Abstract
We contribute to the vastly growing field of machine learning for engineering systems by demonstrating that equivariant graph neural networks have the potential to learn more accurate dynamic-interaction models than their non-equivariant counterparts. We benchmark two well-studied fluid-flow systems, namely 3D decaying Taylor-Green vortex and 3D reverse Poiseuille flow, and evaluate the models based on different performance measures, such as kinetic energy or Sinkhorn distance. In addition, we investigate different embedding methods of physical-information histories for equivariant models. We find that while currently being rather slow to train and evaluate, equivariant models with our proposed history embeddings learn more accurate physical interactions.
Our code will be released under https://github.com/tumaer/sph-hae.
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References
Adami, S., Hu, X., Adams, N.A.: A transport-velocity formulation for smoothed particle hydrodynamics. J. Comput. Phys. 241, 292–307 (2013)
Batatia, I., Kovács, D.P., Simm, G.N.C., Ortner, C., Csányi, G.: Mace: higher order equivariant message passing neural networks for fast and accurate force fields (2022)
Battaglia, P.W., et al.: Relational inductive biases, deep learning, and graph networks. arXiv (2018)
Batzner, S., et al.: E(3)-equivariant graph neural networks for data-efficient and accurate interatomic potentials. Nat. Commun. 13(1), 2453 (2022)
Brachet, M.E., Meiron, D., Orszag, S., Nickel, B., Morf, R., Frisch, U.: The taylor-green vortex and fully developed turbulence. J. Stat. Phys. 34(5–6), 1049–1063 (1984)
Bradbury, J., et al.: JAX: composable transformations of Python+NumPy programs (2018)
Brandstetter, J., Berg, R.V.D., Welling, M., Gupta, J.K.: Clifford neural layers for PDE modeling. arXiv preprint arXiv:2209.04934 (2022)
Brandstetter, J., Hesselink, R., van der Pol, E., Bekkers, E.J., Welling, M.: Geometric and physical quantities improve E(3) equivariant message passing. In: ICLR (2022)
Brandstetter, J., Worrall, D.E., Welling, M.: Message passing neural PDE solvers. In: ICLR (2022)
Cohen, T.S., Welling, M.: Group equivariant convolutional networks. In: Proceedings of the 33rd ICML, ICML 2016, vol. 48, pp. 2990–2999. JMLR.org (2016)
Fedosov, D.A., Caswell, B., Em Karniadakis, G.: Reverse poiseuille flow: the numerical viscometer. In: AIP Conference Proceedings, vol. 1027, pp. 1432–1434. American Institute of Physics (2008)
Gasteiger, J., Becker, F., Günnemann, S.: Gemnet: universal directional graph neural networks for molecules. NeurIPS 34, 6790–6802 (2021)
Gasteiger, J., Groß, J., Günnemann, S.: Directional message passing for molecular graphs. In: ICLR (2020)
Geiger, M., Smidt, T.: e3nn: Euclidean neural networks (2022)
Gilmer, J., Schoenholz, S.S., Riley, P.F., Vinyals, O., Dahl, G.E.: Neural message passing for quantum chemistry. In: ICML, pp. 1263–1272. PMLR (2017)
Gingold, R.A., Monaghan, J.J.: Smoothed particle hydrodynamics: theory and application to non-spherical stars. Mon. Not. R. Astron. Soc. 181(3), 375–389 (1977)
Gupta, J.K., Brandstetter, J.: Towards multi-spatiotemporal-scale generalized PDE modeling. arXiv preprint arXiv:2209.15616 (2022)
Hu, W., Pan, W., Rakhsha, M., Tian, Q., Hu, H., Negrut, D.: A consistent multi-resolution smoothed particle hydrodynamics method. Comput. Methods Appl. Mech. Eng. 324, 278–299 (2017)
Lagrave, P.Y., Tron, E.: Equivariant neural networks and differential invariants theory for solving partial differential equations. In: Physical Sciences Forum, vol. 5, p. 13. MDPI (2022)
