“Physics informed neural fields for smoke reconstruction with sparse data” by Chu, Liu, Zheng, Frankz, Seidel, et al. …

  • ©Mengyu Chu, Lingjie Liu, Quan Zheng, Erik Frankz, Hans-Peter Seidel, Christian Theobalt, and Rhaleb Zayer




    Physics informed neural fields for smoke reconstruction with sparse data



    High-fidelity reconstruction of dynamic fluids from sparse multiview RGB videos remains a formidable challenge, due to the complexity of the underlying physics as well as the severe occlusion and complex lighting in the captured data. Existing solutions either assume knowledge of obstacles and lighting, or only focus on simple fluid scenes without obstacles or complex lighting, and thus are unsuitable for real-world scenes with unknown lighting conditions or arbitrary obstacles. We present the first method to reconstruct dynamic fluid phenomena by leveraging the governing physics (ie, Navier -Stokes equations) in an end-to-end optimization from a mere set of sparse video frames without taking lighting conditions, geometry information, or boundary conditions as input. Our method provides a continuous spatio-temporal scene representation using neural networks as the ansatz of density and velocity solution functions for fluids as well as the radiance field for static objects. With a hybrid architecture that separates static and dynamic contents apart, fluid interactions with static obstacles are reconstructed for the first time without additional geometry input or human labeling. By augmenting time-varying neural radiance fields with physics-informed deep learning, our method benefits from the supervision of images and physical priors. Our progressively growing model with regularization further disentangles the density-color ambiguity in the radiance field, which allows for a more robust optimization from the given input of sparse views. A pretrained density-to-velocity fluid model is leveraged in addition as the data prior to avoid suboptimal velocity solutions which underestimate vorticity but trivially fulfill physical equations. Our method exhibits high-quality results with relaxed constraints and strong flexibility on a representative set of synthetic and real flow captures. Code and sample tests are at https://people.mpi-inf.mpg.de/~mchu/projects/PI-NeRF/.


    1. Bradley Atcheson, Wolfgang Heidrich, and Ivo Ihrke. 2009. An evaluation of optical flow algorithms for background oriented schlieren imaging. Experiments in Fluids 46, 3 (01 Mar 2009), 467–476. Google ScholarCross Ref
    2. Peter Bauer, Alan Thorpe, and Gilbert Brunet. 2015. The quiet revolution of numerical weather prediction. Nature 525, 7567 (2015), 47–55.Google Scholar
    3. Jens Berg and Kaj Nyström. 2018. A unified deep artificial neural network approach to partial differential equations in complex geometries. Neurocomputing 317 (2018), 28–41. Google ScholarCross Ref
    4. Robert Bridson. 2015. Fluid simulation for computer graphics. CRC press.Google ScholarDigital Library
    5. Michael Broxton, John Flynn, Ryan Overbeck, Daniel Erickson, Peter Hedman, Matthew Duvall, Jason Dourgarian, Jay Busch, Matt Whalen, and Paul Debevec. 2020. Immersive light field video with a layered mesh representation. ACM Transactions on Graphics (TOG) 39, 4 (2020), 86–1.Google ScholarDigital Library
    6. Dennis M Bushnell and KJ Moore. 1991. Drag reduction in nature. Annual review of fluid mechanics 23, 1 (1991), 65–79.Google Scholar
