Physics-Informed Neural Operator for Learning Partial Differential Equations
Abstract
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.
Copyright and License
© 2024 Copyright held by the owner/author(s). Publication rights licensed to ACM.
Acknowledgement
The authors want to thank Sifan Wang for meaningful discussions.
Z. Li gratefully acknowledges the inancial support from the Kortschak Scholars, PIMCO Fellows, and Amazon AI4Science Fellows programs. N. Kovachki was partially supported by the Amazon AI4Science Fellowship. A. Anandkumar is supported in part by the Bren endowed chair professorship.
This work was carried out on (1) the NVIDIA NGC as part of Zongyi Li’s internship and (2) the Resnick High-Performance Computing Center, a facility supported by the Resnick Sustainability Institute at the California Institute of Technology.
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Additional details
- California Institute of Technology
- Kortschak Scholars
- Amazon (United States)
- Amazon AI4Science Fellowship
- California Institute of Technology
- Bren Professor of Computing and Mathematical Sciences
- Caltech groups
- Resnick Sustainability Institute