Machine learning methods have recently shown promise in solving partial
differential equations (PDEs). They can be classified into two broad
categories: approximating the solution function and learning the solution
operator. The Physics-Informed Neural Network (PINN) is an example of the
former while the Fourier neural operator (FNO) is an example of the latter.
Both these approaches have shortcomings. The optimization in PINN is
challenging and prone to failure, especially on multi-scale dynamic systems.
FNO does not suffer from this optimization issue since it carries out
supervised learning on a given dataset, but obtaining such data may be too
expensive or infeasible. In this work, we propose the physics-informed neural
operator (PINO), where we combine the operating-learning and
function-optimization frameworks. This integrated approach improves convergence
rates and accuracy over both PINN and FNO models. In the operator-learning
phase, PINO learns the solution operator over multiple instances of the
parametric PDE family. In the test-time optimization phase, PINO optimizes the
pre-trained operator ansatz for the querying instance of the PDE. Experiments
show PINO outperforms previous ML methods on many popular PDE families while
retaining the extraordinary speed-up of FNO compared to solvers. In particular,
PINO accurately solves challenging long temporal transient flows and Kolmogorov
flows where other baseline ML methods fail to converge