Reliable real-time estimation of atmospheric turbulence intensity remains an open challenge for aircraft operating across diverse altitude bands, particularly over oceanic, polar, and data-sparse regions that lack operational nowcasting infrastructure. Classical spectral models encode climatological averages rather than the instantaneous atmospheric state, and generic ML regressors offer adaptivity but provide no guarantee that predictions respect fundamental scaling laws. This paper introduces the Physically-Structured Turbulence Network (PSTNet), a lightweight architecture that embeds physics directly into its structure. PSTNet couples four components: (i) a zero-parameter backbone derived from Monin-Obukhov theory, (ii) a regime-gated mixture of specialist sub-networks supervised by Richardson-number-derived soft targets, (iii) Feature-wise Linear Modulation layers conditioning hidden representations on local air-density ratio, and (iv) a Kolmogorov output layer enforcing inertial-subrange scaling as an architectural constraint. The entire model contains only 552 learnable parameters, requiring fewer than 2.5 kB of storage and executing in under 12s on a Cortex-M7 microcontroller. We validate PSTNet on 340 paired six-degree-of-freedom guidance simulations spanning three vehicle classes (Mach 2.8, 4.5, and 8.0) and six operational categories with real-time satellite weather ingestion. PSTNet achieves a mean miss-distance improvement of +2.8% with a 78% win rate and a statistically significant effect size. Our results demonstrate that encoding domain physics as architectural priors yields a more efficient and interpretable path to turbulence estimation accuracy than scaling model capacity, establishing PSTNet as a viable drop-in replacement for legacy look-up tables in resource-constrained, safety-critical on-board guidance systems.
Figures & tables
Fig. 1: PSTNet architecture overview.
Fig. 2: Training loss convergence for PSTNet (552 parameters). The smooth, monotonic decay to a final MSE of 0.0063 reflects the strong inductive bias provided by the analytical backbone.
Fig. 3: Regime gate activation maps for the four PSTNet experts. Brighter regions indicate higher gate weight αj . The gating network recovers classical stability regimes: convective ( Ri<0 , low altitude), neutral ( Ri≈0 ), stable ( Ri>0.25 , mid-altitude), and stratospheric (high altitude, all Ri ).
Model
Params
Δ%
Win%
d
p
PSTNet (ours)
552
+2.8
78
+0.408
1.96×10−10
Vanilla MLP
627
+1.4
62
+0.185
2.1×10−3
Deep MLP (10 × )
6 819
+1.1
58
+0.142
1.8×10−2
GBT Ensemble
∼ 9 000
+0.9
55
+0.108
6.4×10−2 ns
Dryden Classical
0
+0.6
51
+0.071
1.9×10−1 ns
TABLE I: Overall results across 340 paired simulations
Fig. 4: Improvement (%) over the no-turbulence reference, broken down by scenario category (rows) and model (columns). PSTNet leads in five of six categories; the sole exception is edge cases (E), where the Vanilla MLP benefits from over-fitting to rare conditions.
Regime
Cohen’s d
p -value
Effect
Hypersonic Glide ( M=8.0 )
1.027
<0.001
Large
Supersonic ( M=2.8 )
0.813
<0.001
Large
High Supersonic ( M=4.5 )
0.456
<0.01
Medium
TABLE II: Per-regime effect sizes for PSTNet (paired t -test).
Fig. 5: Cohen’s d by vehicle type for all five models. PSTNet (blue) achieves the largest effect size in every regime. The red vertical line marks d=0 (no effect); shaded bands denote conventional small / medium / large thresholds.
Fig. 6: Screenshot of the PSTNet interactive demonstration ( https://pstnet.boriskriuk-powered.com ). Users select any geographic region on the global turbulence globe and any of the eight available flight levels to obtain live turbulence intensity estimates powered by NASA POWER satellite reanalysis data.
