,\"editable\":false,\"responsive\":true,\"scrollZoom\":true,\"toImageButtonOptions\":{\"filename\":\"neural-designer-chart\",\"format\":\"svg\"}},\"data\":[{\"connectgaps\":false,\"line\":{\"color\":\"#529cc2\",\"dash\":\"solid\",\"width\":2},\"marker\":{\"color\":\"#529cc2\",\"size\":8,\"symbol\":\"circle\"},\"mode\":\"lines+markers\",\"name\":\"Training error\",\"opacity\":1,\"showlegend\":true,\"type\":\"scatter\",\"x\":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],\"y\":[0.20896600186824799,0.2022559940814972,0.19173599779605865,0.1881379932165146,0.18537800014019012,0.1828490048646927,0.1817770004272461,0.1784300059080124,0.17833800613880157,0.17811399698257446,0.178180992603302,0.17673400044441223,0.17680899798870087,0.17672599852085114,0.17675499618053436]},{\"connectgaps\":false,\"line\":{\"color\":\"#e6cd5f\",\"dash\":\"solid\",\"width\":2},\"marker\":{\"color\":\"#e6cd5f\",\"size\":8,\"symbol\":\"circle\"},\"mode\":\"lines+markers\",\"name\":\"Selection error\",\"opacity\":1,\"showlegend\":true,\"type\":\"scatter\",\"x\":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],\"y\":[0.07185029983520508,0.07039809972047806,0.06646890193223953,0.06510269641876221,0.06390219926834106,0.063263900578022,0.0628558024764061,0.0621596984565258,0.06203880161046982,0.06192630156874657,0.06192849949002266,0.061600301414728165,0.06155350059270859,0.06159140169620514,0.061574600636959076]}],\"layout\":{\"autosize\":true,\"font\":{\"color\":\"#30343b\",\"family\":\"Arial, sans-serif\",\"size\":14},\"hovermode\":\"closest\",\"legend\":{\"orientation\":\"h\",\"x\":0.5,\"xanchor\":\"center\",\"y\":-0.24,\"yanchor\":\"top\"},\"margin\":{\"b\":120,\"l\":72,\"pad\":4,\"r\":36,\"t\":112},\"paper_bgcolor\":\"#ffffff\",\"plot_bgcolor\":\"#ffffff\",\"title\":{\"automargin\":true,\"pad\":{\"t\":48},\"text\":\"Growing neurons training\/selection errors plot\",\"x\":0.5,\"xanchor\":\"center\",\"y\":1,\"yanchor\":\"top\",\"yref\":\"container\"},\"xaxis\":{\"automargin\":true,\"gridcolor\":\"#d9dde3\",\"linecolor\":\"#9aa3ad\",\"range\":[1,15],\"title\":{\"text\":\"Neurons number\"},\"zerolinecolor\":\"#aeb6c2\"},\"yaxis\":{\"automargin\":true,\"gridcolor\":\"#d9dde3\",\"linecolor\":\"#9aa3ad\",\"range\":[0,0.25005],\"title\":{\"text\":\"NormalizedSquaredError\"},\"zerolinecolor\":\"#aeb6c2\"}}}<\/script><\/div><figcaption>The selected 13-neuron model balances lower selection error with a still-compact architecture.<\/figcaption><\/figure>\n<h3>Selected architecture<\/h3>\n<p>After neuron selection, the hidden layer is expanded from three to 13 tanh neurons. The resulting 81\u201313\u20131 network contains 1,080 trainable parameters and is the model used for testing, deployment and the downloadable Python export.<\/p>\n<figure class=\"nds-architecture-figure nds-final-architecture\"><img decoding=\"async\" class=\"nds-architecture\" src=\"https:\/\/www.neuraldesigner.com\/wp-content\/uploads\/2026\/08\/superconductivity-final-network-architecture-2026.png\" alt=\"Final neural network with 81 inputs, 13 hidden neurons and one critical-temperature output\"><figcaption>Final 81\u201313\u20131 architecture obtained after neuron selection.<\/figcaption><\/figure>\n<div class=\"nds-table-scroll\">\n<table>\n<thead>\n<tr>\n<th>Reference<\/th>\n<th>Testing RMSE<\/th>\n<th>Interpretation<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<th>Training-mean baseline<\/th>\n<td>34.81 K<\/td>\n<td>Null predictor using one constant value<\/td>\n<\/tr>\n<tr>\n<th>Final neural network<\/th>\n<td>14.88 K<\/td>\n<td>57% lower RMSE than the null baseline<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<\/section>\n<section id=\"6-validation\" class=\"nds-card\">\n<h2>6. Scientific validation<\/h2>\n<p>Final metrics are calculated only on the 4,252 testing rows. R\u00b2 measures explained test variance; RMSE emphasizes large misses; MAE gives a more direct typical error; signed bias checks systematic over- or underprediction.