---
license: mit
---

This repository contains the dataset for ["Breakeven complexity: A new perspective on neural partial differential equation solvers"](https://arxiv.org/abs/2605.15399).

Dataset detail:
- Navier-Stokes: Simulated via [Exponax](https://fkoehler.site/exponax/). Contains field "u". 20,000 training trajectories and 1,000 test trajectories. Shape (N, res, res, T).
- Kuramoto-Sivashinsky: Simulated via [Exponax](https://fkoehler.site/exponax/). Contains field "u". 20,000 training trajectories and 1,000 test trajectories. Shape (N, res, res, T).
- Gray-Scott: Simulated via [Exponax](https://fkoehler.site/exponax/). Contains field "u" and "v". 20,000 training trajectories seperated equally into 10 files, and 1,000 test trajectories. Shape (N, res, res, T).
- BreakFlow: Simulated via [PyFR](https://www.pyfr.org/). Contains 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
| Coarse size | 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
| Coarse size | 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
| Coarse size | 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
| Coarse size | 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 |