Accelerated physics-based simulations of train aerodynamics using machine learning libraries

Abstract

This paper presents the application of a novel AI-based approach, Neural Physics, to produce high-fidelity simulations of train aerodynamics. Neural Physics is built upon convolutional neural networks (CNNs), where the weights are explicitly determined by classical numerical discretisation schemes rather than by training. By leveraging the power of AI technology, this recent approach results in code that can run easily on GPUs and AI processors, achieving high computational speed without sacrificing accuracy. The approach uses an implicit large eddy simulation method based on a non-linear Petrov-Galerkin method to model the unresolved turbulence. Furthermore, for higher-order finite elements, the convolutional finite element method (ConvFEM) is used, which greatly simplifies the implementation of higher-order elements within the NN4DPEs approach. We demonstrate the capability of Neural Physics by simulating a freight Locomotive Class 66 and a partially loaded freight train operating in an open field environment with and without cross wind. This is the first time that ConvFEM has been applied to high-speed fluid flow problems in complex geometries. The results are validated against existing numerical results and experimental measurements, and show good agreement in terms of pressure and velocity distributions around the train body.

Keywords

numerical solutions of partial differential equations(PDEs) / convolutional neural network / computational fluid dynamics / train aerodynamics / graphics processing unit(GPU) / AI processors

References

[1] DFT. Delivering a sustainable railway, 2007, London, UK, Department for TransportVol. 1 [R]

[2] Baker C J, Dalley S J, Johnson Tet al.. The slipstream and wake of a high-speed train [J]. Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit, 2001, 215(2): 83-99

[3] Baker C J, Quinn A, Sima Met al.. Full-scale measurement and analysis of train slipstreams and wakes. Part 1: Ensemble averages [J]. Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit, 2014, 228(5): 451-467

[4] Wang L, Liu T-h, Chen Z-wet al.. Evaluation of the slipstream in different regions around a train with respect to different nose lengths: A comparison study [J]. Journal of Central South University, 2024, 31(9): 3295-3311

[5] Huang Z-x, Li W-h, Chen L. Effects of the Reynolds number on train aerodynamics considering air compressibility: A wind tunnel study [J]. Transportation Safety and Environment, 2024, 6(4): tdae006

[6] Baker C, Jordan S, Gilbert Tet al.. Transient aerodynamic pressures and forces on trackside and overhead structures due to passing trains. Part 1: Model-scale experiments; Part 2: Standards applications [J]. Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit, 2014, 228137-70

[7] Zhou D, Xia C-j, Wu L-let al.. Effect of the wind speed on aerodynamic behaviours during the acceleration of a high-speed train under crosswinds [J]. Journal of Wind Engineering and Industrial Aerodynamics, 2023, 232: 105287

[8] Zeng J-w, Yang M-z, Zhang Let al.. Structural dynamic responses evaluation of pedestrian bridge under effect of aerodynamic disturbance of high-speed train [J]. International Journal of Numerical Methods for Heat & Fluid Flow, 2025, 35(10): 3664-3684

[9] Carassale L, Marrè Brunenghi M. Dynamic response of trackside structures due to the aerodynamic effects produced by passing trains [J]. Journal of Wind Engineering and Industrial Aerodynamics, 2013, 123: 317-324

[10] Liu T-h, Wang L, Gao H-ret al.. Research progress on train operation safety in Xinjiang railway under wind environment [J]. Transportation Safety and Environment, 2022, 4(2): tdac005

[11] Xiong X-h, Li A-h, Liang X-fet al.. Field study on high-speed train induced fluctuating pressure on a bridge noise barrier [J]. Journal of Wind Engineering and Industrial Aerodynamics, 2018, 177: 157-166

[12] Niu J-q, Zhang Y-c, Li Ret al.. Aerodynamic simulation of effects of one- and two-side windbreak walls on a moving train running on a double track railway line subjected to strong crosswind [J]. Journal of Wind Engineering and Industrial Aerodynamics, 2022, 221: 104912

[13] Gilbert T, Baker C, Quinn A. Aerodynamic pressures around high-speed trains: The transition from unconfined to enclosed spaces [J]. Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit, 2013, 227(6): 609-622

[14] Guo D-l, Shang K-m, Zhang Yet al.. Influences of affiliated components and train length on the train wind [J]. Acta Mechanica Sinica, 2016, 32(2): 191-205

[15] Yang Z-y, Xu G, Wu Fet al.. The influence of the leading-edge angle of subgrade on the aerodynamic loads of a high-speed train in a wind tunnel [J]. Transportation Safety and Environment, 2024, 6(2): tdad020

[16] Tang L-b, He X-h, Yan Let al.. Experimental study of aerodynamic characteristics of high-speed train on bridge-tunnel junctions under crosswinds [J]. Journal of Central South University, 2023, 302613-624

[17] Schetz J A. Aerodynamics of high-speed trains [J]. Annual Review of Fluid Mechanics, 2001, 33: 371-414

[18] Zhang J, Ding Y-s, Wang Y-het al.. A novel bionic Coleoptera pantograph deflector for aerodynamic drag reduction of a high-speed train [J]. Journal of Central South University, 2023, 30(6): 2064-2080

