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	---
	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 \|