{"id":3461,"date":"2026-01-31T11:12:59","date_gmt":"2026-01-31T10:12:59","guid":{"rendered":"https:\/\/neuraldesigner.com\/learning\/airfoil-self-noise-prediction\/"},"modified":"2026-09-18T16:07:21","modified_gmt":"2026-09-18T14:07:21","slug":"airfoil-self-noise-prediction","status":"publish","type":"learning","link":"https:\/\/www.neuraldesigner.com\/learning\/examples\/airfoil-self-noise-prediction\/","title":{"rendered":"Predict airfoil self-noise 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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14px}.ndb-executive{padding:24px 20px}.ndb-executive h2{font-size:24px}.ndb-kpis,.ndb-figure-grid,.ndb-calculator-grid,.ndb-flow{grid-template-columns:1fr}.ndb-calculator{padding:20px 16px}.ndb-card table{font-size:13px}.ndb-card th,.ndb-card td{padding:9px 8px}}\n<\/style>\n\n<style>\n.ndb-video{display:flex;justify-content:center;margin:26px 0}.ndb-video iframe{width:min(760px,100%);aspect-ratio:16\/9;height:auto;border:0;border-radius:14px;box-shadow:0 12px 28px rgba(0,18,51,.12)}\n.ndb-architecture--wide{width:min(940px,100%)!important}\n<\/style>\n\n<div class=\"ndb\"><div class=\"ndb-wrap\">\n<section class=\"ndb-executive\">\n<h2>Predict airfoil self-noise for faster aeroacoustic design decisions<\/h2>\n<p>This neural-network surrogate estimates scaled sound pressure level from airfoil geometry and wind-tunnel operating conditions. It provides rapid noise estimates for screening and sensitivity studies before committing to higher-fidelity simulation or acoustic testing.<\/p>\n<div class=\"ndb-kpis\"><div class=\"ndb-kpi\"><strong>1,503<\/strong><span>NASA wind-tunnel observations<\/span><\/div><div class=\"ndb-kpi\"><strong>5<\/strong><span>aeroacoustic inputs<\/span><\/div><div class=\"ndb-kpi\"><strong>0.891<\/strong><span>testing determination<\/span><\/div><div class=\"ndb-kpi\"><strong>19<\/strong><span>selected hidden neurons<\/span><\/div><\/div>\n<div class=\"ndb-actions\"><a href=\"#7-model-deployment\">Review deployment<\/a><a href=\"https:\/\/www.neuraldesigner.com\/wp-content\/uploads\/2023\/10\/airfoil_self_noise.csv\">Download the data<\/a><\/div>\n<\/section>\n<div class=\"ndb-lead\"><p>Airfoil self-noise is relevant to aircraft, rotorcraft, drones, wind turbines, cooling fans and turbomachinery. A compact surrogate makes it possible to explore many operating points quickly, identify influential variables and focus expensive aeroacoustic work on the most promising designs.<\/p><\/div>\n<ul class=\"ndb-toc\"><li><a href=\"#1-industrial-challenge\">Industrial challenge<\/a><\/li><li><a href=\"#2-data-set\">Data set<\/a><\/li><li><a href=\"#3-model\">Model<\/a><\/li><li><a href=\"#4-training\">Training<\/a><\/li><li><a href=\"#5-selection\">Model selection<\/a><\/li><li><a href=\"#6-testing\">Testing<\/a><\/li><li><a href=\"#7-model-deployment\">Deployment<\/a><\/li><li><a href=\"#8-limitations\">Limitations<\/a><\/li><li><a href=\"#references\">References<\/a><\/li><\/ul>\n\n<section id=\"1-industrial-challenge\" class=\"ndb-card\">\n<h2>1. Industrial challenge<\/h2>\n<p>This is an <a href=\"https:\/\/www.neuraldesigner.com\/learning\/tutorials\/neural-networks-applications\/#Approximation\">approximation<\/a> problem. The model maps a measured frequency, angle of attack, chord length, free-stream velocity and suction-side displacement thickness to a continuous acoustic response: <code>scaled_sound_pressure_level<\/code>.<\/p>\n<div class=\"ndb-value-grid\"><div class=\"ndb-value\"><strong>Screen operating conditions<\/strong><span>Estimate which combinations of speed and incidence are associated with higher sound pressure levels.<\/span><\/div><div class=\"ndb-value\"><strong>Prioritize acoustic testing<\/strong><span>Use rapid predictions to select the cases that deserve wind-tunnel or high-fidelity analysis.<\/span><\/div><div class=\"ndb-value\"><strong>Support design trade-offs<\/strong><span>Explore noise sensitivity alongside aerodynamic, structural and performance requirements.<\/span><\/div><\/div>\n<p>Potential users include aeroacoustic engineers, aerodynamicists, rotor and propeller designers, wind-energy teams, fan and turbomachinery engineers, NVH specialists and technical programme managers.