[19] Jiang C, Long J-l, Li Y-set al.. Numerical investigation on the aerodynamic drag reduction based on bottom deflectors and streamlined bogies of a high-speed train [J]. Journal of Central South University, 2024, 31(9): 3312-3328

[20] Liu Z, Soper D, Hemida Het al.. A study of the influence of separation bubbles around a generic freight train on pressure waves inside tunnels using 1D and 3D numerical methods [J]. Journal of Wind Engineering and Industrial Aerodynamics, 2023, 240: 105461

[21] Soper D. The aerodynamics of a container freight train[M], 2016

[22] Bell J R, Burton D, Thompson M C. The boundary-layer characteristics and unsteady flow topology of full-scale operational inter-modal freight trains [J]. Journal of Wind Engineering and Industrial Aerodynamics, 2020, 201: 104164

[23] Soper D, Baker C, Sterling M. Experimental investigation of the slipstream development around a container freight train using a moving model facility [J]. Journal of Wind Engineering and Industrial Aerodynamics, 2014, 135: 105-117

[24] Liu Z, Soper D, Hemida H. Effect of separation bubbles around the head of a freight train on pressure waves inside tunnels using 1D and 3D numerical methods [C]. Proceedings of the Computational Challenges in Railways (CCC), 2022

[25] Iliadis P, Soper D, Baker Cet al.. Experimental investigation of the aerodynamics of a freight train passing through a tunnel using a moving model [J]. Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit, 2019, 233(8): 857-868

[26] Giappino S, Melzi S, Tomasini G. High-speed freight trains for intermodal transportation: Wind tunnel study on the aerodynamic coefficients of container wagons [J]. Journal of Wind Engineering and Industrial Aerodynamics, 2018, 175: 111-119

[27] Li C, Burton D, Kost Met al.. Flow topology of a container train wagon subjected to varying local loading configurations [J]. Journal of Wind Engineering and Industrial Aerodynamics, 2017, 169: 12-29

[28] Maleki S, Burton D, Thompson M C. Flow structure between freight train containers with implications for aerodynamic drag [J]. Journal of Wind Engineering and Industrial Aerodynamics, 2019, 188: 194-206

[29] Jan Ö, Krajnović S. A study of the aerodynamics of a generic container freight wagon using Large-Eddy Simulation [J]. Journal of Fluids and Structures, 2014, 44: 31-51

[30] Hemida H, Baker C, Gao G-j. The calculation of train slipstreams using large-eddy simulation [J]. Proceedings of the Institution of Mechanical Engineers, Part F: Journal of Rail and Rapid Transit, 2014, 228(1): 25-36

[31] Flynn D, Hemida H, Baker C. On the effect of crosswinds on the slipstream of a freight train and associated effects [J]. Journal of Wind Engineering and Industrial Aerodynamics, 2016, 156: 14-28

[32] Maleki S, Burton D, Thompson M C. Assessment of various turbulence models (ELES, SAS, URANS and RANS) for predicting the aerodynamics of freight train container wagons [J]. Journal of Wind Engineering and Industrial Aerodynamics, 2017, 170: 68-80

[33] LIU Zhen, SOPER D, HEMIDA H, et al. Numerical modelling of a partially loaded intermodal container freight train passing through a tunnel [EB/OL]. [2025-09-11]. https://arxiv.org/abs/2509.09591.

[34] Bell J R, Burton D, Thompson M Cet al.. Flow topology and unsteady features of the wake of a generic highspeed train [J]. Journal of Fluids and Structures, 2016, 61: 168-183

[35] Hemida H, Baker C. Large-eddy simulation of the flow around a freight wagon subjected to a crosswind [J]. Computers & Fluids, 2010, 39(10): 1944-1956

[36] He M-z, Huo S, Hemida Het al.. Detached eddy simulation of a closely running lorry platoon [J]. Journal of Wind Engineering and Industrial Aerodynamics, 2019, 193: 103956

[37] Flynn D, Hemida H, Soper Det al.. Detached-eddy simulation of the slipstream of an operational freight train [J]. Journal of Wind Engineering and Industrial Aerodynamics, 2014, 132: 1-12

[38] Brunton S L, Noack B R, Koumoutsakos P. Machine learning for fluid mechanics [J]. Annual Review of Fluid Mechanics, 2020, 52: 477-508

[39] Duraisamy K, Iaccarino G, Xiao H. Turbulence modeling in the age of data [J]. Annual Review of Fluid Mechanics, 2019, 51: 357-377

[40] Paszke A, Gross S, Massa Fet al.. PyTorch: An imperative style, high-performance deep learning library [M/OL]. Advances in Neural Information Processing Systems 32, 2019, Red Hook, NY, USA, Curran Associates, Inc.80248035

[41] Abadi M, Barham P, Chen Jet al.. Tensorflow: A system for large-scale machine learning [C/OL]. 12th USENIX Symposium on Operating Systems Design and Implementation (OSDI 16), 2016265283

[42] Pedregosa F, Varoquaux G, Gramfort Aet al.. Scikit-learn: Machine learning in Python [J/OL]. Journal of Machine Learning Research, 2011, 12: 2825-2830

