Papers
Topics
Authors
Recent
Gemini 2.5 Flash
Gemini 2.5 Flash
139 tokens/sec
GPT-4o
7 tokens/sec
Gemini 2.5 Pro Pro
46 tokens/sec
o3 Pro
4 tokens/sec
GPT-4.1 Pro
38 tokens/sec
DeepSeek R1 via Azure Pro
28 tokens/sec
2000 character limit reached

Equivariant Graph Neural Operator for Modeling 3D Dynamics (2401.11037v2)

Published 19 Jan 2024 in cs.LG, cs.NA, math.NA, and q-bio.QM

Abstract: Modeling the complex three-dimensional (3D) dynamics of relational systems is an important problem in the natural sciences, with applications ranging from molecular simulations to particle mechanics. Machine learning methods have achieved good success by learning graph neural networks to model spatial interactions. However, these approaches do not faithfully capture temporal correlations since they only model next-step predictions. In this work, we propose Equivariant Graph Neural Operator (EGNO), a novel and principled method that directly models dynamics as trajectories instead of just next-step prediction. Different from existing methods, EGNO explicitly learns the temporal evolution of 3D dynamics where we formulate the dynamics as a function over time and learn neural operators to approximate it. To capture the temporal correlations while keeping the intrinsic SE(3)-equivariance, we develop equivariant temporal convolutions parameterized in the Fourier space and build EGNO by stacking the Fourier layers over equivariant networks. EGNO is the first operator learning framework that is capable of modeling solution dynamics functions over time while retaining 3D equivariance. Comprehensive experiments in multiple domains, including particle simulations, human motion capture, and molecular dynamics, demonstrate the significantly superior performance of EGNO against existing methods, thanks to the equivariant temporal modeling. Our code is available at https://github.com/MinkaiXu/egno.