    7. Rohan Chabra, Jan E Lenssen, Eddy Ilg, Tanner Schmidt, Julian Straub, Steven Lovegrove, and Richard Newcombe. 2020. Deep local shapes: Learning local SDF priors for detailed 3D reconstruction. In European Conference on Computer Vision. Springer, 608–625.Google ScholarDigital Library
    8. Jianchuan Chen, Ying Zhang, Di Kang, Xuefei Zhe, Linchao Bao, and Huchuan Lu. 2021. Animatable Neural Radiance Fields from Monocular RGB Video. arXiv:2106.13629 [cs.CV]Google Scholar
    9. Zhiqin Chen and Hao Zhang. 2019. Learning implicit fields for generative shape modeling. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. 5939–5948.Google ScholarCross Ref
    10. Julian Chibane, Gerard Pons-Moll, et al. 2020. Neural Unsigned Distance Fields for Implicit Function Learning. Advances in Neural Information Processing Systems 33 (2020).Google Scholar
    11. Mengyu Chu, Nils Thuerey, Hans-Peter Seidel, Christian Theobalt, and Rhaleb Zayer. 2021. Learning meaningful controls for fluids. ACM Transactions on Graphics (TOG) 40, 4 (2021), 1–13.Google ScholarDigital Library
    12. Mengyu Chu, You Xie, Jonas Mayer, Laura Leal-Taixé, and Nils Thuerey. 2020. Learning temporal coherence via self-supervision for GAN-based video generation. ACM Transactions on Graphics (TOG) 39, 4 (2020), 75–1.Google ScholarDigital Library
    13. Marie-Lena Eckert, Kiwon Um, and Nils Thuerey. 2019. ScalarFlow: a large-scale volumetric data set of real-world scalar transport flows for computer animation and machine learning. ACM Transactions on Graphics (TOG) 38, 6 (2019), 1–16.Google ScholarDigital Library
    14. Gerrit E Elsinga, Fulvio Scarano, Bernhard Wieneke, and Bas W van Oudheusden. 2006. Tomographic particle image velocimetry. Experiments in fluids 41, 6 (2006), 933–947.Google Scholar
    15. SM Ali Eslami, Danilo Jimenez Rezende, Frederic Besse, Fabio Viola, Ari S Morcos, Marta Garnelo, Avraham Ruderman, Andrei A Rusu, Ivo Danihelka, Karol Gregor, et al. 2018. Neural scene representation and rendering. Science 360, 6394 (2018), 1204–1210.Google Scholar
    16. Zahra Forootaninia and Rahul Narain. 2020. Frequency-domain smoke guiding. ACM Transactions on Graphics (TOG) 39, 6 (2020), 1–10.Google ScholarDigital Library
    17. Erik Franz, Barbara Solenthaler, and Nils Thuerey. 2021. Global Transport for Fluid Reconstruction with Learned Self-Supervision. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. 1632–1642.Google ScholarCross Ref
    18. Guy Gafni, Justus Thies, Michael Zollhoefer, and Matthias Niessner. 2020. Dynamic Neural Radiance Fields for Monocular 4D Facial Avatar Reconstruction. arXiv:2012.03065 [cs.CV]Google Scholar
    19. Zhenglin Geng, Daniel Johnson, and Ronald Fedkiw. 2020. Coercing machine learning to output physically accurate results. J. Comput. Phys. 406 (2020), 109099.Google ScholarCross Ref
    20. Frederic Gibou, David Hyde, and Ron Fedkiw. 2019. Sharp interface approaches and deep learning techniques for multiphase flows. J. Comput. Phys. 380 (2019), 442–463.Google ScholarDigital Library
    21. Jonathan Granskog, Fabrice Rousselle, Marios Papas, and Jan Novák. 2020. Compositional neural scene representations for shading inference. ACM Transactions on Graphics (TOG) 39, 4 (2020), 135–1.Google ScholarDigital Library
    22. I. Grant. 1997. Particle Image Velocimetry: a Review. Proceedings of the Institution of Mechanical Engineers 211, 1 (1997), 55–76.Google Scholar
