Table 3.
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The performances of the learned representation on the supervised tasks. Accuracy (relative ℓ2 error) of the PI encoder in PIANO and other baselines using linear evaluation on Burgers’ equation (NSE). CL, SM and PC denote contrastive learning, the split-merge trick and the physics-aware cropping strategy, respectively. The best results in each task are highlighted in bold.

MethodBurgers’ equation (accuracy, ↑)NSE (ℓ2 error, ↓)
PIANO-|${CL}\!\!\!\!\!\!{/}$|0.078 |$\scriptscriptstyle{\pm\ 0.003}$|0.161 |$\scriptscriptstyle{\pm\ 0.034}$|
PIANO-|${SM}\!\!\!\!\!\!{/}$|0.988 |$\scriptscriptstyle{\pm\ 0.008}$|0.086 |$\scriptscriptstyle{\pm\ 0.011}$|
PIANO-|${PC}\!\!\!\!\!\!{/}$|0.955 |$\scriptscriptstyle{\pm\ 0.013}$|0.092 |$\scriptscriptstyle{\pm\ 0.004}$|
PIANO0.997 |$\scriptscriptstyle{\pm\ 0.003}$|0.033 |$\scriptscriptstyle{\pm\ 0.002}$|
MethodBurgers’ equation (accuracy, ↑)NSE (ℓ2 error, ↓)
PIANO-|${CL}\!\!\!\!\!\!{/}$|0.078 |$\scriptscriptstyle{\pm\ 0.003}$|0.161 |$\scriptscriptstyle{\pm\ 0.034}$|
PIANO-|${SM}\!\!\!\!\!\!{/}$|0.988 |$\scriptscriptstyle{\pm\ 0.008}$|0.086 |$\scriptscriptstyle{\pm\ 0.011}$|
PIANO-|${PC}\!\!\!\!\!\!{/}$|0.955 |$\scriptscriptstyle{\pm\ 0.013}$|0.092 |$\scriptscriptstyle{\pm\ 0.004}$|
PIANO0.997 |$\scriptscriptstyle{\pm\ 0.003}$|0.033 |$\scriptscriptstyle{\pm\ 0.002}$|
Table 3.
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The performances of the learned representation on the supervised tasks. Accuracy (relative ℓ2 error) of the PI encoder in PIANO and other baselines using linear evaluation on Burgers’ equation (NSE). CL, SM and PC denote contrastive learning, the split-merge trick and the physics-aware cropping strategy, respectively. The best results in each task are highlighted in bold.

MethodBurgers’ equation (accuracy, ↑)NSE (ℓ2 error, ↓)
PIANO-|${CL}\!\!\!\!\!\!{/}$|0.078 |$\scriptscriptstyle{\pm\ 0.003}$|0.161 |$\scriptscriptstyle{\pm\ 0.034}$|
PIANO-|${SM}\!\!\!\!\!\!{/}$|0.988 |$\scriptscriptstyle{\pm\ 0.008}$|0.086 |$\scriptscriptstyle{\pm\ 0.011}$|
PIANO-|${PC}\!\!\!\!\!\!{/}$|0.955 |$\scriptscriptstyle{\pm\ 0.013}$|0.092 |$\scriptscriptstyle{\pm\ 0.004}$|
PIANO0.997 |$\scriptscriptstyle{\pm\ 0.003}$|0.033 |$\scriptscriptstyle{\pm\ 0.002}$|
MethodBurgers’ equation (accuracy, ↑)NSE (ℓ2 error, ↓)
PIANO-|${CL}\!\!\!\!\!\!{/}$|0.078 |$\scriptscriptstyle{\pm\ 0.003}$|0.161 |$\scriptscriptstyle{\pm\ 0.034}$|
PIANO-|${SM}\!\!\!\!\!\!{/}$|0.988 |$\scriptscriptstyle{\pm\ 0.008}$|0.086 |$\scriptscriptstyle{\pm\ 0.011}$|
PIANO-|${PC}\!\!\!\!\!\!{/}$|0.955 |$\scriptscriptstyle{\pm\ 0.013}$|0.092 |$\scriptscriptstyle{\pm\ 0.004}$|
PIANO0.997 |$\scriptscriptstyle{\pm\ 0.003}$|0.033 |$\scriptscriptstyle{\pm\ 0.002}$|
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