[43] Kochkov D, Smith J A, Alieva Aet al.. Machine learning-accelerated computational fluid dynamics [J]. Proceedings of the National Academy of Sciences of the United States of America, 2021, 118(21): e2101784118

[44] Thuerey N, Weißenow K, Prantl Let al.. Deep learning methods for Reynolds-averaged navier-stokes simulations of airfoil flows [J]. AIAA Journal, 2020, 58125-36

[45] Heaney C E, Wolffs Z, Tómasson J Aet al.. An AI-based non-intrusive reduced-order model for extended domains applied to multiphase flow in pipes [J]. Physics of Fluids, 2022, 34(5): 055111

[46] Benner P, Gugercin S, Willcox K. A survey of projection-based model reduction methods for parametric dynamical systems [J]. SIAM Review, 2015, 574483-531

[47] Raissi M, Perdikaris P, Karniadakis G E. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations [J]. Journal of Computational Physics, 2019, 378: 686-707

[48] BRANDSTETTER J, WORRALL D, WELLING M. Message passing neural PDE solvers [A/OL]. arXiv (2022). https://arxiv.org/abs/2202.03376. DOI: https://doi.org/10.48550/arXiv.2202.03376.

[49] Zhao X-z, Xu T-y, Ye Z-tet al.. A TensorFlow-based new high-performance computational framework for CFD [J]. Journal of Hydrodynamics, 2020, 32(4): 735-746

[50] Wang Q, Ihme M, Chen Y-fet al.. A TensorFlow simulation framework for scientific computing of fluid flows on tensor processing units [J]. Computer Physics Communications, 2022, 274: 108292

[51] Wang Y-t, Sun Z-x, Ju S-jet al.. Robust optimisation of the streamlined shape of a high-speed train in crosswind conditions [J]. Engineering Applications of Computational Fluid Mechanics, 2023, 17: 2234012

[52] Chen X-j, Yin B, Yuan Zet al.. Data-driven learning algorithm to predict full-field aerodynamics of large structures subject to crosswinds [J]. Physics of Fluids, 2024, 36(5): 057105

[53] Han S, Xiang N-s, Huang F-yet al.. On reducing high-speed train slipstream using vortex generators [J]. Physics of Fluids, 2025, 375055115

[54] CHEN Bo-yang, HEANEY C E, PAIN C C. Using AI libraries for incompressible computational fluid dynamics [EB/OL]. [2024-02-27]. https://arxiv.org/abs/2402.17913.

[55] Chen B-y, Heaney C E, Gomes J L M Aet al.. Solving the discretised multiphase flow equations with interface capturing on structured grids using machine learning libraries [J]. Computer Methods in Applied Mechanics and Engineering, 2024, 426: 116974

[56] Chen B-y, Nadimy A, Heaney C Eet al.. Solving the discretised shallow water equations using neural networks [J]. Advances in Water Resources, 2025, 197: 104903

[57] Nadimy A, Chen B-y, Chen Z-met al.. Solving the discretised shallow water equations using non-uniform grids and machine-learning libraries [J]. Environmental Modelling & Software, 2026, 196: 106752

[58] Phillips T R F, Heaney C E, Chen B-yet al.. Solving the discretised neutron diffusion equations using neural networks [J]. International Journal for Numerical Methods in Engineering, 2023, 124214659-4686

[59] PHILLIPS T R F, HEANEY C E, CHEN B, et al. Solving the discretised boltzmann transport equations using neural networks: applications in neutron transport [A/OL]. arXiv (2023). DOI: https://doi.org/10.48550/arXiv.2301.09991.

[60] Naderi S, Chen B-y, Yang T-aet al.. A discrete element solution method embedded within a Neural Network [J]. Powder Technology, 2024, 448: 120258

[61] Li L-f, Xiang J-s, Chen B-yet al.. Implementing the discontinuous-Galerkin finite element method using graph neural networks with application to diffusion equations [J]. Neural Networks, 2025, 185: 107061

[62] Niu J-q, Wang Y-m, Liu Fet al.. Numerical study on comparison of detailed flow field and aerodynamic performance of bogies of stationary train and moving train [J]. Vehicle System Dynamics, 2021, 59(12): 1844-1866

[63] Xu B, Liu T-h, Xia Y-tet al.. Computational fluid dynamics prediction of the aerodynamic difference between stationary and moving trains [J]. Alexandria Engineering Journal, 2023, 70: 685-699

[64] NVIDIA. NVIDIA Collective Communications Library (NCCL) [EB/OL]. [2025-04-07]. https://developer.nvidia.com/nccl.

[65] Jia L-r, Zhou D, Niu J-q. Numerical calculation of boundary layers and wake characteristics of high-speed trains with different lengths [J]. PLoS One, 2017, 1212e0189798

[66] BS EN 14067-5:2021. Railway applications: Aerodynamics-requirements and assessment procedures for aerodynamics in tunnels [S], 2021, Belgium, European Committee for Standardization (CEN) Brussels

RIGHTS & PERMISSIONS

Central South University