1 Introduction
2 Related works
2.1 CNOP
2.2 Neural network-based dimension reduction
2.3 The case of double-gyre variation in ROMS
3 Methods
3.1 FEIA framework
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Step 1: Determine the initial solution x.
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Step 2: Calculate the objective function value f with x.
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Step 3: Judge the iteration condition. If the termination condition is satisfied, output the best solution x*; otherwise, go to Step 4.
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Step 4: Update solution x with the related rules and the objective value f calculated in Step 2. Go to Step 2.
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Step 1: Collect the samples in F. Determine the mapper p and the re-constructor r.
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Step 2: Determine the initial solution x. Map x into w by p.
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Step 3: Reconstruct w into x by r. Calculate the objective function value f with x.
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Step 4: Judge the iteration condition. If the termination condition is satisfied, output the best solution x* = r(w*); otherwise, go to Step 4.
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Step 5: Update solution w with the related rules and the objective value f calculated in Step 3. Go to Step 3.
3.1.1 PCA
3.1.2 PSO
3.2 AE and its variants
3.2.1 AE
3.2.2 SAE
3.2.3 CAE
3.2.4 VAE
3.3 The mapping model-based PCA and neural network
3.3.1 Decoder
3.3.2 GAN
3.4 The coupling of neural network for FEIA framework
3.4.1 Activation function
3.4.2 Re-constructor bias
3.4.3 Weight parameter selection
3.4.4 Training data and validation data
3.4.5 Training process
4 Experiment and results
PCA | No Reduction | |||||
---|---|---|---|---|---|---|
20 | 40 | 60 | 80 | 100 | ||
Result (× 1013 m5s2) | 1.520 | 1.630 | 1.384 | 1.420 | 1.402 | 1.214 |
Dimension | Statistic (× 1013 m5s2) | Frequency (effective) | ||||
---|---|---|---|---|---|---|
Max | Ave | Std | < 1.55 | 1.55 ~ 1.65 | > 1.65 | |
20 | 1.594 | 1.503 | 0.061 | 7(0) | 3(0) | 0(0) |
40 | 1.630 | 1.535 | 0.092 | 5(0) | 5(2) | 0(0) |
60 | 1.574 | 1.468 | 0.070 | 8(0) | 2(0) | 0(0) |
80 | 1.507 | 1.418 | 0.047 | 10(0) | 0(0) | 0(0) |
100 | 1.532 | 1.451 | 0.040 | 10(0) | 0(0) | 0(0) |
4.1 The experiment for the first way
Structure | |
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AE | fc(54,776) – fc(256) – fc(256) – fc(Df) – fc(256) – fc(256) – fc(54,776) |
SAE | fc(54,776) – fc(Df) – fc(54,776) |
