When truth arrives late:
observing Navier–Stokes flow through a delayed channel
A reproducible numerical study
1Introduction
A measurement may be accurate at acquisition and stale at delivery. For a changing state, sampling interval and transport delay therefore control different sources of error. We isolate these effects in a numerically solved flow, where the full reference state is available for evaluation.
Reconstruction from sparse observations is an established problem in Navier–Stokes data assimilation [1–3]. The present experiment supplies a reproducible sample-and-hold baseline. Its interactive figure permits independent changes to reporting interval and delivery delay.
2Flow and observation model
On the periodic square [0, 2π)², vorticity evolves according to the two-dimensional incompressible Navier–Stokes equations,
The reference uses a 48 × 48 Fourier grid, strict 2/3 dealiasing, viscosity ν = 0.01 and fourth-order Runge–Kutta integration with step 0.01. The forcing contains a steady component and two prescribed bursts. The observer starts from the exact initial condition, receives full-state snapshots and holds its most recently delivered state between updates.
(a) True state
Current vorticity, ω(x, t)
(b) Received state
Most recently delivered snapshot
(c) Difference
Current field minus received field
| Quantity | Value | Definition |
|---|---|---|
| Current state error | — | Velocity error / reference velocity RMS |
| Mean tracking error | — | Time-integrated RMS |
| Error-increasing updates | — | Deliveries that increased instantaneous squared error |
3Error at delivery
Let x denote the current true state, a the previously held state and b the arriving snapshot. Holding x fixed at delivery, the change in E = ½‖x − a‖² is exactly
For a current exact measurement, b = x and ΔE ≤ 0. A delayed measurement has no universal sign guarantee. Equation (2) is an elementary identity used to classify delivery events; it is not a new theorem or an implementable acceptance rule when x is unknown.
4Pilot results
Table 1 reports integrated velocity error normalized by the first-half reference velocity RMS. All three runs use the same forcing schedule. Reporting counts exclude the supplied initial state.
Table 1. Relative integrated RMS error. Columns specify reporting interval h and delay ℓ, in nondimensional time units. Integration is checked at the 0.01 solver step.
| Seed | h = 1, ℓ = 1 | h = 0.1, ℓ = 1 | h = 0.1, ℓ = 0 |
|---|---|---|---|
| 7 | 29.15% | 23.45% | 1.54% |
| 19 | 29.74% | 23.88% | 1.56% |
| 41 | 32.49% | 25.15% | 1.57% |
The interactive figure integrates error between saved states using linear interpolation. The full-precision experiment also checks the finer solver grid. This accounts for error between deliveries, including when displayed states coincide immediately after a zero-delay update.
4.1Partial observations and causal reconstruction
A separate development screen compares six causal estimators from a zero initial estimate. Observations contain only low Fourier modes and controlled noise. Gains are selected on two development trajectories, then fixed for three reserved trajectories under a different forcing family. Each evaluation trajectory has twelve combinations of observation resolution, transport delay and model mismatch.
Table 2. Mean integrated velocity error in the partial-observation screen. Normalization uses initial reference velocity RMS, with a two-unit burn-in. These 36 condition cells are correlated; they are not 36 independent trajectories. The candidate mean excludes its two failed runs.
| Estimator | Mean RMS | Failures |
|---|---|---|
| Sample-and-hold | 22.75% | 0 / 36 |
| Linear extrapolation | 25.10% | 0 / 36 |
| Free model | 100.35% | 0 / 36 |
| Stale insertion | 15.42% | 0 / 36 |
| Chronological replay | 7.46% | 0 / 36 |
| First-order candidate | 28.73% | 2 / 36 |
The first-order candidate is rejected by this screen: it fails in two cells and worsens sampled tail error relative to stale insertion in every finite paired case. Chronological replay is a stronger established baseline, not a new contribution. Its reference implementation performs extra, partially redundant integration, so these results do not establish an optimized compute frontier.
The screen uses a 32 × 32 grid and step 0.02. A representative refinement changes final vorticity by 5.38% at 48 × 48, while the velocity-error scores are less sensitive. This is preliminary evidence requiring a full resolution sweep, broader regimes and stronger published baselines before submission. Source, frozen protocol, raw outcomes and an English working manuscript are stored in research/assimilation.
4.2Efficient reconstruction with a finer reference
A second study uses a 64 × 64 reference and a 48 × 48 fine observer, with two new forcing families and four reserved trajectories. Cached replay reuses previously computed states. A split candidate retains the fine analysis while forecasting on a 32 × 32 grid and allowing the remaining high modes to decay viscously. A simpler 32 × 32 replay provides a second cost baseline. These partial-observation estimators are evaluated separately from the full-state hold demonstration in Figure 1.