<\/p>\n<div class=\"ndb-audience\"><span>Aeroacoustics<\/span><span>Aerodynamic design<\/span><span>Rotors &amp; propellers<\/span><span>Wind energy<\/span><span>Fans &amp; turbomachinery<\/span><span>Noise engineering<\/span><\/div>\n<div class=\"ndb-note\"><strong>Model role.<\/strong> This example is a data-driven surrogate for engineering screening inside the represented test envelope. It is not an acoustic certification model and does not replace validated physics-based methods or testing.<\/div>\n<\/section>\n\n<section id=\"2-data-set\" class=\"ndb-card\">\n<h2>2. Data set<\/h2>\n<p>The <a href=\"https:\/\/archive.ics.uci.edu\/dataset\/291\/airfoil%2Bself%2Bnoise\">Airfoil Self-Noise dataset<\/a> contains 1,503 observations from aerodynamic and acoustic tests of NACA 0012 airfoil sections in an anechoic wind tunnel. It has no missing values and contains five inputs and one target.<\/p>\n<div class=\"ndb-downloads\"><a href=\"https:\/\/www.neuraldesigner.com\/wp-content\/uploads\/2023\/10\/airfoil_self_noise.csv\">Download dataset (CSV)<\/a><\/div>\n<table><thead><tr><th>Variable<\/th><th>Engineering meaning<\/th><th>Role<\/th><th>Range<\/th><\/tr><\/thead><tbody>\n<tr><td><code>frequency<\/code><\/td><td>One-third-octave-band centre frequency (Hz)<\/td><td>Input<\/td><td>200 to 20,000<\/td><\/tr>\n<tr><td><code>angle_of_attack<\/code><\/td><td>Airfoil angle of attack (deg)<\/td><td>Input<\/td><td>0.0 to 22.2<\/td><\/tr>\n<tr><td><code>chord_length<\/code><\/td><td>Airfoil chord length (m)<\/td><td>Input<\/td><td>0.0254 to 0.3048<\/td><\/tr>\n<tr><td><code>free_stream_velocity<\/code><\/td><td>Wind-tunnel free-stream velocity (m\/s)<\/td><td>Input<\/td><td>31.7 to 71.3<\/td><\/tr>\n<tr><td><code>suction_side_displacement_thickness<\/code><\/td><td>Suction-side boundary-layer displacement thickness (m)<\/td><td>Input<\/td><td>0.000401 to 0.058411<\/td><\/tr>\n<tr><td><code>scaled_sound_pressure_level<\/code><\/td><td>Scaled sound pressure level (dB)<\/td><td>Target<\/td><td>103.380 to 140.987<\/td><\/tr>\n<\/tbody><\/table>\n<p>The current project configuration assigns 903 observations to training, 300 to model selection and 300 to testing.<\/p>\n<div class=\"ndb-figure-grid\"><figure><div class=\"nd-chart-bundle\" tabindex=\"0\" role=\"group\" aria-label=\"Interactive chart; scroll horizontally on narrow screens\" data-native-chart=\"nd-3461-00\"><div class=\"nd-chart-host\" id=\"nd-3461-00\" role=\"img\" aria-label=\"Interactive chart: Distribution of scaled airfoil sound pressure level\"><\/div><img decoding=\"async\" class=\"nd-chart-fallback\" src=\"https:\/\/www.neuraldesigner.com\/wp-content\/uploads\/2026\/07\/airfoil-self-noise-distribution-2026.png\" alt=\"Distribution of scaled airfoil sound pressure level\"><script type=\"application\/json\" id=\"nd-3461-00-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\":[\"105\",\"109\",\"113\",\"117\",\"120\",\"124\",\"128\",\"132\",\"135\",\"139\"],\"y\":[0.699999988079071,2.9000000953674316,5.300000190734863,10.100000381469727,14.199999809265137,18.100000381469727,22.399999618530273,17.399999618530273,7.900000095367432,1.2000000476837158]}],\"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\":\"scaled_sound_pressure_level histogram\",\"x\":0.5,\"xanchor\":\"center\",\"y\":1,\"yanchor\":\"top\",\"yref\":\"container\"},\"xaxis\":{\"automargin\":true,\"categoryarray\":[\"105\",\"109\",\"113\",\"117\",\"120\",\"124\",\"128\",\"132\",\"135\",\"139\"],\"categoryorder\":\"array\",\"gridcolor\":\"#d9dde3\",\"linecolor\":\"#9aa3ad\",\"title\":{\"text\":\"Bin center\"},\"type\":\"category\",\"zerolinecolor\":\"#aeb6c2\"},\"yaxis\":{\"automargin\":true,\"gridcolor\":\"#d9dde3\",\"linecolor\":\"#9aa3ad\",\"range\":[0,40.01],\"title\":{\"text\":\"% frequency\"},\"zerolinecolor\":\"#aeb6c2\"}}}<\/script><\/div><figcaption><strong>Target distribution.<\/strong> The chart shows the acoustic response represented in the experimental dataset.