Abstract

Known Lie symmetries specify how a partial differential equation (PDE) solution operator should transform under group actions, but finite supervised training data constrain this relationship only at observed input--output pairs. To encourage equivariance without changing the architecture, we introduce \textbf{LOCO} (\textbf{L}ocal \textbf{O}rbit \textbf{Co}nsistency), a lightweight and flexible training objective for neural operators. LOCO samples a small Lie-group transformation during training and explicitly penalizes the mismatch between the operator's prediction on the transformed input and the transformed prediction of the original input. By leveraging known input and output actions, LOCO leaves the underlying neural-operator architecture unchanged and introduces no computational overhead at inference time. We evaluate our method on a periodic $64 \times 64$ two-dimensional Navier--Stokes Galilean benchmark. With 2\% of the training examples labeled, combining data augmentation with LOCO ($\lambda_{\mathrm{orb}} = 0.30$) reduces the orbit out-of-distribution (OOD) relative $L_2$ error by 35.3\% and the equivariance defect by 46.4\% compared to a supervised augmentation baseline matched in the number of model forward evaluations. Furthermore, a lower-weight setting, $\lambda_{\mathrm{orb}} = 0.10$, yields consistent performance gains across varying label fractions and OOD boost radii. Finally, solver-level closure residuals remain negligible, and action-mismatch ablations support the role of the prescribed Galilean input--output action in the observed gains.