Table 3. Thirty-two correlated condition cells from four trajectories. RMS is normalized by initial reference velocity RMS; p95 ratios compare each cell with fine replay, and the worst ratio is shown. Speedup is the ratio of mean observer execution times on this machine, excluding truth scoring. All methods have zero failed runs.
| Estimator | Mean RMS | Worst p95 ratio | Speedup |
|---|---|---|---|
| Fine cached replay | 9.686% | 1.0000 | 1.00× |
| Stale insertion | 21.704% | 19.5262 | 2.48× |
| Coarse replay | 9.714% | 1.0574 | 1.74× |
| Fine analysis / coarse forecast | 9.705% | 1.0494 | 1.05× |
The split candidate meets the prewritten 5% RMS and sampled-tail tolerances in 32/32 cells, while coarse replay meets them in 30/32. Its mean execution-time reduction is approximately 5%, compared with approximately 43% for the simpler coarse method. This is a modest accuracy–cost tradeoff, not evidence of a major computational breakthrough. Both methods use grids selected on development data only.
The nearest threshold case was checked after evaluation at model steps 0.02, 0.01 and 0.005. The split p95 ratios were 1.04940, 1.04847 and 1.04799; the coarse method remained above 1.05. This is a post-hoc sensitivity diagnostic, not independent confirmation. A prescribed 64→96 truth refinement changed final velocity by approximately 0.00096% in one representative case.
The complete v0.2 working manuscript, raw outcomes, repeated timing checks and an elementary cost/residual analysis are in research/assimilation_v2. Replay, reduced models and multiple fidelities are established ideas. Publication still requires a precise contribution relative to prior work and broader validation.
4.3Oracle-specific claim audit
A fixed-delay frequency audit on ETH, BTC and LINK found no whole-window deterioration in 450 nested reporting-policy comparisons. Faster reporting halves the interval; prescribed delays range from zero to eight minutes. The overlapping configurations use previously inspected development data and are not independent trials. These hypothetical policies do not reproduce Chainlink's operational triggers.
Among 4,940 recorded oracle updates, 428 increase squared log-price error throughout the corresponding Binance minute high–low band. Widening that reference band by ±10 log-price basis points reduces the count to 19; ±25 reduces it to 2. These are sensitivity assumptions, not confidence bounds or certified fair values. Minute bars, exchange disagreement and USD/USDT basis prevent an unconditional claim that the updates were harmful.
An exact forced Navier–Stokes shear supplies a mathematical counterexample: four times as many delayed reports yield 3.27324 times the mean squared error for a fixed hold observer. This phase-locked laminar construction establishes possibility, not typical market behavior or a novel theorem. The market-paper claim that more updates generally worsen accuracy is therefore unsupported by this audit. Source, protocol, derivation and full results are in research/oracle_audit.
5Validation and limitations
Verification includes Taylor–Green viscous decay, zero divergence, the semidiscrete energy identity, unforced energy decay, and time-step and spatial refinement. For seed 7 at the final time, halving the step changes vorticity by 8.01 × 10⁻⁹ relative; refining from 48 to 64 grid points changes it by approximately 0.104% after projection.
This is a small resolved two-dimensional experiment. The observer does not reconstruct unobserved modes or integrate a recovery equation. A stronger study must compare causal assimilation methods under partial observations, noise and model mismatch, with separate development and evaluation regimes.
The ETH replay uses Binance ETHUSDT minute opens and Chainlink ETH/USD rounds, assuming USD/USDT parity. It is a selected historical illustration. The controls affect a hypothetical reporting policy; they do not change recorded Chainlink data or establish financial predictive skill.
6Reproducibility
Run python3 research/observability/build.py to regenerate the experiment, and npm run lab:test for its numerical and channel tests. Full-precision results, solver parameters and the next experimental protocol accompany the project.
References
- [1]K. Hayden, E. Olson and E. S. Titi. Discrete data assimilation in the Lorenz and 2D Navier–Stokes equations. Physica D, 2011.
- [2]A. Azouani, E. Olson and E. S. Titi. Continuous data assimilation using general interpolant observables. Journal of Nonlinear Science, 2014.
- [3]C. Foias, C. F. Mondaini and E. S. Titi. A discrete data assimilation scheme for the solutions of the 2D Navier–Stokes equations and their statistics. SIAM Journal on Applied Dynamical Systems, 2016.