<\/figcaption><\/figure><figure><div class=\"nd-chart-bundle\" tabindex=\"0\" role=\"group\" aria-label=\"Interactive chart; scroll horizontally on narrow screens\" data-native-chart=\"nd-3461-01\"><div class=\"nd-chart-host\" id=\"nd-3461-01\" role=\"img\" aria-label=\"Interactive chart: Correlations between airfoil inputs and sound pressure level\"><\/div><img decoding=\"async\" class=\"nd-chart-fallback\" src=\"https:\/\/www.neuraldesigner.com\/wp-content\/uploads\/2026\/07\/airfoil-self-noise-correlations-2026.png\" alt=\"Correlations between airfoil inputs and sound pressure level\"><script type=\"application\/json\" 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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\":[\"frequency\",\"suction_side_displacement_thickness\",\"chord_length\",\"angle_of_attack\",\"free_stream_velocity\"],\"categoryorder\":\"array\",\"gridcolor\":\"#d9dde3\",\"linecolor\":\"#9aa3ad\",\"type\":\"category\",\"zerolinecolor\":\"#aeb6c2\"}}}<\/script><\/div><figcaption><strong>Input\u2013target correlations.<\/strong> Linear correlation is useful for orientation, but it does not capture all aeroacoustic interactions.<\/figcaption><\/figure><\/div>\n<div class=\"nd-chart-bundle\" tabindex=\"0\" role=\"group\" aria-label=\"Interactive chart; scroll horizontally on narrow screens\" data-native-chart=\"nd-3461-02\"><div class=\"nd-chart-host\" id=\"nd-3461-02\" role=\"img\" aria-label=\"Interactive chart: Scatter plots of airfoil self-noise inputs and target\"><\/div><img decoding=\"async\" class=\"nd-chart-fallback\" src=\"https:\/\/www.neuraldesigner.com\/wp-content\/uploads\/2026\/07\/airfoil-self-noise-frequency-scatter-2026.png\" alt=\"Scatter plots of airfoil self-noise inputs and target\"><script type=\"application\/json\" 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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\":\"scaled_sound_pressure_level vs. frequency scatter chart\",\"x\":0.5,\"xanchor\":\"center\",\"y\":1,\"yanchor\":\"top\",\"yref\":\"container\"},\"xaxis\":{\"automargin\":true,\"gridcolor\":\"#d9dde3\",\"linecolor\":\"#9aa3ad\",\"range\":[0,20005],\"title\":{\"text\":\"frequency\"},\"zerolinecolor\":\"#aeb6c2\"},\"yaxis\":{\"automargin\":true,\"gridcolor\":\"#d9dde3\",\"linecolor\":\"#9aa3ad\",\"range\":[100,150.01],\"title\":{\"text\":\"scaled_sound_pressure_level\"},\"zerolinecolor\":\"#aeb6c2\"}}}<\/script><\/div>\n<div class=\"ndb-note ndb-note--warning\"><strong>Validation note.<\/strong> A random row split can place related frequency sweeps or operating conditions in both training and testing. For decision-grade validation, reserve complete test configurations, operating regimes or experimental series.<\/div>\n<\/section>\n\n\n\n<section id=\"3-model\" class=\"ndb-card\"><h2>3. Model<\/h2>\n<p>The initial network receives five mean-and-standard-deviation-scaled inputs, processes them with three tanh neurons and returns one linear acoustic output. This 5\u20133\u20131 baseline contains 22 trainable parameters and applies no output bounding.<\/p>\n<img decoding=\"async\" class=\"ndb-architecture ndb-architecture--wide\" src=\"https:\/\/www.neuraldesigner.com\/wp-content\/uploads\/2026\/07\/airfoil-self-noise-initial-network-2026.png\" alt=\"Updated initial airfoil self-noise network with five inputs, three hidden neurons and one output\"><\/section>\n\n\n\n<section id=\"4-training\" class=\"ndb-card\"><h2>4. Training strategy<\/h2>\n<p>The baseline 5\u20133\u20131 network is trained with adaptive moment estimation (Adam), a learning rate of 0.001, batches of 1,000 samples and L2 regularization with weight 0.001. Training ran on NVIDIA CUDA and stopped after 2,785 epochs when the maximum number of selection-error increases was reached. The recorded final training and selection errors are <strong>0.182 MSE<\/strong> and <strong>0.207 MSE<\/strong> in the scaled training space.