CAE | (55, 110, 4) – conv(28, 28, 32) – conv(7, 7, 64) – fc(256) – fc(0.4 * Df) – (56, 109, 4) – conv(28, 28, 32) – conv(7, 7, 64) – fc(256) – fc(0.4 * Df) – (Df) (56, 110, 1) – conv(28, 28, 32) – conv(7, 7, 64) – fc(256) – fc(0.2 * Df) – – fc(0.4 * Df) – fc(256) – fc(3136) – deconv(28, 28, 32) – deconv(55, 110, 4) – fc(0.4 * Df) – fc(256) – fc(3136) – deconv(28, 28, 32) – deconv(56, 109, 4) – fc(0.2 * Df) – fc(256) – fc(3136) – deconv(28, 28, 32) – deconv(56, 110, 1) |
VAE | fc(54,776) – fc(256) – fc(256) – 2 × fc(Df) – (Df) – fc(256) – fc(256) – fc(54,776) |
Parameter | Related error | Result (× 1013 m5s2) | |||||||||
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BS | LR | 20 | 40 | 60 | 80 | 100 | 20 | 40 | 60 | 80 | 100 |
20 | 10–3 | 0.224 | 0.222 | 0.222 | 0.227 | 0.239 | 1.109 | 1.180 | 1.152 | 1.319 | 1.094 |
10–4 | 0.211 | 0.202 | 0.197 | 0.195 | 0.204 | 1.131 | 1.206 | 1.355 | 1.375 | 1.284 | |
10–5 | 0.214 | 0.193 | 0.188 | 0.188 | 0.190 | 1.149 | 1.364 | 1.280 | 1.217 | 1.275 | |
40 | 10–3 | 0.207 | 0.220 | 0.215 | 0.228 | 0.226 | 1.195 | 1.272 | 1.164 | 1.098 | 1.129 |
10–4 | 0.214 | 0.188 | 0.202 | 0.190 | 0.193 | 1.143 | 1.200 | 1.285 | 1.251 | 1.289 | |
10–5 | 0.209 | 0.196 | 0.197 | 0.192 | 0.193 | 1.125 | 1.231 | 1.297 | 1.305 | 1.267 | |
80 | 10–3 | 0.205 | 0.211 | 0.207 | 0.216 | 0.210 | 1.125 | 1.175 | 1.142 | 1.193 | 1.281 |
10–4 | 0.202 | 0.188 | 0.193 | 0.194 | 0.194 | 1.136 | 1.288 | 1.317 | 1.290 | 1.206 | |
10–5 | 0.207 | 0.195 | 0.189 | 0.197 | 0.187 | 1.208 | 1.366 | 1.390 | 1.265 | 1.250 |
Parameter | Related error | Result (× 1013 m5s2) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
BS | LR | 20 | 40 | 60 | 80 | 100 | 20 | 40 | 60 | 80 | 100 |
20 | 10–3 | 0.168 | 0.169 | 0.155 | 0.196 | 0.177 | 1.451 | 1.490 | 1.538 | 1.428 | 1.440 |
10–4 | 0.157 | 0.127 | 0.127 | 0.125 | 0.130 | 1.298 | 1.441 | 1.579 | 1.415 | 1.421 | |
10–5 | 0.152 | 0.124 | 0.123 | 0.122 | 0.126 | 1.411 | 1.509 | 1.496 | 1.499 | 1.624 | |
40 | 10–3 | 0.159 | 0.150 | 0.142 | 0.138 | 0.152 | 1.475 | 1.435 | 1.324 | 1.720 | 1.541 |
10–4 | 0.157 | 0.123 | 0.124 | 0.124 | 0.124 | 1.473 | 1.566 | 1.461 | 1.358 | 1.746 | |
10–5 | 0.148 | 0.123 | 0.123 | 0.123 | 0.124 | 1.437 | 1.538 | 1.553 | 1.442 | 1.404 | |
80 | 10–3 | 0.158 | 0.137 | 0.138 | 0.137 | 0.147 | 1.518 | 1.546 | 1.426 | 1.620 | 1.526 |
10–4 | 0.153 | 0.124 | 0.124 | 0.122 | 0.122 | 1.445 | 1.516 | 1.480 | 1.451 | 1.581 | |
10–5 | 0.151 | 0.129 | 0.124 | 0.125 | 0.125 | 1.394 | 1.647 | 1.536 | 1.493 | 1.564 |
Parameter | Related error | Result (× 1013 m5s2) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|---|
BS | LR | 20 | 40 | 60 | 80 | 100 | 20 | 40 | 60 | 80 | 100 |
