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.

翻译：本文提出物理信息神经算子（PINO），该方法结合训练数据与物理约束，学习给定参数化偏微分方程族（PDE）的解算子。PINO是首个在不同分辨率下融合数据与PDE约束来学习算子的混合方法。具体而言，PINO将粗分辨率训练数据与高分辨率下施加的PDE约束相结合。对于众多常见PDE族，所得PINO模型能精确逼近真实解算子，即使在零样本超分辨率场景下（即预测超出训练数据分辨率的能力）仍保持精度无衰减。PINO采用傅里叶神经算子（FNO）框架，该框架被证明是任意连续算子的通用近似器，且在网格加密极限下具有离散收敛性。通过在高分辨率下向FNO添加PDE约束，我们实现了对真实算子的高保真重构。此外，PINO在无训练数据、仅施加PDE约束的场景中仍能成功运行，而此前方法如物理信息神经网络（PINN）因优化困难（例如科莫戈洛夫流等多尺度动态系统）而失效。