Li, Z., et al.: Neural operator: graph kernel network for partial differential equations. arXiv preprint arXiv:2003.03485 (2020)
Li, Z., et al.: Fourier neural operator for parametric partial differential equations. In: ICLR (2021)
Lu, L., Jin, P., Karniadakis, G.E.: DeepONet: learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators. arXiv preprint arXiv:1910.03193 (2019)
Lucy, L.B.: A numerical approach to the testing of the fission hypothesis. Astron. J. 82, 1013–1024 (1977)
Mayr, A., Lehner, S., Mayrhofer, A., Kloss, C., Hochreiter, S., Brandstetter, J.: Boundary graph neural networks for 3D simulations. arXiv preprint arXiv:2106.11299 (2021)
Pezzicoli, F.S., Charpiat, G., Landes, F.P.: Se (3)-equivariant graph neural networks for learning glassy liquids representations. arXiv:2211.03226 (2022)
Pfaff, T., Fortunato, M., Sanchez-Gonzalez, A., Battaglia, P.W.: Learning mesh-based simulation with graph networks. arXiv preprint arXiv:2010.03409 (2020)
Pope, S.B.: Turbulent Flows. Cambridge University Press, Cambridge (2000)
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. Comput. Phys. 378, 686–707 (2019)
Ruhe, D., Brandstetter, J., Forré, P.: Clifford group equivariant neural networks. arXiv preprint arXiv:2305.11141 (2023)
Ruhe, D., Gupta, J.K., de Keninck, S., Welling, M., Brandstetter, J.: Geometric clifford algebra networks. arXiv preprint arXiv:2302.06594 (2023)
Sanchez-Gonzalez, A., Godwin, J., Pfaff, T., Ying, R., Leskovec, J., Battaglia, P.: Learning to simulate complex physics with graph networks. In: ICML, pp. 8459–8468. PMLR (2020)
Satorras, V.G., Hoogeboom, E., Welling, M.: E(n) equivariant graph neural networks. In: ICML, pp. 9323–9332. PMLR (2021)
Schoenholz, S.S., Cubuk, E.D.: JAX M.D. a framework for differentiable physics. In: NeurIPS, vol. 33. Curran Associates, Inc. (2020)
Taylor, G.I., Green, A.E.: Mechanism of the production of small eddies from large ones. Proc. Roy. Soc. London Ser. A Math. Phys. Sci. 158(895), 499–521 (1937)
Thomas, N., et al.: Tensor field networks: rotation- and translation-equivariant neural networks for 3D point clouds. CoRR abs/1802.08219 (2018)
Violeau, D., Rogers, B.D.: Smoothed particle hydrodynamics (SPH) for free-surface flows: past, present and future. J. Hydraul. Res. 54(1), 1–26 (2016)
Wang, R., Walters, R., Yu, R.: Incorporating symmetry into deep dynamics models for improved generalization. In: ICLR (2021)
Weiler, M., Geiger, M., Welling, M., Boomsma, W., Cohen, T.S.: 3D steerable CNNs: learning rotationally equivariant features in volumetric data. In: Bengio, S., Wallach, H., Larochelle, H., Grauman, K., Cesa-Bianchi, N., Garnett, R. (eds.) NeurIPS, vol. 31. Curran Associates, Inc. (2018)
Weiler, M., Geiger, M., Welling, M., Boomsma, W., Cohen, T.S.: 3D steerable CNNs: learning rotationally equivariant features in volumetric data. In: Advances in Neural Information Processing Systems, vol. 31 (2018)
Weirather, J., et al.: A smoothed particle hydrodynamics model for laser beam melting of NI-based alloy 718. CMA 78(7), 2377–2394 (2019)
Acknowledgements
We are thankful to the developers of the e3nn-jax library [14], which offers efficient implementations of E(3)-equivariant building blocks.
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Toshev, A.P., Galletti, G., Brandstetter, J., Adami, S., Adams, N.A. (2023). Learning Lagrangian Fluid Mechanics with E(3)-Equivariant Graph Neural Networks. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2023. Lecture Notes in Computer Science, vol 14072. Springer, Cham. https://doi.org/10.1007/978-3-031-38299-4_35
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