  Neural operators (NOs) have emerged as effective tools for modeling complex
physical systems in scientific machine learning. In NOs, a central
characteristic is to learn the governing physical laws directly from data. In
contrast to other machine learning applications, partial knowledge is often
known a priori about the physical system at hand whereby quantities such as
mass, energy and momentum are exactly conserved. Currently, NOs have to learn
these conservation laws from data and can only approximately satisfy them due
to finite training data and random noise. In this work, we introduce
conservation law-encoded neural operators (clawNOs), a suite of NOs that endow
inference with automatic satisfaction of such conservation laws. ClawNOs are
built with a divergence-free prediction of the solution field, with which the
continuity equation is automatically guaranteed. As a consequence, clawNOs are
compliant with the most fundamental and ubiquitous conservation laws essential
for correct physical consistency. As demonstrations, we consider a wide variety
of scientific applications ranging from constitutive modeling of material
deformation, incompressible fluid dynamics, to atmospheric simulation. ClawNOs
significantly outperform the state-of-the-art NOs in learning efficacy,
especially in small-data regimes.