<\/p>\n<div class=\"nd-chart-bundle\" tabindex=\"0\" role=\"group\" aria-label=\"Interactive chart; scroll horizontally on narrow screens\" data-native-chart=\"nd-3461-04\"><div class=\"nd-chart-host\" id=\"nd-3461-04\" role=\"img\" aria-label=\"Interactive chart: Adam training and selection error history for the initial airfoil model\"><\/div><img decoding=\"async\" class=\"nd-chart-fallback\" src=\"https:\/\/www.neuraldesigner.com\/wp-content\/uploads\/2026\/07\/airfoil-self-noise-adam-training-history-2026.png\" alt=\"Adam training and selection error history for the initial airfoil model\"><script type=\"application\/json\" 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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\":\"Adaptive moment estimation error history\",\"x\":0.5,\"xanchor\":\"center\",\"y\":1,\"yanchor\":\"top\",\"yref\":\"container\"},\"xaxis\":{\"automargin\":true,\"gridcolor\":\"#d9dde3\",\"linecolor\":\"#9aa3ad\",\"range\":[0,2784],\"title\":{\"text\":\"Epoch\"},\"zerolinecolor\":\"#aeb6c2\"},\"yaxis\":{\"automargin\":true,\"gridcolor\":\"#d9dde3\",\"linecolor\":\"#9aa3ad\",\"range\":[0,0.6001000000000001],\"title\":{\"text\":\"MeanSquaredError\"},\"zerolinecolor\":\"#aeb6c2\"}}}<\/script><\/div><\/section>\n\n\n\n<section id=\"5-selection\" class=\"ndb-card\"><h2>5. Model selection<\/h2>\n<p>Growing-neuron selection evaluates hidden-layer sizes up to 20 neurons, using Adam with the same 0.001 learning rate and three trials per candidate. The selected architecture uses <strong>19 hidden neurons<\/strong>, with reported optimum training and selection errors of <strong>0.0569 MSE<\/strong> and <strong>0.0565 MSE<\/strong>.<\/p>\n<div class=\"ndb-figure-grid\"><figure><div class=\"nd-chart-bundle\" tabindex=\"0\" role=\"group\" aria-label=\"Interactive chart; scroll horizontally on narrow screens\" data-native-chart=\"nd-3461-05\"><div class=\"nd-chart-host\" id=\"nd-3461-05\" role=\"img\" aria-label=\"Interactive chart: Adam hidden-neuron selection errors for the airfoil self-noise model\"><\/div><img decoding=\"async\" class=\"nd-chart-fallback\" src=\"https:\/\/www.neuraldesigner.com\/wp-content\/uploads\/2026\/07\/airfoil-self-noise-adam-selection-errors-2026.png\" alt=\"Adam hidden-neuron selection errors for the airfoil self-noise model\"><script type=\"application\/json\" id=\"nd-3461-05-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\":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],\"y\":[0.20389999449253082,0.16732899844646454,0.1588120013475418,0.15807600319385529,0.08916159719228745,0.08026350289583206,0.07779549807310104,0.07068630307912827,0.06479959934949875,0.06275749951601028,0.06055939942598343,0.060747601091861725,0.06097419932484627,0.0610169991850853,0.06089479848742485,0.05686270073056221,0.056870900094509125,0.05685630068182945,0.05685789883136749,0.05685390159487724]},{\"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\":[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20],\"y\":[0.24350899457931519,0.21043400466442108,0.20423200726509094,0.20419000089168549,0.1054219976067543,0.09454510360956192,0.08767840266227722,0.0800739973783493,0.07131429761648178,0.06758929789066315,0.06516490131616592,0.06452350318431854,0.06451869755983353,0.06444379687309265,0.06448650360107422,0.056742798537015915,0.05664220079779625,0.0566089004278183,0.05652429908514023,0.05663289874792099]}],\"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\":[1,20],\"title\":{\"text\":\"Neurons number\"},\"zerolinecolor\":\"#aeb6c2\"},\"yaxis\":{\"automargin\":true,\"gridcolor\":\"#d9dde3\",\"linecolor\":\"#9aa3ad\",\"range\":[0,0.25005],\"title\":{\"text\":\"MeanSquaredError\"},\"zerolinecolor\":\"#aeb6c2\"}}}<\/script><\/div><figcaption><strong>Architecture search.<\/strong> Selection error determines the chosen capacity across candidates of up to 20 neurons.<\/figcaption><\/figure><figure><img decoding=\"async\" src=\"https:\/\/www.neuraldesigner.com\/wp-content\/uploads\/2026\/07\/airfoil-self-noise-adam-final-network-2026.png\" alt=\"Final airfoil network with nineteen hidden neurons\"><figcaption><strong>Final model.<\/strong> The 5\u201319\u20131 architecture is used for testing, the browser calculator and Python deployment.<\/figcaption><\/figure><\/div><\/section>\n\n\n\n<section id=\"6-testing\" class=\"ndb-card\"><h2>6. Testing analysis<\/h2>\n<p>The exported Python model was independently recalculated on the 300 observations marked as testing. Neural Designer reports a goodness-of-fit determination of <strong>0.8908<\/strong> (squared prediction\u2013target correlation). MAE and RMSE express the error directly in decibels.