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Exactness Matters for Physical Rule Enforcement

NeurIPS 2026 · Probabilistic and scientific modeling › Dynamical systems · approach: Data assimilation from sparse observations · 0 citations (OpenAlex)

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Shows that in-loop/post hoc constraint repair helps greatly when the projector is exact (periodic NS-128 Fourier projection) but can hurt in approximate boundary-aware cleanup regimes (CFDBench), with target-distortion predicting harm better than solver residual.

Problem

When does enforcing physical constraints (e.g., divergence-free) via repair of each autoregressive prediction improve rollout accuracy, versus becoming a source of distribution shift when the repair operator is only approximate?

Approach

Frames constraint-repair reliability through operator exactness on the target manifold: exact projection (Fourier) is near-identity on valid targets and helps; approximate boundary-preserving Poisson cleanup distorts targets, so stronger solves worsen rollouts. Proposes operator-data alignment as the benchmarking criterion and a divergence-gated adaptive cleanup plus validation-based selection of

Key results

• On NS-128 at horizon 100, Raw-FNO final-step rollout MSE is (9.390±6.290)e-5, reduced to (1.130±0.165)e-6 by post hoc projection and (5.370±0.113)e-7 by in-loop projection.
• Across 12 approximate-regime sweep points, ground-truth target distortion MSE correlates with rollout harm (Pearson 0.905, Spearman 0.909), while Poisson residual is negatively correlated (-0.387 Pearson).
• On the official-split Dam@60 and Cylinder@60, validation selects the raw forecast; Direct cleanup is never selected in the four validation-selected deployment rows.

Surprising findings

• Stronger approximate Poisson solves (e.g., Direct) reduce linear-system residual to near zero yet increase target distortion and worsen rollout error.
• In the approximate regime, the best operating point is often raw or a very mild blend rather than full cleanup.

Limitations

• Evidence strongest for 2D incompressible forecasting; hierarchical study is supporting only.
• Approximate-regime findings rely on boundary-aware Poisson cleanup family; other operators not exhaustively covered.
• Adaptive gating thresholds tuned per dataset on validation split.

Experimental setup

Datasets: periodic NS-128 CFDBench (cavity, tube, dam, cylinder) Traffic Labour TourismLarge OldTourismLarge Wiki2

Benchmarks: NS-128 CFDBench cavity CFDBench tube CFDBench dam CFDBench cylinder HierarchicalForecast hierarchies

Metrics: test MSE rollout MSE final-step MSE@T rollout divergence stable rollout length target-distortion MSE Pearson correlation Spearman correlation

Baselines: Raw-FNO PostHoc-FNO SoftDiv-FNO Jacobi MG SOR CG Direct PostHoc-Screened AdaBlend

Builds on: Fourier neural operator Chorin-Temam projection forecast reconciliation (OLS, minimum trace) CFDBench

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