<\/p>\n<div class=\"nds-table-scroll\">\n<table class=\"nds-metrics\">\n<thead>\n<tr>\n<th>Metric<\/th>\n<th>Testing result<\/th>\n<th>Reading<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<th>R\u00b2<\/th>\n<td>0.817<\/td>\n<td>81.7% of testing variance explained<\/td>\n<\/tr>\n<tr>\n<th>RMSE<\/th>\n<td>14.88 K<\/td>\n<td>Penalizes larger temperature errors<\/td>\n<\/tr>\n<tr>\n<th>MAE<\/th>\n<td>10.32 K<\/td>\n<td>Mean absolute testing error<\/td>\n<\/tr>\n<tr>\n<th>Mean signed error<\/th>\n<td>\u22120.28 K<\/td>\n<td>Small overall bias can hide regional bias<\/td>\n<\/tr>\n<tr>\n<th>Prediction-vs-observation slope<\/th>\n<td>0.801<\/td>\n<td>Compression toward the middle of the range<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<h3>Performance by observed temperature<\/h3>\n<div class=\"nds-table-scroll\">\n<table>\n<thead>\n<tr>\n<th>Observed Tc band<\/th>\n<th>Testing rows<\/th>\n<th>MAE<\/th>\n<th>Mean signed error<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<th>0\u201320 K<\/th>\n<td>2,130<\/td>\n<td>6.57 K<\/td>\n<td>+3.37 K<\/td>\n<\/tr>\n<tr>\n<th>20\u201377 K<\/th>\n<td>1,305<\/td>\n<td>12.24 K<\/td>\n<td>+3.76 K<\/td>\n<\/tr>\n<tr>\n<th>77\u2013120 K<\/th>\n<td>768<\/td>\n<td>16.69 K<\/td>\n<td>\u221215.84 K<\/td>\n<\/tr>\n<tr>\n<th>120\u2013186 K<\/th>\n<td>49<\/td>\n<td>22.61 K<\/td>\n<td>\u221222.61 K<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<div class=\"nds-figure-grid\">\n<figure class=\"nds-figure-full nds-gof\">\n<div class=\"nd-chart-bundle\" tabindex=\"0\" role=\"group\" aria-label=\"Interactive chart; scroll horizontally on narrow screens\" data-native-chart=\"nd-3522-07\">\n<div class=\"nd-chart-host\" id=\"nd-3522-07\" role=\"img\" aria-label=\"Interactive chart: Observed and predicted critical temperature on the testing subset\"><\/div>\n<p><img decoding=\"async\" class=\"nd-chart-fallback\" src=\"https:\/\/www.neuraldesigner.com\/wp-content\/uploads\/2026\/08\/superconductivity-goodness-of-fit-2026.png\" alt=\"Observed and predicted critical temperature on the testing subset\"><script type=\"application\/json\" id=\"nd-3522-07-data\">{\"config\":{\"displayModeBar\":true,\"displaylogo\":false,\"editable\":false,\"responsive\":true,\"scrollZoom\":true,\"toImageButtonOptions\":{\"filename\":\"neural-designer-chart\",\"format\":\"svg\"}},\"data\":[{\"connectgaps\":false,\"marker\":{\"color\":\"#209fdf\",\"size\":8,\"symbol\":\"circle\"},\"mode\":\"markers\",\"name\":\"Predicted 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critical_temp\",\"opacity\":1,\"showlegend\":false,\"type\":\"scatter\",\"x\":[0,200.05],\"y\":[0,200.05]}],\"layout\":{\"autosize\":true,\"font\":{\"color\":\"#30343b\",\"family\":\"Arial, sans-serif\",\"size\":14},\"hovermode\":\"closest\",\"margin\":{\"b\":68,\"l\":72,\"pad\":4,\"r\":36,\"t\":112},\"paper_bgcolor\":\"#ffffff\",\"plot_bgcolor\":\"#ffffff\",\"title\":{\"automargin\":true,\"pad\":{\"t\":48},\"text\":\"critical_temp goodness-of-fit chart\",\"x\":0.5,\"xanchor\":\"center\",\"y\":1,\"yanchor\":\"top\",\"yref\":\"container\"},\"xaxis\":{\"automargin\":true,\"gridcolor\":\"#d9dde3\",\"linecolor\":\"#9aa3ad\",\"range\":[-4.001,204.05100000000002],\"title\":{\"text\":\"Real critical_temp\"},\"zerolinecolor\":\"#aeb6c2\"},\"yaxis\":{\"automargin\":true,\"gridcolor\":\"#d9dde3\",\"linecolor\":\"#9aa3ad\",\"range\":[-4.001,204.05100000000002],\"title\":{\"text\":\"Predicted critical_temp\"},\"zerolinecolor\":\"#aeb6c2\"}}}<\/script><\/div><figcaption>Goodness-of-fit analysis from Neural Designer. Predictions compress the extremes, especially for high-Tc observations.<\/figcaption><\/figure>\n<\/div>\n<div class=\"nds-note\"><strong>Scientific interpretation.