Abstract: Turbulence, a complex multi-scale phenomenon inherent in fluid flow systems, presents critical challenges and opportunities for understanding physical mechanisms across scientific and engineering domains. Although high-resolution (HR) turbulence data remain indispensable for advancing both theoretical insights and engineering solutions, their acquisition is severely limited by prohibitively high computational costs. While deep learning architectures show transformative potential in reconstructing high-fidelity flow representations from sparse measurements, current methodologies suffer from two inherent constraints: strict reliance on perfectly paired training data and inability to perform multi-scale reconstruction within a unified framework. To address these challenges, we propose HADF, a hash-adaptive dynamic fusion implicit network for turbulence reconstruction. Specifically, we develop a low-resolution (LR) consistency loss that facilitates effective model training under conditions of missing paired data, eliminating the conventional requirement for fully matched LR and HR datasets. We further employ hash-adaptive spatial encoding and dynamic feature fusion to extract turbulence features, mapping them with implicit neural representations for reconstruction at arbitrary resolutions. Experimental results demonstrate that HADF achieves superior performance in global reconstruction accuracy and local physical properties compared to state-of-the-art models. It precisely recovers fine turbulence details for partially unpaired data conditions and diverse resolutions by training only once while maintaining robustness against noise.

HADF：面向湍流重构的哈希自适应双融合隐式网络

[1]Bode M, Gauding M, Lian ZY, et al., 2021. Using physics-informed enhanced super-resolution generative adversarial networks for subfilter modeling in turbulent reactive flows. Proc Combust Inst, 38(2):2617-2625.

[2]Brunton SL, Noack BR, Koumoutsakos P, 2020. Machine learning for fluid mechanics. Annu Rev Fluid Mech, 52(1):477-508.

[3]Cant RS, Ahmed U, Fang J, et al., 2022. An unstructured adaptive mesh refinement approach for computational fluid dynamics of reacting flows. J Comput Phys, 468:111480.

[4]Chen XH, Chen RL, Wan Q, et al., 2021a. An improved data-free surrogate model for solving partial differential equations using deep neural networks. Sci Rep, 11(1):19507.

[5]Chen XH, Gong CY, Wan Q, et al., 2021b. Transfer learning for deep neural network-based partial differential equations solving. Adv Aerodynam, 3:36.

[6]Chen XH, Wang ZC, Deng L, et al., 2024. Towards a new paradigm in intelligence-driven computational fluid dynamics simulations. Eng Appl Comput Fluid Mech, 18(1):2407005.

[7]Chen YB, Liu SF, Wang XL, 2021. Learning continuous image representation with local implicit image function. Proc IEEE/CVF Conf on Computer Vision and Pattern Recognition, p.8624-8634.

[8]Chung WT, Mishra AA, Ihme M, 2022. Interpretable data-driven methods for subgrid-scale closure in LES for transcritical LOX/GCH4 combustion. Combust Flame, 239:111758.

[9]Chung WT, Ihme M, Jung KS, et al., 2023a. BLASTNet simulation dataset (2.0). Zenodo.

[10]Chung WT, Akoush B, Sharma P, et al., 2023b. Turbulence in focus: benchmarking scaling behavior of 3D volumetric super-resolution with BLASTNet 2.0 data. Proc 37^th Int Conf on Neural Information Processing Systems, p.77430-77484.

[11]Davidson P, 2015. Turbulence: an Introduction for Scientists and Engineers (2^nd Ed.). Oxford University Press.

[12]Erichson NB, Mathelin L, Yao ZW, et al., 2020. Shallow neural networks for fluid flow reconstruction with limited sensors. Proc Roy Soc A, 476(2238):20200097.

[13]Fukami K, Taira K, 2024. Single-snapshot machine learning for super-resolution of turbulence. J Fluid Mech, 1001:A32.

[14]Fukami K, Taira K, 2025. Observable-augmented manifold learning for multi-source turbulent flow data. J Fluid Mech, 1010:R4.

[15]Fukami K, Fukagata K, Taira K, 2019. Super-resolution reconstruction of turbulent flows with machine learning. J Fluid Mech, 870:106-120.

