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arxiv: 2605.18543 · v1 · pith:L6VN7YDSnew · submitted 2026-05-18 · 💻 cs.RO

Geometry-Aware Surrogate for Real-Time Hydrodynamics Estimation of Autonomous Ground Vehicles in Amphibious Environments

Ammar Waheed , Luke Gallantree , Zohaib Hasnain This is my paper

Pith reviewed 2026-05-20 09:10 UTC · model grok-4.3

classification 💻 cs.RO
keywords hydrodynamics estimationneural network surrogateautonomous ground vehiclesamphibious environmentssigned distance fieldcomputational fluid dynamicsreal-time modelingdrag and buoyancy prediction
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The pith

A per-surface neural network predicts hydrodynamic forces on ground vehicles in water at real-time speeds.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a neural network that estimates the push and lift forces water applies to autonomous ground vehicles moving through shallow water. It uses a signed distance field to describe submersion for each part of the vehicle surface and sums force predictions from individual surfaces. The network is trained only on high-fidelity fluid simulations of two different vehicle shapes. In tests on new simulation data it achieves low error rates and runs fast enough for onboard use. When fed real motion data from a full-scale vehicle wading in water at various depths, the predictions follow the expected physical rules for how drag grows with speed squared and buoyancy grows with depth, without those rules being part of the training objective.

Core claim

The central claim is that a geometry-aware per-surface neural network surrogate, supplied with submergence data from a vehicle-specific signed distance field, accurately predicts longitudinal hydrodynamic forces with 13% sMAPE and vertical forces with 3-12% sMAPE on held-out CFD data while executing in under 0.9 ms. Applied to kinematics from real-world wading trials, the model reproduces quadratic speed scaling of drag (R² ≥ 0.97) and linear depth scaling of buoyancy (R² = 0.973) through the summation of per-surface contributions even though the training loss does not encode these relationships.

What carries the argument

The per-surface neural network surrogate with signed distance field inputs for submergence that sums individual surface force predictions to obtain total hydrodynamic loading.

If this is right

  • The framework allows real-time inclusion of hydrodynamic effects in simulation and planning for autonomous ground vehicles in amphibious settings.
  • The per-surface design resolves force variations with geometry, depth, and flow direction without requiring explicit encoding of scaling laws.
  • Predictions from the surrogate can be validated against physical scaling behaviors observed in full-scale experiments.
  • Training on CFD from a limited set of geometries supports generalization to physical deployment on similar vehicles.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Extending the training set with CFD simulations of additional vehicle geometries would likely improve accuracy for a broader class of autonomous platforms.
  • The method could be adapted to predict forces on other dynamic objects in fluid environments such as underwater robots or surface vessels.
  • Integration with existing vehicle control systems might enable better performance in flood or shallow-water navigation scenarios.

Load-bearing premise

High-fidelity CFD data from only two geometrically distinct vehicles combined with the per-surface summation is sufficient to capture hydrodynamic behavior for real-world vehicle geometries, depths, and flow directions.

What would settle it

Collecting motion and force data from a third vehicle with a different geometry during wading trials and checking whether the surrogate predictions match the observed forces or new CFD results within the reported error bounds.

Figures

Figures reproduced from arXiv: 2605.18543 by Ammar Waheed, Luke Gallantree, Zohaib Hasnain.

Figure 1
Figure 1. Figure 1: Volume fractions at two Z-Y planes; (top) center plane, (bottom) [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Input-centric architecture of the hydrodynamic surrogate. Global features encode bulk flow state and dimensionless groups, while per-surface features [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Experimental setup for water wading tests. [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Experimental setup; (a) shows the OptiTrack active pucks mounted onto the chassis frame, (b) shows the experiment run with 4 inches of water depth, [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: End-to-end single-sample CPU latency distribution for Husky and [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Parity plots for net forces. Rows correspond to [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Mean predicted drag |F pred x | versus v 2 in the planar ground section at each depth. Dashed lines are per-depth origin-constrained fits (|Fx| = CD,eff v 2 ). Each marker represents one trial. buoyancy terms of 280 N, 543 N, and 784 N, as shown in [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Predicted vertical force Fz versus v 2 at each depth with per-depth linear fits (Fz = F0 + CL v 2 ). Diamonds mark the speed-independent buoyancy intercepts F0. predicted patch forces. The depth-dependent increase in CD and the buoyancy intercept ratio exceeding the depth ratio fur￾ther demonstrate that the transfer preserves geometry-specific physical effects. These results demonstrate that the surrogate’… view at source ↗
read the original abstract

