Keywords: Partial Differential Equations, operator learning, physics-informed, PINN, inverse problem, Navier-Stokes Equation
Abstract: Machine learning methods have recently shown promise in solving partial differential equations (PDEs). They can be classified into two broad categories: solution function approximation, and operator learning. 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, and this 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 long temporal transient flows and chaotic Kolmogorov flows, while PINN and other methods fail to converge to a reasonable accuracy.
One-sentence Summary: We combine operator-learning with physics-informed optimization and are able to solve challenging PDEs and inverse problems with ~1000x speedup over standard solvers.
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