[16]Fukami K, Fukagata K, Taira K, 2023. Super-resolution analysis via machine learning: a survey for fluid flows. Theor Comput Fluid Dynam, 37(4):421-444.

[17]Fukami K, Goto S, Taira K, 2024. Data-driven nonlinear turbulent flow scaling with Buckingham Pi variables. J Fluid Mech, 984:R4.

[18]Jung KS, Kim SO, Lu TF, et al., 2021. On the flame stabilization of turbulent lifted hydrogen jet flames in heated coflows near the autoignition limit: a comparative DNS study. Combust Flame, 233:111584.

[19]Liu HC, Sun B, Cai WW, 2019. kHz-rate volumetric flame imaging using a single camera. Opt Commun, 437:33-43.

[20]Liu N, Ma L, 2020. Regularized tomographic PIV for incompressible flows based on conservation of mass. Appl Opt, 59(6):1667-1677.

[21]Morimoto M, Fukami K, Zhang K, et al., 2021. Convolutional neural networks for fluid flow analysis: toward effective metamodeling and low dimensionalization. Theor Comput Fluid Dynam, 35(5):633-658.

[22]Müller T, Evans A, Schied C, et al., 2022. Instant neural graphics primitives with a multiresolution hash encoding. ACM Trans Graph, 41(4):102.

[23]Nguyen QH, Beksi WJ, 2023. Single image super-resolution via a dual interactive implicit neural network. Proc IEEE/CVF Winter Conf on Applications of Computer Vision, p.4925-4934.

[24]Page J, 2025. Super-resolution of turbulence with dynamics in the loss. J Fluid Mech, 1002:R3.

[25]Pan SW, Brunton SL, Kutz JN, 2023. Neural implicit flow: a mesh-agnostic dimensionality reduction paradigm of spatio-temporal data. J Mach Learn Res, 24(1):1607-1666.

[26]Pareja J, Johchi A, Li T, et al., 2019. A study of the spatial and temporal evolution of auto-ignition kernels using time-resolved tomographic OH-LIF. Proc Combust Inst, 37(2):1321-1328.

[27]Park K, Sinha U, Hedman P, et al., 2021a. HyperNerf: a higher-dimensional representation for topologically varying neural radiance fields. ACM Trans Graph, 40(6):238.

[28]Park K, Sinha U, Barron JT, et al., 2021b. Nerfies: deformable neural radiance fields. Proc IEEE/CVF Int Conf on Computer Vision, p.5845-5854.

[29]Savard B, Hawkes ER, Aditya K, et al., 2019. Regimes of premixed turbulent spontaneous ignition and deflagration under gas-turbine reheat combustion conditions. Combust Flame, 208:402-419.

[30]Shen LY, Pauly J, Xing L, et al., 2024. NeRP: implicit neural representation learning with prior embedding for sparsely sampled image reconstruction. IEEE Trans Neur Netw Learn Syst, 35(1):770-782.

[31]Sitzmann V, Martel JNP, Bergman AW, et al., 2020. Implicit neural representations with periodic activation functions. Proc 34^th Int Conf on Neural Information Processing Systems, p.7462-7473.

[32]Wang B, Hui X, Li F, et al., 2022. Research status and key technology analysis of dynamic combustion in aero-engine main combustor. J Aerosp Power, 37(11):2479-2487 (in Chinese).

[33]Xu Q, Zhuang ZJ, Pan YC, et al., 2023. Super-resolution reconstruction of turbulent flows with a Transformer-based deep learning framework. Phys Fluids, 35(5):055130.

[34]Xu WJ, Luo WY, Wang Y, et al., 2020. Data-driven three-dimensional super-resolution imaging of a turbulent jet flame using a generative adversarial network. Appl Opt, 59(19):5729-5736.

[35]Yang Z, Yang H, Yin ZP, 2023. Super-resolution reconstruction for the three-dimensional turbulence flows with a back-projection network. Phys Fluids, 35(5):055123.

[36]Zhang KW, Zhu DD, Min XK, et al., 2023. Implicit neural representation learning for hyperspectral image super-resolution. IEEE Trans Geosci Remote Sens, 61:5500212.