<\/p>\n<table><thead><tr><th>Testing observations<\/th><th>Determination<\/th><th>MAE<\/th><th>RMSE<\/th><th>95th-percentile absolute error<\/th><th>Maximum absolute error<\/th><\/tr><\/thead><tbody><tr><td>300<\/td><td>0.8908<\/td><td>1.767 dB<\/td><td>2.408 dB<\/td><td>4.973 dB<\/td><td>10.975 dB<\/td><\/tr><\/tbody><\/table>\n<p>Compared with the previous quasi-Newton run, Adam substantially improves agreement across the represented acoustic range. The remaining largest errors still justify treating the model as an engineering screening surrogate rather than a certification tool.<\/p>\n<div class=\"nd-chart-bundle\" tabindex=\"0\" role=\"group\" aria-label=\"Interactive chart; scroll horizontally on narrow screens\" data-native-chart=\"nd-3461-07\"><div class=\"nd-chart-host\" id=\"nd-3461-07\" role=\"img\" aria-label=\"Interactive chart: Adam model predictions versus measured scaled sound pressure level on the testing set\"><\/div><img decoding=\"async\" class=\"nd-chart-fallback\" src=\"https:\/\/www.neuraldesigner.com\/wp-content\/uploads\/2026\/07\/airfoil-self-noise-adam-goodness-of-fit-2026.png\" alt=\"Adam model predictions versus measured scaled sound pressure level on the testing set\"><script type=\"application\/json\" 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scaled_sound_pressure_level\"},\"zerolinecolor\":\"#aeb6c2\"}}}<\/script><\/div>\n<div class=\"ndb-note\"><strong>Metric note.<\/strong> The conventional residual R\u00b2, calculated as 1 \u2212 SSE\/SST on the same rows, is 0.8897. Reporting error in dB alongside determination avoids relying on a single dimensionless score.<\/div><\/section>\n\n\n\n<section id=\"7-model-deployment\" class=\"ndb-card\"><h2>7. Model deployment<\/h2>\n<p>The surrogate can support acoustic trade studies and test planning when inputs are checked against both their ranges and the combinations represented by the experiment.<\/p>\n<div class=\"ndb-flow\"><div>Airfoil and operating condition<\/div><div>Range and configuration checks<\/div><div>Aeroacoustic surrogate<\/div><div>Noise estimate and engineering review<\/div><\/div>\n\n<div class=\"ndb-calculator\" id=\"airfoil-calculator\">\n<h3>Try the aeroacoustic surrogate<\/h3>\n<p>Enter a condition inside the experimental ranges. The calculation runs locally using the exact weights and preprocessing of the exported Python model.<\/p>\n<form id=\"airfoil-calculator-form\"><div class=\"ndb-calculator-grid\">\n<div class=\"ndb-field\"><label for=\"af-f\">Frequency (Hz)<\/label><input id=\"af-f\" type=\"number\" min=\"200\" max=\"20000\" step=\"any\" value=\"3150\"><small>200 to 20,000<\/small><\/div>\n<div class=\"ndb-field\"><label for=\"af-a\">Angle of attack (deg)<\/label><input id=\"af-a\" type=\"number\" min=\"0\" max=\"22.2\" step=\"any\" value=\"9.5\"><small>0.0 to 22.2<\/small><\/div>\n<div class=\"ndb-field\"><label for=\"af-c\">Chord length (m)<\/label><input id=\"af-c\" type=\"number\" min=\"0.0254\" max=\"0.3048\" step=\"any\" value=\"0.0254\"><small>0.0254 to 0.3048<\/small><\/div>\n<div class=\"ndb-field\"><label for=\"af-v\">Free-stream velocity (m\/s)<\/label><input id=\"af-v\" type=\"number\" min=\"31.7\" max=\"71.3\" step=\"any\" value=\"39.6\"><small>31.7 to 71.3<\/small><\/div>\n<div class=\"ndb-field\"><label for=\"af-d\">Displacement thickness (m)<\/label><input id=\"af-d\" type=\"number\" min=\"0.000400682\" max=\"0.0584113\" step=\"any\" value=\"0.0044982\"><small>0.000401 to 0.058411<\/small><\/div>\n<\/div><div class=\"ndb-calc-actions\"><button type=\"submit\">Calculate sound pressure<\/button><button type=\"button\" id=\"airfoil-reset\">Reset example<\/button><\/div><\/form>\n<div class=\"ndb-output\" aria-live=\"polite\"><span>Predicted scaled sound pressure level<\/span><strong id=\"af-output\">\u2014<\/strong><\/div>\n<p class=\"ndb-calc-status\" id=\"airfoil-status\">Demonstration surrogate \u2014 not a certified control, protection or acoustic-certification system.<\/p>\n<\/div>\n<script>\n(function(){\nconst form=document.getElementById(\"airfoil-calculator-form\");if(!form)return;\nconst ids=[\"af-f\",\"af-a\",\"af-c\",\"af-v\",\"af-d\"],defaults=[3150,9.5,.0254,39.6,.0044982];\nfunction calculate(){\nconst fields=ids.map(id=>document.getElementById(id)),x=fields.map(f=>Number(f.value));\nif(x.some(v=>!Number.isFinite(v))){document.getElementById(\"airfoil-status\").textContent=\"Enter a valid number in every field.