研究物理约束修复的精确性如何影响预测精度，发现精确修复更优但需匹配数据分布。
Exactness Matters for Physical Rule Enforcement
- 通过修复算子的几何对齐度衡量物理规则强制的精确性
- 在周期流中精确投影使100步预测误差降低至5.37×10⁻⁷
- 非周期流中强修复可能引入偏差，需根据数据对齐评估强度
自回归科学预测模型常通过修复每一步预测状态来施加物理或结构约束。然而，强约束何时可靠、何时导致分布偏移尚不明确。本文通过算子精确性（即修复映射在目标流形上是否为恒等且与目标几何一致）探究此问题。对比原始预测、事后修复与循环内修复在周期不可压缩纳维-斯托克斯方程、非周期CFDBench流场及分层预测任务中的表现。在精确周期情形下，傅里叶投影显著提升滚动预测精度：在NS-128基准上，原始FNO的100步滚动均方误差为(9.390±6.290)×10⁻⁵，事后与循环内投影分别降至(1.130±0.165)×10⁻⁶和(5.370±0.113)×10⁻⁷。但当仅能进行近似边界保持清理时，效果反转——基于泊松的强清理虽减少发散，却恶化预测误差；目标畸变均方误差比线性系统残差更能预示损害。受控错配、筛选清理、自适应门控与外部骨干检查表明，最佳近似情形下的最优策略可能是原始或近恒等修复。分层预测亦呈现相同模式。精确预报校正为稳定基线，而验证调参的自上而下混合修复则依赖数据集。因此，约束强化应在强化前评估算子与数据的对齐程度。
原文摘要 · Abstract (English)
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.
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