Autoregressive scientific forecasters often enforce physical or structural constraints by repairing each predicted state before feeding it back into the model. However, it remains unclear when stronger physical rule enforcement becomes reliable and when it becomes a source of distribution shift. We study this question through operator exactness, meaning whether the repair map is the identity on the target manifold and is aligned with the target geometry. We compare raw forecasting, post hoc repair, and in-loop repair across periodic incompressible Navier--Stokes, non-periodic CFDBench flows, and a hierarchical-forecasting support task. In the exact periodic regime, Fourier projection substantially improves rollout accuracy. On the NS-128 benchmark, a strong Raw-FNO has a final-step rollout MSE at horizon 100 of $(9.390 \pm 6.290)\times 10^{-5}$, and post hoc and in-loop projection reduce it to $(1.130 \pm 0.165)\times 10^{-6}$ and $(5.370 \pm 0.113)\times 10^{-7}$. However, once an exact projection is unavailable and only approximate boundary-preserving cleanup is available, the ordering changes. Across cavity, tube, dam, and cylinder flow, stronger Poisson-based cleanup can reduce divergence while worsening rollout error; target-distortion MSE predicts this harm far better than a linear-system residual. Controlled mismatch, screened cleanup, adaptive gating, and external-backbone checks show that the best approximate-regime operating point can be raw or near-identity. Hierarchical forecasting gives the same broader pattern. Exact forecast reconciliation is a stable baseline, whereas blended top-down repair, a validation-tuned interpolation toward historical-proportion top-down reconciliation, is dataset-dependent. Thus, constraint enforcement should be benchmarked by operator--data alignment before enforcement strength.

翻译：自回归科学预测模型常通过修复每个预测状态后将其反馈回模型来强制执行物理或结构约束。然而，目前尚不清楚何时更强物理规则执行变得可靠，何时会成为分布偏移的来源。我们通过算子精确性来研究这一问题，即修复映射是否在目标流形上为单位映射，并与目标几何结构对齐。我们比较了原始预测、事后修复和回路内修复在周期不可压缩纳维-斯托克斯方程、非周期CFDBench流以及层次预测支持任务中的表现。在精确周期情况下，傅里叶投影显著提升了滚动预测精度。在NS-128基准上，强原始FNO在100步预测时间水平上的最终步滚动均方误差为$(9.390 \pm 6.290)\times 10^{-5}$，而事后投影和回路内投影将其降低至$(1.130 \pm 0.165)\times 10^{-6}$和$(5.370 \pm 0.113)\times 10^{-7}$。然而，一旦无法获得精确投影，仅剩近似保边清理手段时，排序发生改变。在腔体流、管流、坝流和圆柱绕流中，更强的基于泊松方程的清理可降低散度，但会恶化滚动误差；目标畸变均方误差比线性系统残差能更好预测这种恶化。通过受控失配、筛选清理、自适应门控和外部骨干网络检查发现，近似机制下的最佳操作点可能是原始或近单位映射。层次预测呈现相同的宏观模式。精确预测协调是稳定基线，而混合自上而下修复（一种向历史比例自上而下协调插值的验证调优方法）则依赖于数据集。因此，约束执行应在强制执行强度之前通过算子-数据对齐进行基准测试。