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breakeven_complexity

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 license: mit
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 ---
 license: mit
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+---
+license: mit
+---
+This repository contains the dataset for "Breakeven complexity: A new perspective on neural partial differential equation solvers".
+Dataset detail:
+- Navier-Stokes: field "u". 20,000 training trajectories and 1,000 test trajectories. Shape (N, res, res, T).
+- Kuramoto-Sivashinsky: field "u". 20,000 training trajectories and 1,000 test trajectories. Shape (N, res, res, T).
+- Gray-Scott: field "u" and "v". 20,000 training trajectories seperated equally into 10 files, and 1,000 test trajectories. Shape (N, res, res, T).
+- BreakFlow: field "velocity_x", ""velocity_y", "pressure", "vorticity", and "mask". 2,000 training trajectories and 50 test trajectories for each Reynolds bin. The combined folder is a shuffled collection of the data. The OOD(out of distribution) folder contrains train set of a shuffled collection of bin1 and bin2, and bin3 as test set. Shape (N, T, res, res).
+Here are time matching statistics for the datasets:
+# Navier-Stokes
+| Resolution | nRMSE | Time (seconds) |
+| :--- | :--- | :--- |
+| **10** | 0.710526 | 0.020756 |
+| **20** | 0.274556 | 0.020724 |
+| **30** | 0.079504 | 0.022734 |
+| **40** | 0.036205 | 0.025200 |
+| **50** | 0.018495 | 0.027116 |
+| **60** | 0.009136 | 0.032884 |
+| **70** | 0.005968 | 0.036457 |
+| **80** | 0.003813 | 0.041083 |
+| **90** | 0.002139 | 0.041915 |
+| **100** | 0.001344 | 0.043408 |
+| **256** | 0.000000 | 0.043537 |
+# Kuramoto-Sivashinsky 1D
+| Resolution | nRMSE | Time (seconds) |
+| :--- | :--- | :--- |
+| **8** | 0.840487 | 0.000209 |
+| **10** | 0.694043 | 0.000236 |
+| **12** | 0.579952 | 0.000225 |
+| **14** | 0.579953 | 0.000234 |
+| **16** | 0.576410 | 0.000200 |
+| **18** | 0.590125 | 0.000230 |
+| **20** | 0.590130 | 0.000318 |
+| **22** | 0.516423 | 0.000243 |
+| **24** | 0.430445 | 0.000232 |
+| **26** | 0.430447 | 0.000254 |
+| **28** | 0.320877 | 0.000243 |
+| **30** | 0.188443 | 0.000238 |
+| **32** | 0.188440 | 0.000237 |
+| **34** | 0.076120 | 0.000240 |
+| **36** | 0.046457 | 0.000212 |
+| **38** | 0.046294 | 0.000243 |
+| **40** | 0.023428 | 0.000240 |
+| **42** | 0.011915 | 0.000250 |
+| **44** | 0.011915 | 0.000243 |
+| **46** | 0.006401 | 0.000256 |
+| **48** | 0.003399 | 0.000243 |
+| **50** | 0.003400 | 0.000251 |
+| **52** | 0.001929 | 0.000369 |
+| **54** | 0.001090 | 0.000250 |
+| **56** | 0.001089 | 0.000250 |
+| **58** | 0.000658 | 0.000290 |
+| **60** | 0.000426 | 0.000262 |
+| **62** | 0.000412 | 0.000373 |
+| **64** | 0.000000 | 0.000310 |
+# Kuramoto-Sivashinsky 2D
+| Resolution | nRMSE | Time (seconds) |
+| :--- | :--- | :--- |
+| **54** | 0.504471 | 0.015612 |
+| **60** | 0.217395 | 0.017008 |
+| **64** | 0.148039 | 0.017242 |
+| **70** | 0.070031 | 0.017442 |
+| **74** | 0.045836 | 0.019862 |
+| **80** | 0.015524 | 0.021907 |
+| **84** | 0.004999 | 0.023360 |
+| **90** | 0.001333 | 0.023940 |
+| **94** | 0.000871 | 0.027558 |
+| **100** | 0.000704 | 0.030900 |
+| **256** | 0.000000 | 0.066734 |
+# Kuramoto-Sivashinsky 3D
+| Resolution | nRMSE | Time (seconds) |
+| :--- | :--- | :--- |
+| **48** | 0.439662 | 1.144816 |
+| **50** | 0.444682 | 1.199737 |
+| **52** | 0.171244 | 1.654107 |
+| **54** | 0.073160 | 1.338130 |
+| **56** | 0.074237 | 1.356794 |
+| **58** | 0.031996 | 1.912379 |
+| **60** | 0.013426 | 1.338568 |
+| **62** | 0.012842 | 2.050640 |
+| **64** | 0.000000 | 2.711006 |
+# Gray-Scott
+| Resolution | nRMSE | Time (seconds) |
+| :--- | :--- | :--- |
+| **10** | 0.803211 | 0.037729 |
+| **20** | 0.722904 | 0.041896 |
+| **30** | 0.635374 | 0.047774 |
+| **40** | 0.587572 | 0.051580 |
+| **50** | 0.546942 | 0.061180 |
