  In this paper, we propose physics-informed neural operators (PINO) that
combine training data and physics constraints to learn the solution operator of
a given family of parametric Partial Differential Equations (PDE). PINO is the
first hybrid approach incorporating data and PDE constraints at different
resolutions to learn the operator. Specifically, in PINO, we combine
coarse-resolution training data with PDE constraints imposed at a higher
resolution. The resulting PINO model can accurately approximate the
ground-truth solution operator for many popular PDE families and shows no
degradation in accuracy even under zero-shot super-resolution, i.e., being able
to predict beyond the resolution of training data. PINO uses the Fourier neural
operator (FNO) framework that is guaranteed to be a universal approximator for
any continuous operator and discretization-convergent in the limit of mesh
refinement. By adding PDE constraints to FNO at a higher resolution, we obtain
a high-fidelity reconstruction of the ground-truth operator. Moreover, PINO
succeeds in settings where no training data is available and only PDE
constraints are imposed, while previous approaches, such as the
Physics-Informed Neural Network (PINN), fail due to optimization challenges,
e.g., in multi-scale dynamic systems such as Kolmogorov flows.