\";return}\nlet outside=false;fields.forEach((f,i)=>{const bad=x[i]<Number(f.min)||x[i]>Number(f.max);f.setAttribute(\"aria-invalid\",bad?\"true\":\"false\");outside=outside||bad});\nconst frequency=x[0],angle_of_attack=x[1],chord_length=x[2],free_stream_velocity=x[3],suction_side_displacement_thickness=x[4];\nconst scaled_frequency = (frequency-2886.379883)\/3152.570068;\nconst scaled_angle_of_attack = (angle_of_attack-6.782299995)\/5.918129921;\nconst scaled_chord_length = (chord_length-0.1365479976)\/0.09354069829;\nconst scaled_free_stream_velocity = (free_stream_velocity-50.8606987)\/15.57279968;\nconst scaled_suction_side_displacement_thickness = (suction_side_displacement_thickness-0.01113990042)\/0.01315020025;\nconst dense_layer_1_output_0 = Math.tanh( -0.9604790807 + (-1.310786486*scaled_frequency) + (1.263388753*scaled_angle_of_attack) + (-0.1101618335*scaled_chord_length) + (0.1569218487*scaled_free_stream_velocity) + (-0.001130336314*scaled_suction_side_displacement_thickness) );\nconst dense_layer_1_output_1 = Math.tanh( 0.124989666 + (0.9621517062*scaled_frequency) + (0.6966676712*scaled_angle_of_attack) + (0.3965918422*scaled_chord_length) + (-0.2541334927*scaled_free_stream_velocity) + (-0.4751221836*scaled_suction_side_displacement_thickness) );\nconst dense_layer_1_output_2 = Math.tanh( 0.1647574008 + (-0.1035819426*scaled_frequency) + (0.1730318218*scaled_angle_of_attack) + (0.1441895217*scaled_chord_length) + (-0.151797682*scaled_free_stream_velocity) + (-0.02187636681*scaled_suction_side_displacement_thickness) );\nconst dense_layer_1_output_3 = Math.tanh( 0.5744281411 + (-0.1213378385*scaled_frequency) + (0.8309887052*scaled_angle_of_attack) + (-0.3976085186*scaled_chord_length) + (-0.04205592722*scaled_free_stream_velocity) + (-0.3463357985*scaled_suction_side_displacement_thickness) );\nconst dense_layer_1_output_4 = Math.tanh( -0.1344737113 + (-0.03355310112*scaled_frequency) + (-0.09533648193*scaled_angle_of_attack) + (0.08076989651*scaled_chord_length) + (0.1628522426*scaled_free_stream_velocity) + (0.03944684193*scaled_suction_side_displacement_thickness) );\nconst dense_layer_1_output_5 = Math.tanh( 0.4099901319 + (0.6195399761*scaled_frequency) + (-0.3383997381*scaled_angle_of_attack) + (0.1113912761*scaled_chord_length) + (0.2950099111*scaled_free_stream_velocity) + (0.1290292889*scaled_suction_side_displacement_thickness) );\nconst dense_layer_1_output_6 = Math.tanh( 0.6288656592 + (-0.1553706676*scaled_frequency) + (0.2719035745*scaled_angle_of_attack) + (0.43776685*scaled_chord_length) + (-0.09311809391*scaled_free_stream_velocity) + (-0.06990084797*scaled_suction_side_displacement_thickness) );\nconst dense_layer_1_output_7 = Math.tanh( 0.4638142288 + (1.23462522*scaled_frequency) + (0.6685234308*scaled_angle_of_attack) + (0.2542767823*scaled_chord_length) + (-0.184428066*scaled_free_stream_velocity) + (0.9800260067*scaled_suction_side_displacement_thickness) );\nconst dense_layer_1_output_8 = Math.tanh( 2.172823191 + (2.348927498*scaled_frequency) + (0.1242148131*scaled_angle_of_attack) + (0.5253459215*scaled_chord_length) + (-0.1378752887*scaled_free_stream_velocity) + (0.5607316494*scaled_suction_side_displacement_thickness) );\nconst dense_layer_1_output_9 = Math.tanh( 0.6019958854 + (-0.3946969509*scaled_frequency) + (-0.5452480912*scaled_angle_of_attack) + (1.198436379*scaled_chord_length) + (0.1205590963*scaled_free_stream_velocity) + (0.3654276431*scaled_suction_side_displacement_thickness) );\nconst dense_layer_1_output_10 = Math.tanh( 1.263896346 + (2.705039978*scaled_frequency) + (-0.2224836648*scaled_angle_of_attack) + (0.1572250724*scaled_chord_length) + (0.006727933884*scaled_free_stream_velocity) + (0.5076295137*scaled_suction_side_displacement_thickness) );\nconst dense_layer_1_output_11 = Math.tanh( 0.1729726046 + (-0.1026126966*scaled_frequency) + (0.1765154451*scaled_angle_of_attack) + (0.1509633511*scaled_chord_length) + (-0.1461141407*scaled_free_stream_velocity) + (-0.02212109044*scaled_suction_side_displacement_thickness) );\nconst