    23. James Gregson, Ivo Ihrke, Nils Thuerey, and Wolfgang Heidrich. 2014. From capture to simulation: connecting forward and inverse problems in fluids. ACM Transactions on Graphics (TOG) 33, 4 (2014), 1–11.Google ScholarDigital Library
    24. Jinwei Gu, S.K. Nayar, E. Grinspun, P.N. Belhumeur, and R. Ramamoorthi. 2013. Compressive Structured Light for Recovering Inhomogeneous Participating Media. PAMI 35, 3 (2013), 555–567.Google Scholar
    25. S. He, K. Reif, and R. Unbehauen. 2000. Multilayer Neural Networks for Solving a Class of Partial Differential Equations. Neural Netw. 13, 3 (apr 2000), 385–396. Google ScholarDigital Library
    26. Yu Ji, Jinwei Ye, and Jingyi Yu. 2013. Reconstructing Gas Flows Using Light-Path Approximation. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition (CVPR).Google ScholarDigital Library
    27. Theodore Kim, Nils Thuerey, Doug James, and Markus Gross. 2008. Wavelet turbulence for fluid simulation. ACM Transactions on Graphics (TOG) 27, 3 (2008), 1–6.Google ScholarDigital Library
    28. Youngjoong Kwon, Dahun Kim, Duygu Ceylan, and Henry Fuchs. 2021. Neural Human Performer: Learning Generalizable Radiance Fields for Human Performance Rendering. NeurIPS (2021).Google Scholar
    29. I.E. Lagaris, A. Likas, and D.I. Fotiadis. 1998. Artificial neural networks for solving ordinary and partial differential equations. IEEE Transactions on Neural Networks 9, 5 (1998), 987–1000. Google ScholarDigital Library
    30. Christian Ledig, Lucas Theis, Ferenc Huszár, Jose Caballero, Andrew Cunningham, Alejandro Acosta, Andrew Aitken, Alykhan Tejani, Johannes Totz, Zehan Wang, et al. 2017. Photo-realistic single image super-resolution using a generative adversarial network. In Proceedings of the IEEE conference on computer vision and pattern recognition. 4681–4690.Google ScholarCross Ref
    31. Z. Li, Simon Niklaus, Noah Snavely, and Oliver Wang. 2020. Neural Scene Flow Fields for Space-Time View Synthesis of Dynamic Scenes. ArXiv abs/2011.13084 (2020).Google Scholar
    32. Lingjie Liu, Marc Habermann, Viktor Rudnev, Kripasindhu Sarkar, Jiatao Gu, and Christian Theobalt. 2021. Neural Actor: Neural Free-view Synthesis of Human Actors with Pose Control. ACM Trans. Graph.(ACM SIGGRAPH Asia) (2021).Google ScholarDigital Library
    33. Stephen Lombardi, Tomas Simon, Jason Saragih, Gabriel Schwartz, Andreas Lehrmann, and Yaser Sheikh. 2019. Neural volumes: learning dynamic renderable volumes from images. ACM Transactions on Graphics (TOG) 38, 4 (2019), 1–14.Google ScholarDigital Library
    34. Stephen Lombardi, Tomas Simon, Gabriel Schwartz, Michael Zollhoefer, Yaser Sheikh, and Jason Saragih. 2021. Mixture of Volumetric Primitives for Efficient Neural Rendering. arXiv:2103.01954 [cs.GR]Google Scholar
    35. Nam Mai-Duy and Thanh Tran-Cong. 2003. Approximation of function and its derivatives using radial basis function networks. Applied Mathematical Modelling 27, 3 (2003), 197–220. Google ScholarCross Ref
    36. Ben Mildenhall, Pratul P Srinivasan, Rodrigo Ortiz-Cayon, Nima Khademi Kalantari, Ravi Ramamoorthi, Ren Ng, and Abhishek Kar. 2019. Local light field fusion: Practical view synthesis with prescriptive sampling guidelines. ACM Transactions on Graphics (TOG) 38, 4 (2019), 1–14.Google ScholarDigital Library
    37. Ben Mildenhall, Pratul P. Srinivasan, Matthew Tancik, Jonathan T. Barron, Ravi Ramamoorthi, and Ren Ng. 2020. NeRF: Representing Scenes as Neural Radiance Fields for View Synthesis. In ECCV.Google Scholar
