Physics-Informed Neural Operator for Learning Partial Differential Equations

TL;DR

PINO introduces a physics-informed neural operator framework that blends data with high-resolution PDE constraints to learn the solution operator for parametric PDEs, enabling accurate extrapolation and zero-shot super-resolution. Built on the Fourier neural operator backbone, PINO comprises an operator-learning phase followed by instance-wise fine-tuning, with explicit derivatives computed in function space to enforce PDE losses. The approach demonstrates improved accuracy and speed across Burgers, Darcy, and Navier–Stokes problems, handles inverse problems, and shows robust transfer across Reynolds numbers and resolutions. This hybrid method mitigates key limitations of pure data-driven operators and PINNs, offering data-efficient training, strong generalization, and practical applicability to complex, multi-scale PDEs. The work also discusses discretization-convergence properties and outlines avenues for extending PINO to higher dimensions and diverse geometries, potentially via a shared model library.

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.

Figures (10)

• Figure 1: PINO uses both training data and PDE loss function and perfectly extrapolates to unseen frequencies in Kolmogorov Flows. FNO uses only training data and does not have further information on higher frequencies, but still follows the general trend of the ground-truth spectrum. On the other hand, using a trained UNet with trilinear interpolation (NN+Interpolation) has severe distortions at higher frequencies. Details in Section \ref{['sec:solve-eqn']}.
• Figure 2: PINO trains neural operator with both training data and PDE loss function. The figure shows the neural operator architecture with the lifting point-wise operator that receives input function $a$ and outputs function $v_0$ with a larger co-dimension. This operation is followed by $L$ blocks that compute linear integral operators followed by non-linearity, and the last layer of which outputs the function $v_L$. The pointwise projection operator projects $v_L$ to output function $u$. Both $v_L$ and $u$ are functions and all their derivatives ($\mathrm{D} v_{\mathrm{L}}$, $\mathrm{D}u$) can be computed in their exact forms at any query points $x$.
• Figure 3: solve for one specific instance verse learn the entire solution operator
• Figure 4: Plot of test relative $L_2$ error versus runtime step for the Kolmogorov flow with Re500, T=0.5s. Left: resolution $64\times 64 \times 65$; right: resolution $128\times 128\times 129$. Averaged over 20 instances. LAAF-PINN: PINN with locally adaptive activation functions. SA-PINN: self-adaptive PINN.
• Figure 5: PINO on Kolmogorov flow (left) and Lid-cavity flow (right)

Theorems & Definitions (3)

• Definition 1: Neural operator $\mathcal{G}_\theta$
• Definition 2: Kernel Integral Operators
• Definition 3: Fourier convolution operator