Table 1 Benchmark on the Navier–Stokes and Burgers’ equations
From: Learning integral operators via neural integral equations
Navier–Stokes | Burgers’ | |||||||||
|---|---|---|---|---|---|---|---|---|---|---|
t = 3 | t = 5 | t = 10 | t = 20 | t = 10 | t = 15 | t = 25 | ||||
s = 256 | s = 512 | s = 256 | s = 512 | s = 256 | s = 512 | |||||
LSTM | 0.1384 | 0.2337 | 0.1422 | 0.2465 | − | − | − | − | − | − |
ResNet | − | − | − | − | 0.0295 | 0.0309 | 0.0280 | 0.0232 | 0.0194 | 0.0204 |
Conv1DLSTM | − | − | − | − | 0.0132 | 0.0133 | 0.0132 | 0.0136 | 0.0124 | 0.0134 |
Conv2DLSTM | 0.4935 | 0.4393 | 0.3931 | 0.2999 | − | − | − | − | − | − |
FNO1D | − | − | − | − | 0.0088 | 0.088 | 0.0087 | 0.087 | 0.083 | 0.086 |
Galerkin | − | − | − | − | 0.525 | NA | 0.521 | NA | 0.518 | NA |
FNO2D | 0.2795 | 0.2724 | NA | NA | − | − | − | − | − | − |
FNO3D | NA | NA | 0.1751 | 0.701 | − | − | − | − | − | − |
ViT | 0.1093 | 0.877 | 0.2473 | 0.2367 | 0.430 | 0.423 | 0.423 | 0.422 | 0.422 | 0.424 |
ViTsmall | 0.926 | 0.702 | 0.677 | 0.655 | 0.429 | 0.429 | 0.426 | 0.427 | 0.417 | 0.424 |
ViTparallel | 0.2901 | 0.2660 | 0.2475 | 0.2368 | 0.433 | 0.702 | 0.573 | 0.861 | 0.435 | 0.700 |
ViT3D | 0.360 | 0.365 | 0.433 | 0.406 | − | − | − | − | − | − |
ANIE (this work) | 0.0194 | 0.0220 | 0.0193 | 0.0117 | 0.0024 | 0.0026 | 0.0024 | 0.0024 | 0.0022 | 0.0023 |
