  Learning physical simulations has been an essential and central aspect of
many recent research efforts in machine learning, particularly for
Navier-Stokes-based fluid mechanics. Classic numerical solvers have
traditionally been computationally expensive and challenging to use in inverse
problems, whereas Neural solvers aim to address both concerns through machine
learning. We propose a general formulation for continuous convolutions using
separable basis functions as a superset of existing methods and evaluate a
large set of basis functions in the context of (a) a compressible 1D SPH
simulation, (b) a weakly compressible 2D SPH simulation, and (c) an
incompressible 2D SPH Simulation. We demonstrate that even and odd symmetries
included in the basis functions are key aspects of stability and accuracy. Our
broad evaluation shows that Fourier-based continuous convolutions outperform
all other architectures regarding accuracy and generalization. Finally, using
these Fourier-based networks, we show that prior inductive biases, such as
window functions, are no longer necessary. An implementation of our approach,
as well as complete datasets and solver implementations, is available at
https://github.com/tum-pbs/SFBC.