<\/strong> Aggregate bias is close to zero, but it is not uniform: the model overpredicts many low-temperature records and underpredicts the high-Tc region. Candidate ranking near the upper tail therefore needs temperature-band diagnostics and expert review, not R\u00b2 alone.<\/div>\n<\/section>\n<section id=\"7-inference\" class=\"nds-card\">\n<h2>7. Inference and reproducibility<\/h2>\n<p>A credible deployment begins with a formula or composition record, generates the same 81 descriptors, validates names, order and applicability ranges, calculates Tc and routes the estimate\u2014together with model version and warnings\u2014to scientific review.<\/p>\n<div class=\"nds-flow\">\n<div>Formula and composition<\/div>\n<div>81 descriptor calculation<\/div>\n<div>Schema and domain checks<\/div>\n<div>Critical-temperature estimate<\/div>\n<div>Experimental prioritization<\/div>\n<\/div>\n<h3>Three representative inference cases<\/h3>\n<p>Neural Designer&#8217;s output-data task was run on three descriptor vectors taken from the primary table. Their formulas were recovered from the row-aligned auxiliary composition table, and the same inputs were then evaluated with the exported Python model.<\/p>\n<div class=\"nds-table-scroll\">\n<table>\n<thead>\n<tr>\n<th>Material<\/th>\n<th>Formula<\/th>\n<th>Observed Tc<\/th>\n<th>Predicted Tc<\/th>\n<th>Signed error<\/th>\n<\/tr>\n<\/thead>\n<tbody>\n<tr>\n<th>Mercury<\/th>\n<td>Hg<\/td>\n<td>4.16 K<\/td>\n<td>4.83 K<\/td>\n<td>+0.67 K<\/td>\n<\/tr>\n<tr>\n<th>Magnesium diboride<\/th>\n<td>MgB<sub>2<\/sub><\/td>\n<td>39.00 K<\/td>\n<td>31.22 K<\/td>\n<td>\u22127.78 K<\/td>\n<\/tr>\n<tr>\n<th>YBCO<\/th>\n<td>YBa<sub>2<\/sub>Cu<sub>3<\/sub>O<sub>7<\/sub><\/td>\n<td>92.00 K<\/td>\n<td>73.77 K<\/td>\n<td>\u221218.23 K<\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<\/div>\n<p>These are transparent calculation examples, not an additional validation set. They make the operational pattern visible: the low-Tc case is close, while the high-Tc YBCO case is substantially underestimated. Multiple experimental records can share the same composition-derived vector but report different Tc values, because conditions absent from the descriptors can matter.<\/p>\n<h3>Reproduce the calculation<\/h3>\n<p>The Python package contains the exact final export, the ordered 81-input schema, the three deployment rows and their calculated outputs. The compact Neural Designer project excludes obsolete embedded data from an earlier configuration.<\/p>\n<pre><code>import pandas as pd\nfrom model import NeuralNetwork\n\ncases = pd.read_csv(\"deployment_cases.csv\", sep=\";\")\nmodel = NeuralNetwork()\nassert list(cases.columns) == model.input_names\npredicted_tc_K = model.calculate_batch_output(\n    cases.to_numpy(dtype=float)\n)[:, 0]<\/code><\/pre>\n<div class=\"nds-note nds-note--warning\"><strong>Deployment guard.<\/strong> The raw export can return a negative value for outlying inputs. Production integration must validate descriptor ranges and reject or flag non-physical output; a numerical result is not evidence that an arbitrary material will superconduct.<\/div>\n<div class=\"nds-downloads\"><a href=\"https:\/\/www.neuraldesigner.com\/wp-content\/uploads\/2026\/08\/superconductivity-python-model-2026.zip\">Download Python model (ZIP)<\/a><br \/>\n<a href=\"https:\/\/www.neuraldesigner.com\/wp-content\/uploads\/2026\/08\/superconductivity-neural-designer-project-2026-v2.zip\">Download Neural Designer project (ZIP)<\/a><br \/>\n<a href=\"https:\/\/www.neuraldesigner.com\/wp-content\/uploads\/2023\/10\/superconductor.csv\">Download model data (CSV)<\/a><br \/>\n<a href=\"https:\/\/www.neuraldesigner.com\/wp-content\/uploads\/2023\/10\/chemical_compounds.csv\">Download composition table (CSV)<\/a><\/div>\n<\/section>\n<section id=\"8-validity\" class=\"nds-card\">\n<h2>8. Validity, uncertainty and limitations<\/h2>\n<ul>\n<li><strong>Applicability is restricted.