  Neural operators can learn nonlinear mappings between function spaces and
offer a new simulation paradigm for real-time prediction of complex dynamics
for realistic diverse applications as well as for system identification in
science and engineering. Herein, we investigate the performance of two neural
operators, and we develop new practical extensions that will make them more
accurate and robust and importantly more suitable for industrial-complexity
applications. The first neural operator, DeepONet, was published in 2019, and
the second one, named Fourier Neural Operator or FNO, was published in 2020. In
order to compare FNO with DeepONet for realistic setups, we develop several
extensions of FNO that can deal with complex geometric domains as well as
mappings where the input and output function spaces are of different
dimensions. We also endow DeepONet with special features that provide inductive
bias and accelerate training, and we present a faster implementation of
DeepONet with cost comparable to the computational cost of FNO. We consider 16
different benchmarks to demonstrate the relative performance of the two neural
operators, including instability wave analysis in hypersonic boundary layers,
prediction of the vorticity field of a flapping airfoil, porous media
simulations in complex-geometry domains, etc. The performance of DeepONet and
FNO is comparable for relatively simple settings, but for complex geometries
and especially noisy data, the performance of FNO deteriorates greatly. For
example, for the instability wave analysis with only 0.1% noise added to the
input data, the error of FNO increases 10000 times making it inappropriate for
such important applications, while there is hardly any effect of such noise on
the DeepONet. We also compare theoretically the two neural operators and obtain
similar error estimates for DeepONet and FNO under the same regularity
assumptions.
