NSFNet4¶
# VP_NSFNet4
# linux
wget -c https://paddle-org.bj.bcebos.com/paddlescience/datasets/NSFNet/NSF4_data.zip -P ./data/
unzip ./data/NSF4_data.zip
# windows
# curl https://paddle-org.bj.bcebos.com/paddlescience/datasets/NSFNet/NSF4_data.zip --create-dirs -o ./data/NSF4_data.zip
# unzip ./data/NSF4_data.zip
python VP_NSFNet4.py data_dir=./data/
# VP_NSFNet4
# linux
wget -c https://paddle-org.bj.bcebos.com/paddlescience/datasets/NSFNet/NSF4_data.zip -P ./data/
unzip ./data/NSF4_data.zip
# windows
# curl https://paddle-org.bj.bcebos.com/paddlescience/datasets/NSFNet/NSF4_data.zip --create-dirs -o ./data/NSF4_data.zip
# unzip ./data/NSF4_data.zip
python VP_NSFNet4.py mode=eval data_dir=./data/ EVAL.pretrained_model_path=https://paddle-org.bj.bcebos.com/paddlescience/models/nsfnet/nsfnet4.pdparams
# VP_NSFNet4
# linux
wget -c https://paddle-org.bj.bcebos.com/paddlescience/datasets/NSFNet/NSF4_data.zip -P ./data/
unzip ./data/NSF4_data.zip
# windows
# curl https://paddle-org.bj.bcebos.com/paddlescience/datasets/NSFNet/NSF4_data.zip --create-dirs -o ./data/NSF4_data.zip
# unzip ./data/NSF4_data.zip
python VP_NSFNet4.py mode=infer
1. Background Introduction¶
In recent years, deep learning has achieved remarkable achievements in many fields, especially in computer vision and natural language processing. Inspired by the rapid development of deep learning and based on the powerful function approximation ability of deep learning, neural networks have also achieved success in the field of scientific computing. Current research is mainly divided into two categories. One is to add physical information and physical constraints to the loss function to train neural networks, represented by PINN and Deep Ritz Net. The other is data-driven deep neural network operators, represented by FNO and DeepONet. These methods have been widely used in scientific practice, such as weather forecasting, quantum chemistry, biological engineering, and computational fluid dynamics. In order to fully explore the ability of PINN to solve fluid equations, the author of this reproduction paper designed NSFNets, and successively used two-dimensional and three-dimensional Navier-Stokes equations with analytical or numerical solutions, as well as datasets solved with high precision using the DNS method as references, to perform forward problem solving training. The paper experiments show that PINN has excellent numerical solving capabilities for incompressible Navier-Stokes equations. The main goal of this project is to use PaddleScience to reproduce the code for high-precision solving of Navier-Stokes equations implemented in the paper.
2. Problem Definition¶
The classic PINN model is used for this problem, so I won't go into details.
Mainly introduce several types of Navier-Stokes equations solved:
The incompressible Navier-Stokes equation can be expressed as:
\[\frac{\partial \mathbf{u}}{\partial t}+(\mathbf{u} \cdot \nabla) \mathbf{u} =-\nabla p+\frac{1}{Re} \nabla^2 \mathbf{u} \quad \text { in } \Omega, \]
\[\nabla \cdot \mathbf{u} =0 \quad \text { in } \Omega, \]
\[\mathbf{u} =\mathbf{u}_{\Gamma} \quad \text { on } \Gamma_D, \]
\[\frac{\partial \mathbf{u}}{\partial n} =0 \quad \text { on } \Gamma_N.\]
2.1 JHTDB Dataset¶
The dataset is a high-precision dataset of three-dimensional incompressible forced isotropic turbulence at Re=999.35 solved using DNS. Detailed parameters can be found in readme.
3. Problem Solving¶
3.1 Model Construction¶
This paper uses the classic PINN MLP model for training.
3.2 Data Generation¶
Boundary points, initial value points, and internal points for calculating residuals are taken successively (see section 3.3 of paper for specific selection method) and test points are generated.
3.3 Normalization¶
To change the selected smaller rectangular area into a cubic area, we embed the normalization function before the network training.
3.4 Constraint Construction¶
Since our boundary points and initial value points have analytical solutions, we use supervised constraints, where alpha and beta are the weights of the loss function, which are both taken as 100 in this code, consistent with the description in the paper.
Use internal points to construct residual constraints of Navier-Stokes equations
3.5 Validator Construction¶
Use the test set generated during data generation for model evaluation:
3.6 Optimizer Construction¶
Consistent with the description in the paper, we use piecewise learning rate to construct the Adam optimizer, where the number of training epochs can be adjusted by adjusting epoch_list.
3.7 Model Training and Evaluation¶
After completing the above settings, you only need to pass the instantiated objects to ppsci.solver.Solver.