Autonomous ground vehicles operating in shallow water or flood-prone terrains require dynamic models that account for hydrodynamic forces. However, the simulation and planning tools currently available either lack the physical fidelity or are too computationally expensive to run in real time. This work presents a per-surface neural network surrogate that bridges this gap by predicting geometry-resolved hydrodynamic forces at real-time rates, trained entirely on high-fidelity CFD data from two geometrically distinct vehicles. A vehicle specific Signed Distance Field (SDF) provides per-surface submergence inputs, allowing the model to resolve how loading varies with vehicle geometry, depth, and flow direction. On held-out CFD data, the surrogate achieves a longitudinal-force symmetric MAPE (sMAPE) of 13\% and a vertical-force sMAPE of 3-12\%, with inference running under 0.9\,ms per sample. To evaluate the model under real-world conditions, water wading trials of a full-scale vehicle at different submersion depths are used. Motion capture derived kinematics serve as the surrogate inputs, and the resulting predictions are tested to reproduce known physical relationships between force, speed, and depth. The predicted drag follows quadratic speed scaling ($R^2 \geq 0.97$) and the buoyancy intercepts scale linearly with depth ($R^2 = 0.973$). Neither relationship is encoded in the model training loss, both emerge from the per-surface architecture summing individually predicted surface forces. The resulting framework provides a pathway for embedding physically grounded hydrodynamics into the simulation and planning loops that autonomous ground vehicles depend on in amphibious environments.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 1 minor

Summary. The manuscript introduces a geometry-aware neural network surrogate for estimating hydrodynamic forces on autonomous ground vehicles in shallow water. Trained on CFD simulations from two distinct vehicle geometries using per-surface Signed Distance Field inputs, the model predicts forces in real time (<0.9 ms). It achieves sMAPE of 13% for longitudinal and 3-12% for vertical forces on held-out CFD data. Real-world wading trials with motion-capture inputs demonstrate that the model reproduces quadratic drag scaling with speed (R² ≥ 0.97) and linear buoyancy scaling with depth (R² = 0.973), properties that emerge from the per-surface force summation without being explicitly trained.

Significance. If the results hold, this provides a practical bridge between high-fidelity but slow CFD and real-time needs for AGV planning in amphibious environments. Credit is due for the emergence of quadratic and linear scaling laws from the additive per-surface architecture (neither encoded in the loss), which supplies direct evidence that the surrogate captures underlying hydrodynamics rather than fitting superficial patterns. The real-time inference speed and held-out CFD metrics further support utility for embedding into simulation and control loops.

major comments (1)
  1. [Real-world wading trials] Real-world wading trials (abstract and evaluation section): validation consists only of consistency with expected quadratic speed scaling (R² ≥ 0.97) and linear depth scaling (R² = 0.973) derived from motion-capture kinematics; no direct sensor-based force measurements are reported for quantitative error assessment under physical conditions. This indirect evidence supports emergence but is load-bearing for the generalization claim from two-vehicle CFD training to full-scale trials.
minor comments (1)
  1. [Training data and model architecture] Additional detail on the geometric distinctions between the two CFD training vehicles and explicit discussion of how the per-surface SDF inputs enable extrapolation to unseen submersion depths or flow angles would strengthen the geometry-awareness claim.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the constructive review and positive recommendation for minor revision. The significance of the emergent scaling laws is well noted. We address the major comment on real-world validation below.

read point-by-point responses

  1. Referee: [Real-world wading trials] Real-world wading trials (abstract and evaluation section): validation consists only of consistency with expected quadratic speed scaling (R² ≥ 0.97) and linear depth scaling (R² = 0.973) derived from motion-capture kinematics; no direct sensor-based force measurements are reported for quantitative error assessment under physical conditions. This indirect evidence supports emergence but is load-bearing for the generalization claim from two-vehicle CFD training to full-scale trials.