dense_layer_1_output_12 = Math.tanh( -0.4648934603 + (-0.7258431315*scaled_frequency) + (-0.4270290434*scaled_angle_of_attack) + (0.6284768581*scaled_chord_length) + (0.04178196564*scaled_free_stream_velocity) + (0.1321227103*scaled_suction_side_displacement_thickness) );\nconst dense_layer_1_output_13 = Math.tanh( -0.01286269631 + (-0.3812293112*scaled_frequency) + (-0.8895414472*scaled_angle_of_attack) + (-0.7039260864*scaled_chord_length) + (0.183639586*scaled_free_stream_velocity) + (0.1123513356*scaled_suction_side_displacement_thickness) );\nconst dense_layer_1_output_14 = Math.tanh( 0.8827536702 + (-0.1830069572*scaled_frequency) + (0.2051407546*scaled_angle_of_attack) + (0.4543457627*scaled_chord_length) + (0.2320585549*scaled_free_stream_velocity) + (0.06668931991*scaled_suction_side_displacement_thickness) );\nconst dense_layer_1_output_15 = Math.tanh( 0.7195150256 + (0.753831923*scaled_frequency) + (0.1019449383*scaled_angle_of_attack) + (1.598658919*scaled_chord_length) + (-0.01267192047*scaled_free_stream_velocity) + (0.1549862027*scaled_suction_side_displacement_thickness) );\nconst dense_layer_1_output_16 = Math.tanh( 0.0008453064947 + (-0.06175771728*scaled_frequency) + (-0.05051426217*scaled_angle_of_attack) + (0.08238738775*scaled_chord_length) + (0.0238923952*scaled_free_stream_velocity) + (0.08052080125*scaled_suction_side_displacement_thickness) );\nconst dense_layer_1_output_17 = Math.tanh( -0.0006235885085 + (0.05380342528*scaled_frequency) + (-0.09766328335*scaled_angle_of_attack) + (0.0920253545*scaled_chord_length) + (-0.04489162937*scaled_free_stream_velocity) + (0.03699810058*scaled_suction_side_displacement_thickness) );\nconst dense_layer_1_output_18 = Math.tanh( 2.319896885e-05 + (0.03222060204*scaled_frequency) + (0.02603542432*scaled_angle_of_attack) + (-0.006000517402*scaled_chord_length) + (-0.03213799372*scaled_free_stream_velocity) + (0.06178404391*scaled_suction_side_displacement_thickness) );\nconst approximation_layer_output_0 = ( -0.3187816143 + (1.109885335*dense_layer_1_output_0) + (1.337363124*dense_layer_1_output_1) + (-0.4112312198*dense_layer_1_output_2) + (0.8016780019*dense_layer_1_output_3) + (0.2391184717*dense_layer_1_output_4) + (0.876634419*dense_layer_1_output_5) + (-0.7909697294*dense_layer_1_output_6) + (-0.7052898407*dense_layer_1_output_7) + (2.902696133*dense_layer_1_output_8) + (0.9276366234*dense_layer_1_output_9) + (-1.319486976*dense_layer_1_output_10) + (-0.4155459404*dense_layer_1_output_11) + (1.129656076*dense_layer_1_output_12) + (0.9416408539*dense_layer_1_output_13) + (-1.015141726*dense_layer_1_output_14) + (-1.211652875*dense_layer_1_output_15) + (-0.01843057387*dense_layer_1_output_16) + (0.09675262123*dense_layer_1_output_17) + (0.005167976022*dense_layer_1_output_18) );\nconst unscaling_layer_output_0=approximation_layer_output_0*6.898656845+124.8359451;\ndocument.getElementById(\"af-output\").textContent=unscaling_layer_output_0.toFixed(2)+\" dB\";\ndocument.getElementById(\"airfoil-status\").textContent=outside?\"Warning: one or more inputs are outside the training range; this prediction should not be trusted.\":\"All inputs are inside the individual ranges represented in the dataset.\";\n}\nform.addEventListener(\"submit\",e=>{e.preventDefault();calculate()});\ndocument.getElementById(\"airfoil-reset\").addEventListener(\"click\",()=>{ids.forEach((id,i)=>document.getElementById(id).value=defaults[i]);calculate()});\ncalculate();\n})();\n<\/script>\n<h3>Directional response<\/h3>\n<p>The updated chart varies frequency around a reference condition of 5.4\u00b0 angle of attack, 0.1524 m chord, 55.5 m\/s free-stream velocity and 0.0043329 m displacement thickness.