    38. Thomas Müller, Alex Evans, Christoph Schied, and Alexander Keller. 2022. Instant Neural Graphics Primitives with a Multiresolution Hash Encoding. arXiv preprint arXiv:2201.05989 (2022).Google Scholar
    39. Michael Niemeyer, Lars Mescheder, Michael Oechsle, and Andreas Geiger. 2020. Differentiable volumetric rendering: Learning implicit 3d representations without 3d supervision. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. 3504–3515.Google ScholarCross Ref
    40. Samira Pakravan, Pouria A. Mistani, Miguel A. Aragon-Calvo, and Frederic Gibou. 2021. Solving inverse-PDE problems with physics-aware neural networks. J. Comput. Phys. 440 (2021), 110414.Google ScholarCross Ref
    41. Jeong Joon Park, Peter Florence, Julian Straub, Richard Newcombe, and Steven Lovegrove. 2019. DeepSDF: Learning continuous signed distance functions for shape representation. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. 165–174.Google ScholarCross Ref
    42. Keunhong Park, Utkarsh Sinha, Jonathan T. Barron, Sofien Bouaziz, Dan B Goldman, Steven M. Seitz, and Ricardo Martin-Brualla. 2020. Deformable Neural Radiance Fields. arXiv preprint arXiv:2011.12948 (2020).Google Scholar
    43. Keunhong Park, Utkarsh Sinha, Peter Hedman, Jonathan T. Barron, Sofien Bouaziz, Dan B Goldman, Ricardo Martin-Brualla, and Steven M. Seitz. 2021. HyperNeRF: A Higher-Dimensional Representation for Topologically Varying Neural Radiance Fields. ACM Trans. Graph. 40, 6, Article 238 (dec 2021).Google ScholarCross Ref
    44. Sida Peng, Junting Dong, Qianqian Wang, Shangzhan Zhang, Qing Shuai, Hujun Bao, and Xiaowei Zhou. 2021a. Animatable Neural Radiance Fields for Human Body Modeling. ICCV (2021).Google Scholar
    45. Songyou Peng, Michael Niemeyer, Lars Mescheder, Marc Pollefeys, and Andreas Geiger. 2020. Convolutional occupancy networks. In Computer Vision-ECCV 2020: 16th European Conference, Glasgow, UK, August 23–28, 2020, Proceedings, Part III 16. Springer, 523–540.Google Scholar
    46. Sida Peng, Yuanqing Zhang, Yinghao Xu, Qianqian Wang, Qing Shuai, Hujun Bao, and Xiaowei Zhou. 2021b. Neural Body: Implicit Neural Representations with Structured Latent Codes for Novel View Synthesis of Dynamic Humans. In CVPR.Google Scholar
    47. Julien Philip, Michaël Gharbi, Tinghui Zhou, Alexei A Efros, and George Drettakis. 2019. Multi-view relighting using a geometry-aware network. ACM Transactions on Graphics (TOG) 38, 4 (2019), 1–14.Google ScholarDigital Library
    48. Sheng Qiu, Chen Li, Changbo Wang, and Hong Qin. 2021. A Rapid, End-to-end, Generative Model for Gaseous Phenomena from Limited Views. (2021). Google ScholarCross Ref
    49. M. Raissi, P. Perdikaris, and G.E. Karniadakis. 2019. Physics-informed neural networks: A deep learning framework for solving forward and inverse problems involving nonlinear partial differential equations. J. Comput. Phys. 378 (2019), 686–707. Google ScholarCross Ref
    50. Maziar Raissi, Alireza Yazdani, and George Em Karniadakis. 2020. Hidden fluid mechanics: Learning velocity and pressure fields from flow visualizations. Science 367, 6481 (2020), 1026–1030.Google Scholar
    51. Amit Raj, Michael Zollhoefer, Tomas Simon, Jason Saragih, Shunsuke Saito, James Hays, and Stephen Lombardi. 2021. PVA: Pixel-aligned Volumetric Avatars. arXiv:2101.02697 [cs.CV]Google Scholar