Atmospheric turbulence degrades image quality in long-range imaging. The degradation is generally modeled with two spatio-temporally varying components, tilt and blur. Recent unsupervised turbulence mitigation methods have demonstrated the effectiveness of representing a sharp image as an implicit neural representation (INR). However, their tilt and blur representations remain frame-dependent, and the number of optimization parameters grows with the number of input frames. To address this limitation, we propose TriINR, an unsupervised turbulence mitigation method that recovers a sharp image from multiple degraded frames. TriINR represents the sharp image, tilt, and blur as three implicit neural representations. By modeling tilt and blur as spatio-temporal INRs, TriINR achieves frame-count-independent degradation representation and memory-efficient optimization. We further incorporate physical and statistical properties of atmospheric turbulence as constraints on the INRs to improve restoration accuracy. Experiments on synthesized turbulence data with three degradation strengths show that TriINR restores images more robustly across varying turbulence strengths than prior TM methods and achieves the highest accuracy among the evaluated unsupervised methods. Our method also demonstrates robustness on real long-range scenes containing additional non-ideal motion.
Neural surrogates have emerged as fast alternatives to the numerical simulation of three-dimensional turbulence. However, training them at high resolution remains challenging, since the memory of full-field models grows with the resolution. In addition, full-resolution training data are expensive to simulate and store, and therefore scarce. We introduce ScaleSplit-NO (Scale-Split Neural Operator), which exploits the scale structure of turbulence with two neural operators: a Parent predicts the global coarse field at the next time step, and a Child predicts full-resolution local patches conditioned on this prediction. Neither model operates on the full-resolution field. The Child is pretrained alone and then attached to the Parent's coarse prediction through zero-initialized connections. On two complex high-resolution turbulence benchmarks, ScaleSplit-NO surpasses all competing baselines in both prediction accuracy and data efficiency. On the higher-resolution dataset JHTDB256 (2563), its normalized mean squared error (NMSE) is 53% lower than that of the strongest baseline, and its training memory is 79% lower than that of the most memory-efficient baseline. We further demonstrate its effectiveness for urban wind prediction in a real district of Montreal on a 500×150×500 grid, reducing one-step NMSE by 65.8% relative to the baseline. Moreover, swapping in a Parent trained on additional coarse fields improves prediction without retraining the Child, providing further accuracy gains at a small storage cost.
Shaoxiang Qin, Yucheng Zhao, Zongyi Li +2
University of Tennessee, Knoxville · McGill University · New York University +1
Physics-informed neural networks (PINNs) reduce or eliminate the need for large labelled datasets by enforcing governing equations during training. Physics-informed Deep Operator Networks (PI-DeepONets) extend this approach to learning solution operators, enabling rapid prediction across families of problems. Despite these advantages, we show that their application to higher-dimensional problems remains limited by the computational cost of training. To address this issue, we propose a modified PI-DeepONet architecture that reduces this cost by removing redundancy in how the Branch network processes input functions across collocation points. The proposed formulation is mathematically equivalent to the original architecture but reduces input tensor sizes and the number of Branch network evaluations. We analyse the computational complexity of both architectures and derive theoretical scaling laws for training time with respect to the number of collocation points and input functions. The proposed architecture is evaluated on 1D, 2D, and 3D linear advection problems. In all cases, it achieves comparable predictive accuracy to the original PI-DeepONet while consistently reducing training time. In 2D experiments, a maximum speed-up of 3.75× is observed, with empirical training-time scaling agreeing with the predicted linear dependence on collocation points and input functions. Furthermore, training a 3D model on an NVIDIA A100 GPU is only feasible using the proposed architecture, as the original formulation exceeds memory limits. These results demonstrate that careful consideration of how PI-DeepONets are implemented can yield substantial gains, enhancing the practical performance of PI-DeepONets and enabling their application to otherwise intractable physics-informed learning problems.
Nahum Banks, Gustavo A. M. de Almeida, Sergio Maldonado
Faculty of Engineering and Physical Sciences, University of Southampton, Southampton, UK