<\/strong> The source contains known superconductors; the model neither discovers superconductivity nor establishes that an arbitrary compound has a nonzero Tc.<\/li>\n<li><strong>Composition is incomplete physics.<\/strong> Crystal structure, pressure, synthesis route, stoichiometric uncertainty and measurement conditions are absent.<\/li>\n<li><strong>The row-random split is optimistic.<\/strong> About 33.4% of testing rows have an exact engineered descriptor vector in training. On unseen descriptor vectors, R\u00b2 falls to 0.801 and RMSE rises to 15.32 K.<\/li>\n<li><strong>Repeated formulas are not independent experiments.<\/strong> Formula-level or material-family grouping is preferable when estimating transfer to new chemistries.<\/li>\n<li><strong>High-Tc estimates are systematically compressed.<\/strong> The 77\u2013120 K and 120\u2013186 K bands show mean underprediction of 15.84 K and 22.61 K respectively.<\/li>\n<li><strong>Raw output requires a physical guard.<\/strong> The stored test predictions include 202 negative values; these must be flagged as non-physical and investigated rather than silently accepted.<\/li>\n<li><strong>No external material-family validation is shown.<\/strong> Before laboratory use, compare against grouped cross-validation, uncertainty estimates and truly external compounds.<\/li>\n<\/ul>\n<\/section>\n<section id=\"references\" class=\"nds-card\">\n<h2>References<\/h2>\n<ul>\n<li><a href=\"https:\/\/archive.ics.uci.edu\/dataset\/464\/superconductivty+data\">UCI Machine Learning Repository: Superconductivity Data<\/a> (DOI: <a href=\"https:\/\/doi.org\/10.24432\/C53P47\">10.24432\/C53P47<\/a>; CC BY 4.0).<\/li>\n<li>K. Hamidieh, <a href=\"https:\/\/www.sciencedirect.com\/science\/article\/pii\/S0927025618304877\">A data-driven statistical model for predicting the critical temperature of a superconductor<\/a>, Computational Materials Science 154 (2018), 346\u2013354.<\/li>\n<li><a href=\"https:\/\/mdr.nims.go.jp\/collections\/5712m6524\">National Institute for Materials Science: SuperCon data resources<\/a>.<\/li>\n<\/ul>\n<\/section>\n<\/div>\n<\/div>\n<p><script data-nd-chart-migration=\"1\" data-noptimize=\"1\" data-cfasync=\"false\" src=\"https:\/\/cdn.plot.ly\/plotly-basic-4.0.0.min.js\"><\/script><script data-nd-chart-migration=\"1\" data-noptimize=\"1\">(()=>{const start=()=>{if(!window.Plotly)return;document.querySelectorAll('.nd-chart-bundle').forEach(bundle=>{if(bundle.dataset.started)return;bundle.dataset.started='1';const host=bundle.querySelector('.nd-chart-host'),source=bundle.querySelector('script[type=\"application\/json\"]');if(!host||!source)return;const figure=JSON.parse(source.textContent);window.Plotly.newPlot(host,figure.data,figure.layout,figure.config).then(async()=>{const title=host.querySelector('.gtitle');if(title)host.style.minWidth=Math.ceil(title.getComputedTextLength()+48)+'px';await window.Plotly.Plots.resize(host);bundle.classList.add('is-ready');if(window.ResizeObserver){let timer;new ResizeObserver(()=>{clearTimeout(timer);timer=setTimeout(()=>window.Plotly.Plots.resize(host),80);}).observe(bundle);}}).catch(error=>{bundle.dataset.error=String(error);console.error(error);});});};const ready=()=>{if(window.Plotly){start();return;}let count=0;const timer=setInterval(()=>{if(window.Plotly){clearInterval(timer);start();}else if(++count>200)clearInterval(timer);},50);};if(document.readyState==='loading')document.addEventListener('DOMContentLoaded',ready,{once:true});else ready();})();<\/script><\/p>\n","protected":false},"author":11,"featured_media":1476,"template":"","categories":[29],"tags":[],"class_list":["post-3522","learning","type-learning","status-publish","has-post-thumbnail","hentry","category-examples"],"acf":[],"yoast_head":"<!-- This site is optimized with the Yoast SEO plugin v26.4 - https:\/\/yoast.com\/wordpress\/plugins\/seo\/ -->\n<title>Model superconductors&#039; 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