An arXiv preprint reports that a neural model called CoKo-UNO had the lowest mean relative L2 rollout error—the study’s main measure of how far predicted flow trajectories were from their references—across all four unsteady-flow benchmark problems it tested. Relative to the strongest baseline in each case, the reported reductions were 76.76% and 40.80% for the two Kolmogorov-flow cases, 17.32% for flow past a cylinder and 19.60% for shallow-water dam-break flow. In the ν=10−3 comparison with KNO2d, CoKo-UNO’s mean error was 1.09×10−2 instead of 7.63×10−2, an 85.71% reduction.
The main accuracy figures were reported as means with standard deviations from three independent training runs. The paper did not report formal p-values for those comparisons.
What the model is trying to fix
The model is built around a specific possibility: finite-dimensional Koopman truncation—a limited representation of flow dynamics—may leave a state-dependent residual, or leftover prediction error, that accumulates during an autoregressive rollout, in which each forecast is used to produce the next one. The study asks whether explicitly compensating for that residual can improve stability. CoKo-UNO combines a U-shaped spectral backbone and Koopman-dominated latent propagation with selective state-space/Mamba residual compensation, resolution-adaptive skip connections and an overlapping-warmup rollout.
The evaluation covered Kolmogorov flow at two viscosities, flow past a cylinder and shallow-water radial dam-break flow. Each of the four datasets provided 1,000 training samples and 200 test samples. Baselines were retrained under the same protocol: 200 epochs with Adam, an initial learning rate of 10−3 and weight decay of 10−4. Evaluation windows shared five time steps and used a stride of five, and the experiments were repeated with seeds 42, 2025 and 3407.
The error grew more slowly
On Kolmogorov flow at ν=10−3, CoKo-UNO also had the lowest listed fitted step-wise error-growth rate, γ=2.96×10−4. This rate summarizes how the relative L2 error changed as the rollout advanced, but the paper did not report an uncertainty estimate for it.
Tests that removed or changed parts of the design favored the full setup, although the reported ablation results were point estimates without uncertainty ranges. At 32×32 resolution, full compensation had a mean rollout error of 1.35×10−2, compared with 1.43×10−2 for partial compensation, a 5.6% reduction. Removing warmup raised the error to 2.15×10−2, while replacing partial compensation with an all-linear version produced 1.72×10−2 instead of 1.43×10−2, a 20.3% degradation. Against UNO T-batch, the reported full model’s error was 19.6% lower: 1.35×10−2 versus 1.68×10−2.
A mixed computational trade-off
Accuracy did not translate into a single cost advantage. Across the remaining tasks, CoKo-UNO’s average time per training epoch was about 41.40% of RNO’s, while its rollout errors were lower. In the Kolmogorov ν=10−3 complexity comparison, it used 43.47% less peak memory than FNO2d and had 64.76% lower per-step inference time than RNO, although it used more peak memory than RNO.
The paper also checked physical diagnostics alongside rollout error. For Kolmogorov flow at ν=10−4, the mean kinetic-energy discrepancy was 2.40×10−3 and the final-step discrepancy was 1.63×10−2. In the resolved spectral range 1≤k≤10, mean relative spectral discrepancy was 24.30% at ν=10−3 and 13.10% at ν=10−4. In the cylinder wake, mean velocity-magnitude error was approximately 6.30×10−3, staying near 5.80–5.90×10−3 during the first five steps and rising to about 6.70–6.80×10−3 later. For shallow-water flow, mean relative water-mass drift was about 4.20×10−5; it reached 6.00×10−5, then remained below 5.20×10−5 and ended at about 5.10×10−5 without monotonic accumulation.
The boundary of the result
The sharpest warning came from a resolution-transfer test. A model trained at 32×32 and evaluated without fine-tuning saw its relative L2 error rise from 1.43×10−2 at the training resolution to 2.87×10−1 at 64×64 and 4.21×10−1 at 128×128. The results therefore do not demonstrate zero-shot resolution generalization.
The document is identified as arXiv:2608.25879v1 in physics.flu-dyn and dated 26 August 2026. It says the research was supported by the Jiangxi Provincial Natural Science Foundation under grant S20254688.
Paper data and sources
Original title: A Compensated Koopman Neural Operator with Selective State-Space Dynamics for Unsteady Flows
Authors: Tangying Lv, Yuanjun Dai, Zhenxu Sun
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text