    Authors: We thank the referee for this observation. We agree that direct sensor-based force measurements would enable stronger quantitative error assessment in physical conditions. Such measurements on a full-scale vehicle during wading trials are practically challenging due to the need for waterproof, high-bandwidth load cells integrated without altering vehicle dynamics or introducing additional hydrodynamic interference; our experimental setup relied on motion-capture kinematics as surrogate inputs because this instrumentation was available and provided accurate 6-DoF trajectories. The validation demonstrates that the per-surface model, trained only on CFD from two geometries, produces forces obeying quadratic drag and linear buoyancy scaling without these relations appearing in the loss function. This emergent physical consistency, together with the held-out CFD sMAPE results, supports generalization to real-world amphibious operation. We acknowledge the indirect nature of the evidence as a limitation for the generalization claim. In the revised manuscript we have added a dedicated paragraph in the Evaluation section explicitly discussing the practical barriers to direct force sensing and the rationale for the chosen indirect checks. revision: yes

Circularity Check

0 steps flagged

No significant circularity; scaling laws emerge independently from per-surface summation

full rationale

The paper trains a per-surface neural surrogate exclusively on CFD data from two vehicles and validates generalization by confirming that predicted forces reproduce quadratic speed scaling (R² ≥ 0.97) and linear depth scaling (R² = 0.973) in real-world wading trials. The text explicitly states these relationships are not encoded in the training loss and instead arise from summing individually predicted surface forces given SDF inputs. This constitutes an independent physical-consistency check rather than a fitted input renamed as prediction, a self-definition, or a load-bearing self-citation. No equations or claims reduce the reported performance metrics or scaling reproduction to tautological inputs by construction; the derivation chain remains self-contained against external CFD benchmarks and known hydrodynamics.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard neural-network approximation capabilities and the assumption that CFD data faithfully represents real hydrodynamics; no new physical entities are postulated.

free parameters (1)
  • Neural network weights and biases
    All model parameters are fitted to CFD force data from the two training vehicles.
axioms (2)
  • standard math Neural networks with sufficient capacity can approximate the mapping from per-surface SDF inputs to local hydrodynamic forces
    Invoked by the choice of NN surrogate architecture.
  • domain assumption High-fidelity CFD simulations provide accurate ground-truth hydrodynamic forces for the two vehicle geometries
    Training and held-out evaluation depend on this.

pith-pipeline@v0.9.0 · 5828 in / 1583 out tokens · 64900 ms · 2026-05-20T09:10:37.063310+00:00 · methodology

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Reference graph

Works this paper leans on

36 extracted references · 36 canonical work pages

  1. [1]

    CARLA: An open urban driving simulator,

    A. Dosovitskiy, G. Ros, F. Codevilla, A. Lopez, and V . Koltun, “CARLA: An open urban driving simulator,” inProceedings of the 1st Annual Conference on Robot Learning, ser. Proceedings of Machine Learning Research, S. Levine, V . Vanhoucke, and K. Goldberg, Eds., vol. 78. PMLR, 13–15 Nov 2017, pp. 1–16. [Online]. Available: https://proceedings.mlr.press/v...

    work page 2017

  2. [2]

    On the use of simulation in robotics: Opportunities, challenges, and suggestions for moving forward,

    H. Choi, C. Crump, C. Duriez, A. Elmquist, G. Hager, D. Han, F. Hearl, J. Hodgins, A. Jain, F. Leveet al., “On the use of simulation in robotics: Opportunities, challenges, and suggestions for moving forward,”Pro- ceedings of the National Academy of Sciences, vol. 118, no. 1, p. e1907856118, 2021

    work page 2021

  3. [3]

