[Paper Review] Physics-Informed Neural Operator for Learning Partial Differential Equations

Zongyi Li, Hongkai Zheng+6 more|arXiv (Cornell University)|Nov 6, 2021

Model Reduction and Neural Networks37 references157 citations

TL;DR

PINO combines data with higher-resolution PDE constraints to learn the solution operator of parametric PDEs, enabling accurate, multi-resolution extrapolation and efficient instance-wise fine-tuning. It outperforms purely data-driven methods and overcomes PINN optimization challenges in multi-scale problems.

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.

Motivation & Objective

• Motivate learning the solution operator for families of parametric PDEs rather than solving a single instance.
• Address data scarcity by incorporating PDE constraints to complement limited training data.
• Achieve high-fidelity operator learning with the ability to extrapolate to higher resolutions beyond training data.
• Improve optimization over purely data-driven or purely physics-based approaches, especially for multi-scale dynamics.

Proposed method

• Use Fourier neural operator (FNO) as the base neural operator to learn the solution operator.
• Train with a data loss when data is available and impose a PDE loss at a higher resolution to regularize the operator.
• Fine-tune the learned operator instance-wise on a given PDE instance using the PDE loss, with an optional anchor (operator) loss to bound changes.
• Compute derivatives required for the PDE loss directly in Fourier space to enable efficient, exact gradients.
• Handle forward operator learning (data and PDE loss) and inverse problems (learn forward/inverse operators or use optimization with PDE constraints).
• Leverage multi-resolution training to achieve high-fidelity recovery of the ground-truth operator and enable zero-shot super-resolution.

Experimental results

Research questions

• RQ1 Can physics-informed neural operators learn the ground-truth solution operator across a family of parametric PDEs with limited data?
• RQ2 Do higher-resolution PDE constraints improve generalization and enable accurate extrapolation to unseen frequencies or resolutions?
• RQ3 Can PINO solve inverse problems more efficiently than traditional solvers or purely data-driven methods?
• RQ4 How does instance-wise fine-tuning affect accuracy and computational cost across different PDEs and Reynolds numbers?

Key findings

• PINO achieves high-fidelity approximation of the ground-truth operator across multiple PDE families.
• PINO extrapolates to unseen frequencies in Kolmogorov flows, outperforming data-only FNO and unet-based interpolation in high-frequency regimes.
• PINO reduces relative error by about 7% on transient and Kolmogorov flows on average compared to data-only baselines, while maintaining 400x speedups over GPU-based solvers.
• Navier–Stokes transfer across Reynolds numbers (100–500) via instance-wise fine-tuning demonstrates transferability to different flow regimes.
• Inverse problems solved with PINO can be significantly faster than traditional MCMC-based approaches (e.g., 3000x faster in a Darcy flow setting).
• PINO can operate with little or no training data by leveraging PDE constraints, addressing optimization challenges seen in purely PINN frameworks.