<\/p>\n<div class=\"nd-chart-bundle\" tabindex=\"0\" role=\"group\" aria-label=\"Interactive chart; scroll horizontally on narrow screens\" data-native-chart=\"nd-3461-08\"><div class=\"nd-chart-host\" id=\"nd-3461-08\" role=\"img\" aria-label=\"Interactive chart: Adam model scaled sound pressure level as a function of frequency\"><\/div><img decoding=\"async\" class=\"nd-chart-fallback\" src=\"https:\/\/www.neuraldesigner.com\/wp-content\/uploads\/2026\/07\/airfoil-self-noise-adam-directional-output-2026.png\" alt=\"Adam model scaled sound pressure level as a function of frequency\"><script type=\"application\/json\" id=\"nd-3461-08-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\",\"shape\":\"spline\",\"width\":2},\"mode\":\"lines\",\"name\":\"scaled_sound_pressure_level\",\"opacity\":1,\"showlegend\":false,\"type\":\"scatter\",\"x\":[0,400,800,1200,1600,2000,2400,2800,3200,3600,4000,4400,4800,5200,5600,6000,6400,6800,7200,7600,8000,8400,8800,9200,9600,10000,10400,10800,11200,11600,12000,12400,12800,13200,13600,14000,14400,14800,15200,15600,16000,16400,16800,17200,17600,18000,18400,18800,19200,19600,20000],\"y\":[125,128,131,132,130,128,126,124,122,121,120,119,118,117,117,116,116,115,115,114,114,113,113,112,112,112,112,111,111,111,111,110,110,110,110,110,110,110,110,110,110,110,110,110,110,110,110,110,111,111,111]},{\"connectgaps\":false,\"marker\":{\"color\":\"#3c3c3c\",\"size\":8,\"symbol\":\"circle\"},\"mode\":\"markers\",\"name\":\"scaled_sound_pressure_level\",\"opacity\":1,\"showlegend\":false,\"type\":\"scatter\",\"x\":[2500],\"y\":[125.45600128173828]}],\"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\":\"scaled_sound_pressure_level - frequency directional output\",\"x\":0.5,\"xanchor\":\"center\",\"y\":1,\"yanchor\":\"top\",\"yref\":\"container\"},\"xaxis\":{\"automargin\":true,\"gridcolor\":\"#d9dde3\",\"linecolor\":\"#9aa3ad\",\"range\":[0,20005],\"title\":{\"text\":\"frequency\"},\"zerolinecolor\":\"#aeb6c2\"},\"yaxis\":{\"automargin\":true,\"gridcolor\":\"#d9dde3\",\"linecolor\":\"#9aa3ad\",\"range\":[100,150.01],\"title\":{\"text\":\"scaled_sound_pressure_level\"},\"zerolinecolor\":\"#aeb6c2\"}}}<\/script><\/div>\n<h3>Download and reproduce<\/h3>\n<p>The deployment package contains the current Python model. The project package contains the Neural Designer project and the matching CSV dataset.<\/p>\n<div class=\"ndb-downloads\"><a href=\"https:\/\/www.neuraldesigner.com\/wp-content\/uploads\/2026\/07\/airfoil-self-noise-python-model-adam-2026.zip\">Download Python model (ZIP)<\/a><a href=\"https:\/\/www.neuraldesigner.com\/wp-content\/uploads\/2026\/07\/airfoil-self-noise-project-adam-2026.zip\">Download Neural Designer project (ZIP)<\/a><a href=\"https:\/\/www.neuraldesigner.com\/wp-content\/uploads\/2023\/10\/airfoil_self_noise.csv\">Download dataset (CSV)<\/a><\/div>\n<h3>Tutorial video<\/h3><div class=\"ndb-video\"><iframe src=\"https:\/\/www.youtube.com\/embed\/GwZ0Ko70eHA\" title=\"Airfoil self-noise prediction tutorial\" allowfullscreen><\/iframe><\/div>\n<\/section>\n\n<section id=\"8-limitations\" class=\"ndb-card\">\n<h2>8. Scope and limitations<\/h2>\n<ul><li>The data describes NACA 0012 airfoil sections and the ranges shown above, not arbitrary modern blade or airfoil geometries.<\/li><li>The model is a steady experimental surrogate and does not explicitly resolve individual self-noise mechanisms or transient flow.<\/li><li>The target is scaled sound pressure level under the source experiment&#8217;s measurement definition; it is not a complete certification or community-noise metric.<\/li><li>Predictions outside the observed ranges, or for a different tunnel, observer arrangement, surface condition or Reynolds\/Mach regime, require new validation.<\/li><li>A random row split can overstate performance on genuinely new configurations; grouped or campaign-level holdouts are preferable.<\/li><li>Engineering decisions should be confirmed with appropriate aeroacoustic analysis, testing and applicable standards.<\/li><\/ul>\n<\/section>\n\n<section id=\"references\" class=\"ndb-card\"><h2>References<\/h2><ul><li>Brooks, T. F., Pope, D. S., &amp; Marcolini, M. A. (1989). <a href=\"https:\/\/ntrs.nasa.gov\/citations\/19890016302\">Airfoil Self-Noise and Prediction<\/a>. NASA Reference Publication 1218.<\/li><li>Brooks, T., Pope, D., &amp; Marcolini, M. (1989). <a href=\"https:\/\/archive.ics.uci.edu\/dataset\/291\/airfoil%2Bself%2Bnoise\">Airfoil Self-Noise dataset<\/a>. UCI Machine Learning Repository. DOI: 10.24432\/C5VW2C.<\/li><\/ul><\/section>\n<\/div><\/div>\n\n<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 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