Finally start training:
4. Complete Code¶
| NSFNet.py | |
|---|---|
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 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 122 123 124 125 126 127 128 129 130 131 132 133 134 135 136 137 138 139 140 141 142 143 144 145 146 147 148 149 150 151 152 153 154 155 156 157 158 159 160 161 162 163 164 165 166 167 168 169 170 171 172 173 174 175 176 177 178 179 180 181 182 183 184 185 186 187 188 189 190 191 192 193 194 195 196 197 198 199 200 201 202 203 204 205 206 207 208 209 210 211 212 213 214 215 216 217 218 219 220 221 222 223 224 225 226 227 228 229 230 231 232 233 234 235 236 237 238 239 240 241 242 243 244 245 246 247 248 249 250 251 252 253 254 255 256 257 258 259 260 261 262 263 264 265 266 267 268 269 270 271 272 273 274 275 276 277 278 279 280 281 282 283 284 285 286 287 288 289 290 291 292 293 294 295 296 297 298 299 300 301 302 303 304 305 306 307 308 309 310 311 312 313 314 315 316 317 318 319 320 321 322 323 324 325 326 327 328 329 330 331 332 333 334 335 336 337 338 339 340 341 342 343 344 345 346 347 348 349 350 351 352 353 354 355 356 357 358 359 360 361 362 363 364 365 366 367 368 369 370 371 372 373 374 375 376 377 378 379 380 381 382 383 384 385 386 387 388 389 390 391 392 393 394 395 396 397 398 399 400 401 402 403 404 405 406 407 408 409 410 411 412 413 414 415 416 417 418 419 420 421 422 423 424 425 426 427 428 429 430 431 432 433 434 435 436 437 438 439 440 441 442 443 444 445 446 447 448 449 450 451 452 453 454 455 456 457 458 459 460 461 462 463 464 465 466 467 468 469 470 471 472 473 474 475 476 477 478 479 480 481 482 483 484 485 486 487 488 489 490 491 492 493 494 495 496 497 498 499 500 501 502 503 504 505 506 507 508 509 510 511 512 513 514 515 516 517 518 519 520 521 522 523 524 525 526 527 528 529 530 531 532 533 534 535 536 537 538 539 540 541 542 543 544 545 546 547 548 549 550 551 552 553 554 555 556 557 558 559 560 561 562 563 564 565 566 567 568 569 570 571 572 573 574 575 576 577 578 579 580 581 582 583 584 585 586 587 588 589 590 591 592 593 594 595 596 597 598 599 600 601 602 603 604 605 606 607 608 609 610 611 612 613 614 615 616 617 618 619 620 621 622 623 624 625 626 627 628 629 630 631 | |
5. Result Display¶
NSFNet4¶
As shown in the figure, the error of NSFNet in time is relatively stable, and the error accumulation problem often found in traditional methods does not appear. Among them, although the velocities in the three directions were not weighted during the training process, it can be seen from the training results that the neural network has the best approximation effect on the first velocity direction u, followed by the third velocity direction w, and the second velocity v has the worst approximation effect and a relatively obvious error accumulation phenomenon appears.
As shown in the figure, in the contour map of the y-z plane at x=12.47, the first one is the contour map of velocity u, the second is the contour map of velocity v, the third is the contour map of velocity w, and the fourth is the contour map of velocity p. It can be seen that the contour map of velocity u is relatively smoother than v, w, p.
As shown in the figure, in the contour map of the x-y plane at z=4.61, the first one is the contour map of velocity u, the second is the contour map of velocity v, the third is the contour map of velocity w, and the fourth is the contour map of velocity p. It can be seen that the contour map of velocity u is relatively smoother than v, w, p.
In summary, although u, v, w three velocity directions all require neural network training, for the JHTDB dataset, u direction data is smoother and easier to be learned by neural networks. Therefore, in subsequent research, we can try to divide and conquer the components in three different directions, increase the training intensity of complex component directions, and reduce the training intensity of simple component directions.
6. Results Description¶
We use PINN to numerically solve the incompressible Navier-Stokes equations. In PINN, randomly selected time and space coordinates are used as input values, corresponding velocity fields and pressure fields are used as output values, and initial values, boundary conditions are used as supervised constraints and the Navier-Stokes equation itself is used as unsupervised constraints added to the loss function for training. We use high-precision JHTDB dataset for training. Through the decrease of the loss function, the convergence of the neural network in solving the Navier-Stokes equation can be proven, indicating that PINN possesses the ability to solve incompressible forced isotropic turbulence. The experimental results show that PINN can well approximate the corresponding high-precision incompressible forced isotropic turbulence dataset, and we found that increasing the weights of boundary constraints and initial value constraints can enable the neural network to have better approximation effects. In contrast, within the allowable error range, using PINN to solve the Navier-Stokes equation is faster than the original DNS method inference speed.