    52. Shunsuke Saito, Zeng Huang, Ryota Natsume, Shigeo Morishima, Angjoo Kanazawa, and Hao Li. 2019. PIFu: Pixel-aligned implicit function for high-resolution clothed human digitization. In Proceedings of the IEEE/CVF International Conference on Computer Vision. 2304–2314.Google ScholarCross Ref
    53. Katja Schwarz, Yiyi Liao, Michael Niemeyer, and Andreas Geiger. 2020. GRAF: Generative Radiance Fields for 3D-Aware Image Synthesis. Advances in Neural Information Processing Systems 33 (2020).Google Scholar
    54. Kuangyu Shi, Michael Souvatzoglou, Sabrina T Astner, Peter Vaupel, Fridtjof Nüsslin, Jan J Wilkens, and Sibylle I Ziegler. 2010. Quantitative assessment of hypoxia kinetic models by a cross-study of dynamic 18F-FAZA and 15O-H2O in patients with head and neck tumors. Journal of Nuclear Medicine 51, 9 (2010), 1386–1394.Google ScholarCross Ref
    55. Justin Sirignano and Konstantinos Spiliopoulos. 2018. DGM: A deep learning algorithm for solving partial differential equations. J. Comput. Phys. 375 (Dec 2018), 1339–1364. Google ScholarCross Ref
    56. Vincent Sitzmann, Julien N.P. Martel, Alexander W. Bergman, David B. Lindell, and Gordon Wetzstein. 2020. Implicit Neural Representations with Periodic Activation Functions. In Proc. NeurIPS.Google Scholar
    57. Vincent Sitzmann, Justus Thies, Felix Heide, Matthias Nießner, Gordon Wetzstein, and Michael Zollhofer. 2019a. DeepVoxels: Learning persistent 3D feature embeddings. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. 2437–2446.Google ScholarCross Ref
    58. Vincent Sitzmann, Michael Zollhöfer, and Gordon Wetzstein. 2019b. Scene representation networks: Continuous 3D-structure-aware neural scene representations. In Advances in Neural Information Processing Systems. 1121–1132.Google Scholar
    59. Pratul P Srinivasan, Boyang Deng, Xiuming Zhang, Matthew Tancik, Ben Mildenhall, and Jonathan T Barron. 2021. NeRV: Neural reflectance and visibility fields for relighting and view synthesis. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. 7495–7504.Google ScholarCross Ref
    60. Pratul P Srinivasan, Richard Tucker, Jonathan T Barron, Ravi Ramamoorthi, Ren Ng, and Noah Snavely. 2019. Pushing the boundaries of view extrapolation with multiplane images. In Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. 175–184.Google ScholarCross Ref
    61. Shih-Yang Su, Frank Yu, Michael Zollhoefer, and Helge Rhodin. 2021. A-NeRF: Surface-free Human 3D Pose Refinement via Neural Rendering. arXiv preprint arXiv:2102.06199 (2021).Google Scholar
    62. Matthew Tancik, Pratul P. Srinivasan, Ben Mildenhall, Sara Fridovich-Keil, Nithin Raghavan, Utkarsh Singhal, Ravi Ramamoorthi, Jonathan T. Barron, and Ren Ng. 2020. Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains. NeurIPS (2020).Google Scholar
    63. Ayush Tewari, Ohad Fried, Justus Thies, Vincent Sitzmann, Stephen Lombardi, Kalyan Sunkavalli, Ricardo Martin-Brualla, Tomas Simon, Jason Saragih, Matthias Nießner, et al. 2020. State of the art on neural rendering. In Computer Graphics Forum, Vol. 39. Wiley Online Library, 701–727.Google ScholarCross Ref
    64. Nils Thuerey and Tobias Pfaff. 2018. MantaFlow. http://mantaflow.com.Google Scholar
    65. Edgar Tretschk, Ayush Tewari, Vladislav Golyanik, Michael Zollhöfer, Christoph Lassner, and Christian Theobalt. 2021. Non-Rigid Neural Radiance Fields: Reconstruction and Novel View Synthesis of a Dynamic Scene From Monocular Video. In IEEE International Conference on Computer Vision (ICCV). IEEE.Google ScholarCross Ref
