Learning neural operators on Riemannian manifolds

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Table 1

Performance comparison for the five case studies^a

Case	Metric	GNN	DeepONet	POD-DeepONet	FNO	WNO	NORM
1. Darcy problem	MME	0.140(0.002)	0.045(4×10^-4)	0.044(3×10^-4)	0.094(0.002)	0.077(2×10^-4)	0.039(4×10^-4)
	$E_{L_{2}}$ (%)	6.732(0.053)	1.358(0.013)	1.296(0.023)	3.826(0.077)	1.090(0.030)	1.046(0.020)
2. Pipe turbulence	MME	2.358(0.125)	0.960(0.002)	0.241(0.017)	0.896(0.001)	0.892(0.001)	0.116(0.003)
	$E_{L_{2}}$ (%)	23.583(1.411)	9.358(0.107)	2.587(0.275)	3.801(0.002)	3.786(0.003)	1.008(0.020)
3. Heat transfer	MME	–	3.038(0.156)	1.304(0.045)	–	–	1.599(0.096)
	$E_{L_{2}}$ (%)	–	0.072(0.002)	0.057(0.001)	–	–	0.027(0.002)
4. Composite	MME	0.882(0.029)	0.157(0.002)	0.077(0.003)	–	–	0.051(0.002)
	$E_{L_{2}}$ (%)	20.908(0.050)	1.880(0.034)	1.437(0.060)	–	–	0.999(0.027)
5. Blood flow	MME	–	0.899(0.010)	0.488(0.002)	–	–	0.093(0.003)
	$E_{L_{2}}$ (%)	–	60.613(0.961)	33.256(0.126)	–	–	4.822(0.061)

a: The values A(B) represent the mean and standard deviation of five repeated runs, respectively.

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