    Sim-to-real transfer in deep reinforcement learning for robotics: a survey,

    W. Zhao, J. P. Queralta, and T. Westerlund, “Sim-to-real transfer in deep reinforcement learning for robotics: a survey,” in2020 IEEE Symposium Series on Computational Intelligence (SSCI), 2020, pp. 737–744

    work page 2020

  4. [4]

    Drag, added mass and radiation damping of oscillating vertical cylindrical bodies in heave and surge in still water,

    H. Gu, P. Stansby, T. Stallard, and E. Carpintero Moreno, “Drag, added mass and radiation damping of oscillating vertical cylindrical bodies in heave and surge in still water,”Journal of Fluids and Structures, vol. 82, pp. 343–356, 2018. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S0889974617306552

    work page 2018

  5. [5]

    Unav-sim: A visually realistic underwater robotics simulator and synthetic data-generation framework,

    A. Amer, O. ´Alvarez-Tu˜n´on, H. ˙I. U ˘gurlu, J. Le Fevre Sejersen, Y . Brodskiy, and E. Kayacan, “Unav-sim: A visually realistic underwater robotics simulator and synthetic data-generation framework,” in2023 21st International Conference on Advanced Robotics (ICAR), 2023, pp. 570–576

    work page 2023

  6. [6]

    Hydrodynamics of semi-submersible vehicle hulls with variable height–width ratio in deep and shallow water,

    K. I. Matveev, “Hydrodynamics of semi-submersible vehicle hulls with variable height–width ratio in deep and shallow water,”Journal of Offshore Mechanics and Arctic Engineering, vol. 147, no. 6, p. 061402, 2025

    work page 2025

  7. [7]

    A review on drag reduction technology: Focusing on amphibious vehicles,

    D. Pan, X. Xu, B. Liu, H. Xu, and X. Wang, “A review on drag reduction technology: Focusing on amphibious vehicles,” Ocean Engineering, vol. 280, p. 114618, 2023. [Online]. Available: https://www.sciencedirect.com/science/article/pii/S0029801823010028

    work page 2023

  8. [8]

    Cfd analyses on the water entry process of a freefall lifeboat,

    L. Huang, S. Tavakoli, M. Li, A. Dolatshah, B. Pena, B. Ding, and A. Dashtimanesh, “Cfd analyses on the water entry process of a freefall lifeboat,”Ocean engineering, vol. 232, p. 109115, 2021

    work page 2021

  9. [9]

    Study of water impact and entry of a free falling wedge using computational fluid dynamics simulations,

    A. Kamath, H. Bihs, and Ø. A. Arntsen, “Study of water impact and entry of a free falling wedge using computational fluid dynamics simulations,”Journal of Offshore Mechanics and Arctic Engineering, vol. 139, no. 3, p. 031802, 2017

    work page 2017

  10. [10]

    W. B. Horne and R. C. Dreher,Phenomena of pneumatic tire hydroplan- ing. National Aeronautics and Space Administration, 1963, vol. 2056

    work page 1963

  11. [11]

    Transient, 3d cfd, moving mesh simulation of vehicle water wading in a water tunnel with inclined entry-exit,

    M. Varshney, A. Ballani, S. Pasunurthi, D. Maiti, S. Dhar, and H. Ding, “Transient, 3d cfd, moving mesh simulation of vehicle water wading in a water tunnel with inclined entry-exit,” SAE Technical Paper, Tech. Rep., 2022

    work page 2022

  12. [12]

    Under- water robotic simulators review for autonomous system development,

    S. Aldhaheri, Y . Hu, Y . Xie, P. Wu, D. Kanoulas, and Y . Liu, “Under- water robotic simulators review for autonomous system development,” arXiv preprint arXiv:2504.06245, 2025

    work page arXiv 2025

  13. [13]