    66. Kiwon Um, Robert Brand, Yun Raymond Fei, Philipp Holl, and Nils Thuerey. 2020. Solver-in-the-loop: Learning from differentiable physics to interact with iterative pde-solvers. Advances in Neural Information Processing Systems 33 (2020), 6111–6122.Google Scholar
    67. Ziyan Wang, Timur Bagautdinov, Stephen Lombardi, Tomas Simon, Jason Saragih, Jessica Hodgins, and Michael Zollhöfer. 2020. Learning Compositional Radiance Fields of Dynamic Human Heads. arXiv:2012.09955 [cs.CV]Google Scholar
    68. Wenqi Xian, Jia-Bin Huang, Johannes Kopf, and Changil Kim. 2020. Space-time Neural Irradiance Fields for Free-Viewpoint Video. arXiv:2011.12950 [cs.CV]Google Scholar
    69. You Xie, Erik Franz, Mengyu Chu, and Nils Thuerey. 2018. tempoGAN: A Temporally Coherent, Volumetric GAN for Super-resolution Fluid Flow. ACM Transactions on Graphics (TOG) 37, 4 (2018), 95.Google ScholarDigital Library
    70. Jinhui Xiong, Ramzi Idoughi, Andres A Aguirre-Pablo, Abdulrahman B Aljedaani, Xiong Dun, Qiang Fu, Sigurdur T Thoroddsen, and Wolfgang Heidrich. 2017. Rainbow particle imaging velocimetry for dense 3D fluid velocity imaging. ACM Transactions on Graphics (TOG) 36, 4 (2017), 1–14.Google ScholarDigital Library
    71. Lior Yariv, Yoni Kasten, Dror Moran, Meirav Galun, Matan Atzmon, Basri Ronen, and Yaron Lipman. 2020. Multiview Neural Surface Reconstruction by Disentangling Geometry and Appearance. Advances in Neural Information Processing Systems 33 (2020).Google Scholar
    72. Wang Yifan, Lukas Rahmann, and Olga Sorkine-hornung. 2022. Geometry-Consistent Neural Shape Representation with Implicit Displacement Fields. In International Conference on Learning Representations.Google Scholar
    73. Wang Yifan, Felice Serena, Shihao Wu, Cengiz Öztireli, and Olga Sorkine-Hornung. 2019. Differentiable surface splatting for point-based geometry processing. ACM Transactions on Graphics (TOG) 38, 6 (2019), 1–14.Google ScholarDigital Library
    74. Alex Yu, Ruilong Li, Matthew Tancik, Hao Li, Ren Ng, and Angjoo Kanazawa. 2021. PlenOctrees for Real-time Rendering of Neural Radiance Fields. In ICCV.Google Scholar
    75. Guangming Zang, Ramzi Idoughi, Congli Wang, Anthony Bennett, Jianguo Du, Scott Skeen, William L. Roberts, Peter Wonka, and Wolfgang Heidrich. 2020. TomoFluid: Reconstructing Dynamic Fluid from Sparse View Videos. In Proceedings of the IEEE Conference on Computer Vision and Pattern Recognition. IEEE. https://repository.kaust.edu.sa/handle/10754/662380Google ScholarCross Ref
    76. Xiuming Zhang, Sean Fanello, Yun-Ta Tsai, Tiancheng Sun, Tianfan Xue, Rohit Pandey, Sergio Orts-Escolano, Philip Davidson, Christoph Rhemann, Paul Debevec, et al. 2021. Neural light transport for relighting and view synthesis. ACM Transactions on Graphics (TOG) 40, 1 (2021), 1–17.Google ScholarDigital Library
    77. Quan Zheng, Gurprit Singh, and Hans-Peter Seidel. 2021. Neural Relightable Participating Media Rendering. Advances in Neural Information Processing Systems 34 (2021).Google Scholar
    78. Tinghui Zhou, Richard Tucker, John Flynn, Graham Fyffe, and Noah Snavely. 2018. Stereo magnification: learning view synthesis using multiplane images. ACM Transactions on Graphics (TOG) 37, 4 (2018), 1–12.Google ScholarDigital Library

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