+| **60** | 0.453108 | 0.068281 |
+| **70** | 0.326739 | 0.074282 |
+| **80** | 0.245069 | 0.081521 |
+| **90** | 0.180130 | 0.101329 |
+| **100** | 0.129753 | 0.114296 |
+| **110** | 0.091766 | 0.131567 |
+| **120** | 0.064967 | 0.143684 |
+| **130** | 0.045852 | 0.172376 |
+| **140** | 0.034773 | 0.189323 |
+| **150** | 0.024973 | 0.213734 |
+| **160** | 0.016908 | 0.224823 |
+| **170** | 0.015451 | 0.361601 |
+| **180** | 0.014250 | 0.283169 |
+| **190** | 0.010380 | 0.354393 |
+| **200** | 0.003622 | 0.379086 |
+| **210** | 0.002213 | 0.383786 |
+| **220** | 0.002397 | 0.425519 |
+| **230** | 0.002240 | 0.473417 |
+| **256** | 0.000000 | 0.475097 |
+# BreakFlow Combined
+| Resolution | nRMSE | Time (seconds) |
+| :--- | :--- | :--- |
+| **1.0000** | 0.000000 | 136.90 |
+| **1.0443** | 0.128982 | 116.90 |
+| **1.0905** | 0.129075 | 112.56 |
+| **1.1388** | 0.130949 | 108.54 |
+| **1.1892** | 0.131635 | 106.62 |
+| **1.4142** | 0.148803 | 80.38 |
+| **1.6818** | 0.161297 | 61.96 |
+| **2.0000** | 0.183323 | 45.88 |
+| **2.3784** | 0.185847 | 42.64 |
+| **2.8284** | 0.188891 | 41.28 |
+| **3.3636** | 0.222432 | 28.84 |
+| **4.0000** | 0.225344 | 27.60 |
+| **4.7568** | 0.240020 | 26.86 |
+| **5.6569** | 0.240704 | 26.84 |
+| **6.7272** | 0.271237 | 17.96 |
+| **8.0000** | 0.276092 | 17.64 |
+| **9.5137** | 0.288030 | 15.78 |
+| **11.3137** | 0.292035 | 15.72 |
+| **13.4543** | 0.307576 | 15.68 |
+| **16.0000** | 0.310202 | 15.66 |
+| **19.0274** | 0.331415 | 15.00 |
+# BreakFlow Bin1
+| Resolution | nRMSE | Time (seconds) |
+| :--- | :--- | :--- |
+| **1.0000** | 0.000000 | 95.00 |
+| **1.0443** | 0.002162 | 90.68 |
+| **1.0905** | 0.002408 | 87.58 |
+| **1.1388** | 0.002191 | 84.58 |
+| **1.1892** | 0.002310 | 82.16 |
+| **1.4142** | 0.002670 | 62.50 |
+| **1.6818** | 0.004234 | 48.10 |
+| **2.0000** | 0.006039 | 36.38 |
+| **2.3784** | 0.007088 | 33.78 |
+| **2.8284** | 0.007740 | 32.42 |
+| **3.3636** | 0.017143 | 23.14 |
+| **4.0000** | 0.018088 | 22.24 |
+| **4.7568** | 0.019215 | 21.78 |
+| **5.6569** | 0.022276 | 21.58 |
+| **6.7272** | 0.042654 | 14.48 |
+| **8.0000** | 0.048285 | 14.26 |
+| **9.5137** | 0.063298 | 12.96 |
+| **11.3137** | 0.072127 | 12.88 |
+| **13.4543** | 0.092477 | 12.86 |
+| **16.0000** | 0.093449 | 12.78 |
+# BreakFlow Bin2
+| Resolution | nRMSE | Time (seconds) |
+| :--- | :--- | :--- |
+| **1.0000** | 0.000000 | 99.24 |
+| **1.0443** | 0.054530 | 96.30 |
+| **1.0905** | 0.058209 | 92.84 |
+| **1.1388** | 0.048698 | 89.30 |
+| **1.1892** | 0.055428 | 87.10 |
+| **1.4142** | 0.081105 | 66.50 |
+| **1.6818** | 0.088567 | 51.16 |
+| **2.0000** | 0.123205 | 39.36 |
+| **2.3784** | 0.126006 | 36.22 |
+| **2.8284** | 0.125205 | 35.32 |
+| **3.3636** | 0.179135 | 24.54 |
+| **4.0000** | 0.179902 | 24.02 |
+| **4.7568** | 0.199868 | 23.18 |
+| **5.6569** | 0.211182 | 23.32 |
+| **6.7272** | 0.250170 | 16.12 |
+| **8.0000** | 0.255241 | 15.60 |
+| **9.5137** | 0.285511 | 13.48 |
+| **11.3137** | 0.278287 | 13.66 |
+| **13.4543** | 0.314424 | 13.52 |
+| **16.0000** | 0.314749 | 13.80 |
+# BreakFlow Bin3 and OOD
+| Resolution | nRMSE | Time (seconds) |
+| :--- | :--- | :--- |
+| **1.0000** | 0.000000 | 167.48 |
+| **1.0443** | 0.330253 | 163.74 |
+| **1.0905** | 0.332230 | 157.28 |
+| **1.1388** | 0.336337 | 151.76 |
+| **1.1892** | 0.337167 | 150.62 |
+| **1.4142** | 0.362633 | 112.16 |
+| **1.6818** | 0.391089 | 86.62 |
+| **2.0000** | 0.420725 | 61.88 |
+| **2.3784** | 0.424446 | 57.90 |
+| **2.8284** | 0.433728 | 56.10 |
+| **3.3636** | 0.471017 | 38.82 |
+| **4.0000** | 0.478041 | 36.52 |
+| **4.7568** | 0.500976 | 35.62 |
+| **5.6569** | 0.488654 | 35.62 |
+| **6.7272** | 0.520888 | 23.28 |
+| **8.0000** | 0.524751 | 23.08 |
+| **9.5137** | 0.515281 | 20.32 |
+| **11.3137** | 0.525692 | 20.64 |
+| **13.4543** | 0.515828 | 20.64 |
+| **16.0000** | 0.522407 | 20.40 |