    Simu2vita: A general purpose under- water vehicle simulator,

    P. D. de Cerqueira Gava, C. L. Nascimento J ´unior, J. R. Belchior de Franc ¸a Silva, and G. J. Adabo, “Simu2vita: A general purpose under- water vehicle simulator,”Sensors, vol. 22, no. 9, p. 3255, 2022

    work page 2022

  14. [14]

    Dave aquatic virtual environment: Toward a general underwater robotics simulator,

    M. M. Zhang, W.-S. Choi, J. Herman, D. Davis, C. V ogt, M. McCarrin, Y . Vijay, D. Dutia, W. Lew, S. Peters, and B. Bingham, “Dave aquatic virtual environment: Toward a general underwater robotics simulator,” in 2022 IEEE/OES Autonomous Underwater Vehicles Symposium (AUV), 2022, pp. 1–8

    work page 2022

  15. [15]

    Stonefish: An advanced open-source simulation tool de- signed for marine robotics, with a ros interface,

    P. Cie ´slak, “Stonefish: An advanced open-source simulation tool de- signed for marine robotics, with a ros interface,” inOCEANS 2019 - Marseille, 2019, pp. 1–6

    work page 2019

  16. [16]

    Oceansim: A gpu-accelerated underwa- ter robot perception simulation framework,

    J. Song, H. Ma, O. Bagoren, A. V . Sethuraman, Y . Zhang, and K. A. Skinner, “Oceansim: A gpu-accelerated underwater robot perception simulation framework,” 2025. [Online]. Available: https://arxiv.org/abs/2503.01074

    work page arXiv 2025

  17. [17]

    Stability and drag analysis of wheeled amphibious vehicle using cfd and model testing techniques,

    R. More, P. Adhav, K. Senthilkumar, and M. Trikande, “Stability and drag analysis of wheeled amphibious vehicle using cfd and model testing techniques,”Applied Mechanics and Materials, vol. 592, pp. 1210–1219, 2014

    work page 2014

  18. [18]

    Criterion of vehicle stability in floodwaters based on theoretical and experimental studies,

    J. Xia, R. A. Falconer, X. Xiao, and Y . Wang, “Criterion of vehicle stability in floodwaters based on theoretical and experimental studies,” Natural hazards, vol. 70, no. 2, pp. 1619–1630, 2014

    work page 2014

  19. [19]

    Experimental testing to determine stability thresholds for partially submerged vehicles at different flow orientations,

    X. Hu, J. Li, W. Wang, and X. Fang, “Experimental testing to determine stability thresholds for partially submerged vehicles at different flow orientations,”Journal of Hydrology, vol. 620, p. 129525, 2023

    work page 2023

  20. [20]

    Review and analysis of vehicle stability models during floods and proposal for future improvements,

    R. A. Bocanegra, F. J. Vall ´es-Mor´an, and F. Franc ´es, “Review and analysis of vehicle stability models during floods and proposal for future improvements,”Journal of flood risk management, vol. 13, p. e12551, 2020

    work page 2020

  21. [21]

    Resistance reduction optimization of an amphibious transport vehicle,

    B. Liu, X. Xu, and D. Pan, “Resistance reduction optimization of an amphibious transport vehicle,”Ocean Engineering, vol. 280, p. 114854, 2023

    work page 2023

  22. [22]

    Modeling of vehicle mobility in shallow water with data- driven hydrodynamics model,

    H. Yamashita, J. E. Martin, N. Tison, A. Grunin, P. Jayakumar, and H. Sugiyama, “Modeling of vehicle mobility in shallow water with data- driven hydrodynamics model,”Journal of computational and nonlinear dynamics, vol. 19, no. 7, p. 071010, 2024

    work page 2024

  23. [23]

    Numerical methods for fluid- structure interaction—a review,

    G. Hou, J. Wang, and A. Layton, “Numerical methods for fluid- structure interaction—a review,”Communications in Computational Physics, vol. 12, no. 2, pp. 337–377, 2012

    work page 2012

  24. [24]

    J. N. Newman,Marine hydrodynamics. MIT press, 2018

    work page 2018

  25. [25]