20 | 10–3 | 0.255 | 0.269 | 0.250 | 0.239 | 0.251 | 1.190 | 1.192 | 1.199 | 1.261 | 1.166 |
10–4 | 0.350 | 0.298 | 0.315 | 0.325 | 0.293 | 1.095 | 1.199 | 1.275 | 1.419 | 1.195 | |
10–5 | 0.429 | 0.386 | 0.375 | 0.401 | 0.398 | 1.107 | 1.152 | 1.253 | 1.162 | 1.223 | |
40 | 10–3 | 0.267 | 0.220 | 0.223 | 0.232 | 0.211 | 1.139 | 1.325 | 1.254 | 1.253 | 1.167 |
10–4 | 0.346 | 0.331 | 0.328 | 0.326 | 0.319 | 1.156 | 1.144 | 1.415 | 1.116 | 1.242 | |
10–5 | 0.389 | 0.394 | 0.380 | 0.421 | 0.391 | 9.771 | 1.128 | 1.358 | 1.262 | 1.119 | |
80 | 10–3 | 0.255 | 0.215 | 0.218 | 0.227 | 0.220 | 1.134 | 1.374 | 1.232 | 1.311 | 1.242 |
10–4 | 0.352 | 0.334 | 0.346 | 0.319 | 0.307 | 1.125 | 1.243 | 1.140 | 1.223 | 1.286 | |
10–5 | 0.432 | 0.385 | 0.376 | 0.380 | 0.397 | 1.121 | 1.180 | 1.352 | 1.241 | 1.263 |
Dimension | Statistic (× 1013 m5s2) | Frequency (effective) | ||||
---|---|---|---|---|---|---|
Max | Ave | Std | < 1.55 | 1.55 ~ 1.65 | > 1.65 | |
20 | 1.502 | 1.455 | 0.040 | 10(0) | 0(0) | 0(0) |
40 | 1.681 | 1.576 | 0.082 | 2(0) | 6(2) | 2(2) |
60 | 1.710 | 1.527 | 0.109 | 6(0) | 2(1) | 2(2) |
80 | 1.662 | 1.514 | 0.093 | 7(0) | 2(0) | 1(1) |
100 | 1.746 | 1.519 | 0.123 | 6(0) | 3(0) | 1(1) |
4.2 The experiment for the second way
Structure | |
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Decoder | fc(Df) – EL × (fc(256) –) fc(54,776) |
GAN | Generator: fc(Df) – EL × (fc(256) –) fc(54,776) |
Discriminator: fc(54,776) – fc(256) – fc(256) – fc(1) |
Parameter | Related error | Result (× 1013 m5s2) | ||||||||
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EL | 20 | 40 | 60 | 80 | 100 | 20 | 40 | 60 | 80 | 100 |
0 | 0.197 | 0.122 | 0.115 | 0.111 | 0.112 | 1.529 | 1.409 | 1.476 | 1.413 | 1.466 |
1 | 0.204 | 0.149 | 0.146 | 0.150 | 0.147 | 1.358 | 1.605 | 1.425 | 1.360 | 1.384 |
2 | 0.209 | 0.159 | 0.163 | 0.167 | 0.175 | 1.318 | 1.241 | 1.329 | 1.237 | 1.265 |
Parameter | Related error | Result (× 1013 m5s2) | ||||||||
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EL | 20 | 40 | 60 | 80 | 100 | 20 | 40 | 60 | 80 | 100 |
0 | 0.222 | 0.122 | 0.116 | 0.114 | 0.112 | 1.489 | 1.688 | 1.499 | 1.501 | 1.640 |
1 | 0.205 | 0.147 | 0.147 | 0.141 | 0.147 | 1.239 | 1.553 | 1.396 | 1.334 | 1.469 |
2 | 0.210 | 0.170 | 0.172 | 0.164 | 0.172 | 1.201 | 1.296 | 1.429 | 1.369 | 1.220 |
Dimension | Statistic (× 1013 m5s2) | Frequency (effective) | ||||
---|---|---|---|---|---|---|
Max | Ave | Std | < 1.55 | 1.55 ~ 1.65 | > 1.65 | |
20 | 1.491 | 1.440 | 0.043 | 10(0) | 0(0) | 0(0) |
40 | 1.820 | 1.566 | 0.135 | 6(0) | 1(0) | 3(3) |
60 | 1.683 | 1.541 | 0.076 | 6(0) | 2(0) | 2(2) |
80 | 1.657 | 1.495 | 0.071 | 8(0) | 1(0) | 1(1) |
100 | 1.640 | 1.530 | 0.096 | 5(0) | 5(2) | 0(0) |