  Fourier neural operators (FNOs) can learn highly nonlinear mappings between
function spaces, and have recently become a popular tool for learning responses
of complex physical systems. However, to achieve good accuracy and efficiency,
FNOs rely on the Fast Fourier transform (FFT), which is restricted to modeling
problems on rectangular domains. To lift such a restriction and permit FFT on
irregular geometries as well as topology changes, we introduce domain agnostic
Fourier neural operator (DAFNO), a novel neural operator architecture for
learning surrogates with irregular geometries and evolving domains. The key
idea is to incorporate a smoothed characteristic function in the integral layer
architecture of FNOs, and leverage FFT to achieve rapid computations, in such a
way that the geometric information is explicitly encoded in the architecture.
In our empirical evaluation, DAFNO has achieved state-of-the-art accuracy as
compared to baseline neural operator models on two benchmark datasets of
material modeling and airfoil simulation. To further demonstrate the capability
and generalizability of DAFNO in handling complex domains with topology
changes, we consider a brittle material fracture evolution problem. With only
one training crack simulation sample, DAFNO has achieved generalizability to
unseen loading scenarios and substantially different crack patterns from the
trained scenario. Our code and data accompanying this paper are available at
https://github.com/ningliu-iga/DAFNO.