    V olume of fluid (vof) method for the dynamics of free boundaries,

    C. W. Hirt and B. D. Nichols, “V olume of fluid (vof) method for the dynamics of free boundaries,”Journal of computational physics, vol. 39, no. 1, pp. 201–225, 1981

    work page 1981

  26. [26]

    Chrono: An open source multi-physics dynamics engine,

    A. Tasora, R. Serban, H. Mazhar, A. Pazouki, D. Melanz, J. Fleis- chmann, M. Taylor, H. Sugiyama, and D. Negrut, “Chrono: An open source multi-physics dynamics engine,” ininternational conference on high performance computing in science and engineering. Springer, 2015, pp. 19–49

    work page 2015

  27. [27]

    Gazebo fluids: Sph-based simulation of fluid interaction with articulated rigid body dynamics,

    E. Angelidis, J. Bender, J. Arreguit, L. Gleim, W. Wang, C. Axenie, A. Knoll, and A. Ijspeert, “Gazebo fluids: Sph-based simulation of fluid interaction with articulated rigid body dynamics,” in2022 IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), 2022, pp. 11 238–11 245

    work page 2022

  28. [28]

    Unified particle physics for real-time applications,

    M. Macklin, M. M ¨uller, N. Chentanez, and T.-Y . Kim, “Unified particle physics for real-time applications,”ACM Transactions on Graphics (TOG), vol. 33, no. 4, pp. 1–12, 2014

    work page 2014

  29. [29]

    Water buoyancy component in unreal engine,

    Epic Games, “Water buoyancy component in unreal engine,” https://dev.epicgames.com/documentation/en-us/unreal-engine/ water-buoyancy-component-in-unreal-engine, 2026, epic Developer Community, Unreal Engine 5.7 Documentation

    work page 2026

  30. [30]

    Uuv simulator: A gazebo-based package for under- water intervention and multi-robot simulation,

    M. M. M. Manh ˜aes, S. A. Scherer, M. V oss, L. R. Douat, and T. Rauschenbach, “Uuv simulator: A gazebo-based package for under- water intervention and multi-robot simulation,” inOceans 2016 Mts/Ieee Monterey. Ieee, 2016, pp. 1–8

    work page 2016

  31. [31]

    T. I. Fossen,Handbook of marine craft hydrodynamics and motion control. John wiley & sons, 2011

    work page 2011

  32. [32]

    Quantifying the sim2real gap: Model-based verification and validation in autonomous ground systems,

    A. Waheed, M. Areti, L. Gallantree, and Z. Hasnain, “Quantifying the sim2real gap: Model-based verification and validation in autonomous ground systems,”IEEE Robotics and Automation Letters, 2025

    work page 2025

  33. [33]

    The prediction of external flow field and hydrodynamic force with limited data using deep neural network,

    T.-s. Wang, G. Xi, Z.-g. Sun, and Z. Huang, “The prediction of external flow field and hydrodynamic force with limited data using deep neural network,”Journal of Hydrodynamics, vol. 35, no. 3, pp. 549–570, 2023

    work page 2023

  34. [34]

    Data-driven prediction of experimental hydrodynamic data of the manta ray robot using deep learning method,

    J. Bai, Q. Huang, G. Pan, and J. He, “Data-driven prediction of experimental hydrodynamic data of the manta ray robot using deep learning method,”Journal of Marine Science and Engineering, vol. 10, no. 9, p. 1285, 2022

    work page 2022

  35. [35]

    Physics-inspired architecture for neural network modeling of forces and torques in particle-laden flows,

    A. Seyed-Ahmadi and A. Wachs, “Physics-inspired architecture for neural network modeling of forces and torques in particle-laden flows,” Computers & Fluids, vol. 238, p. 105379, 2022

    work page 2022

  36. [36]

    Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations,

    M. Raissi, P. Perdikaris, and G. E. Karniadakis, “Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations,” Journal of Computational physics, vol. 378, pp. 686–707, 2019

    work page 2019