Physics-Aligned Canonical Equivariant Fourier Neural Operator under Symmetry-Induced Shifts

Neural operators approximate PDE solution maps, but they need not respect the symmetries of the governing equation. In out-of-distribution (OOD) regimes, a standard neural operator must often learn coordinate alignment and physical evolution within a single map, which can hurt generalization. We use known continuous symmetries of evolution equations on periodic domains to separate these two roles. We propose the Physics-Aligned Canonical Equivariant Fourier Neural Operator (PACE-FNO), which estimates the input frame with a Lie-algebra coordinate estimator, maps the field to a reference frame, applies a standard Fourier Neural Operator (FNO), and restores the prediction to the target frame. We train alignment and operator prediction jointly using bounded symmetry perturbations, with an optional low-dimensional refinement step that updates the estimated frame at inference. Equivariance is enforced by the input and output transformations, while the FNO architecture remains unchanged. Across 1-D and 2-D Burgers, shallow-water, and Navier-Stokes equations on periodic domains, PACE-FNO matches the in-distribution (ID) accuracy of standard neural operators and reduces out-of-distribution (OOD) relative error by up to 12x over FNO with symmetry augmentation (FNO+Aug) under translations and Galilean shifts, with smaller gains for coupled rotation-translation shifts. Ablations show that aligning the input and restoring the output frame account for most OOD gains; inference-time refinement provides a smaller correction.

PIP$^2$ Net: Physics-informed Partition Penalty Deep Operator Network

Operator learning has become a powerful tool for accelerating the solution of parameterized partial differential equations (PDEs), enabling rapid prediction of full spatiotemporal fields for new initial conditions or forcing functions. Existing architectures such as DeepONet and the Fourier Neural Operator (FNO) show strong empirical performance but often require large training datasets, lack explicit physical structure, and may suffer from instability in their trunk-network features, where mode imbalance or collapse can hinder accurate operator approximation. Motivated by the stability and locality of classical partition-of-unity (PoU) methods, we investigate PoU-based regularization techniques for operator learning and develop a revised formulation of the existing POU--PI--DeepONet framework. The resulting \emph{P}hysics-\emph{i}nformed \emph{P}artition \emph{P}enalty Deep Operator Network (PIP$^{2}$ Net) introduces a simplified and more principled partition penalty that improved the coordinated trunk outputs that leads to more expressiveness without sacrificing the flexibility of DeepONet. We evaluate PIP$^{2}$ Net on three nonlinear PDEs: the viscous Burgers equation, the Allen--Cahn equation, and a diffusion--reaction system. The results show that it consistently outperforms DeepONet, PI-DeepONet, and POU-DeepONet in prediction accuracy and robustness.