Definition Search Book Streamline Icon: https://streamlinehq.com
References (43)
  1. Interaction networks for learning about objects, relations and physics. arXiv preprint arXiv:1612.00222, 2016.
  2. Message passing neural PDE solvers. In International Conference on Learning Representations, 2022. URL https://openreview.net/forum?id=vSix3HPYKSU.
  3. Machine learning of accurate energy-conserving molecular force fields. Science advances, 3(5):e1603015, 2017.
  4. CMU. Carnegie-mellon motion capture database. 2003. URL http://mocap.cs.cmu.edu.
  5. Spherical CNNs. In International Conference on Learning Representations, 2018. URL https://openreview.net/forum?id=Hkbd5xZRb.
  6. SE(3) equivariant graph neural networks with complete local frames. In Chaudhuri, K., Jegelka, S., Song, L., Szepesvari, C., Niu, G., and Sabato, S. (eds.), Proceedings of the 39th International Conference on Machine Learning, volume 162 of Proceedings of Machine Learning Research, pp.  5583–5608. PMLR, 17–23 Jul 2022. URL https://proceedings.mlr.press/v162/du22e.html.
  7. Se (3)-transformers: 3d roto-translation equivariant attention networks. arXiv preprint arXiv:2006.10503, 2020.
  8. Neural message passing for quantum chemistry. In International conference on machine learning, pp. 1263–1272. PMLR, 2017.
  9. Equivariant graph hierarchy-based neural networks. Advances in Neural Information Processing Systems, 35:9176–9187, 2022.
  10. Deep residual learning for image recognition. In Proceedings of the IEEE conference on computer vision and pattern recognition, pp.  770–778, 2016.
  11. Equivariant graph mechanics networks with constraints. In International Conference on Learning Representations, 2022. URL https://openreview.net/forum?id=SHbhHHfePhP.
  12. Adam: A method for stochastic optimization. arXiv preprint arXiv:1412.6980, 2014.
  13. Neural relational inference for interacting systems. arXiv preprint arXiv:1802.04687, 2018.
  14. Semi-supervised classification with graph convolutional networks. In International Conference on Learning Representations, 2017. URL https://openreview.net/forum?id=SJU4ayYgl.
  15. Equivariant flows: sampling configurations for multi-body systems with symmetric energies. arXiv preprint arXiv:1910.00753, 2019.
  16. On universal approximation and error bounds for fourier neural operators. Journal of Machine Learning Research, 22:Art–No, 2021a.
  17. Neural operator: Learning maps between function spaces. arXiv preprint arXiv:2108.08481, 2021b.
  18. Neural operator: Graph kernel network for partial differential equations. arXiv preprint arXiv:2003.03485, 2020.
  19. Fourier neural operator for parametric partial differential equations. In International Conference on Learning Representations, 2021. URL https://openreview.net/forum?id=c8P9NQVtmnO.
  20. Spherical message passing for 3d molecular graphs. In International Conference on Learning Representations, 2022. URL https://openreview.net/forum?id=givsRXsOt9r.
  21. Development and current status of the charmm force field for nucleic acids. Biopolymers: original Research on biomolecules, 56(4):257–265, 2000.
  22. Scalable graph networks for particle simulations. AAAI, 2021.
  23. Flexible neural representation for physics prediction. arXiv preprint arXiv:1806.08047, 2018.
  24. Learning mesh-based simulation with graph networks. arXiv preprint arXiv:2010.03409, 2020.
  25. MDAnalysis: A Python Package for the Rapid Analysis of Molecular Dynamics Simulations. In Sebastian Benthall and Scott Rostrup (eds.), Proceedings of the 15th Python in Science Conference, pp.  98 – 105, 2016. doi: 10.25080/Majora-629e541a-00e.
  26. Hamiltonian graph networks with ode integrators. arXiv preprint arXiv:1909.12790, 2019.
  27. Learning to simulate complex physics with graph networks. In International Conference on Machine Learning, pp. 8459–8468. PMLR, 2020.
  28. E(n) equivariant graph neural networks. arXiv preprint arXiv:2102.09844, 2021.
  29. Implicit transfer operator learning: Multiple time-resolution surrogates for molecular dynamics. arXiv preprint arXiv:2305.18046, 2023.
  30. Serre, J.-P. et al. Linear representations of finite groups, volume 42. Springer, 1977.
  31. Molecular dynamics trajectory for benchmarking mdanalysis. URL: https://figshare. com/articles/Molecular_dynamics_ trajectory_for_benchmarking_MDAnalysis/5108170, doi, 10:m9, 2017.
  32. Learning gradient fields for molecular conformation generation. arXiv preprint arXiv:2105.03902, 2021.
  33. How to grow a mind: Statistics, structure, and abstraction. science, 331(6022):1279–1285, 2011.
  34. Tensor field networks: Rotation-and translation-equivariant neural networks for 3d point clouds. arXiv preprint arXiv:1802.08219, 2018.
  35. Lagrangian fluid simulation with continuous convolutions. In International Conference on Learning Representations, 2019.
  36. Attention is all you need. Advances in neural information processing systems, 30, 2017.
  37. Spatial attention kinetic networks with e(n)-equivariance. In The Eleventh International Conference on Learning Representations, 2023. URL https://openreview.net/forum?id=3DIpIf3wQMC.
  38. Accelerating carbon capture and storage modeling using fourier neural operators. arXiv preprint arXiv:2210.17051, 2022.
  39. Pointconv: Deep convolutional networks on 3d point clouds, 2018. URL https://arxiv.org/abs/1811.07246.
  40. Geodiff: A geometric diffusion model for molecular conformation generation. In International Conference on Learning Representations, 2022. URL https://openreview.net/forum?id=PzcvxEMzvQC.
  41. Geometric latent diffusion models for 3d molecule generation. In International Conference on Machine Learning, pp. 38592–38610. PMLR, 2023.
  42. Seismic wave propagation and inversion with neural operators. The Seismic Record, 1(3):126–134, 2021.
  43. Fast sampling of diffusion models via operator learning. In International Conference on Machine Learning, pp. 42390–42402. PMLR, 2023.
Citations (12)
View on Semantic Scholar

Summary

  • The paper introduces a trajectory-based model (EGNO) that captures full temporal evolution of 3D dynamics instead of relying on one-step predictions.
  • It employs SE(3)-equivariance through innovative Fourier-based temporal convolutions, ensuring physically consistent predictions across rotations and translations.
  • Experimental results show EGNO reduces error in simulating particle dynamics, human motion, and molecular interactions, demonstrating its scalable efficiency.

Equivariant Graph Neural Operator for Modeling 3D Dynamics: An Overview

In the paper of dynamic systems, whether they be molecular, mechanical, or cosmological, accurately modeling 3D dynamics is crucial. Traditional numerical methods, although precise, are computationally expensive and often infeasible for large-scale systems. On the other hand, machine learning approaches, especially those utilizing graph neural networks (GNNs), present a more computationally efficient alternative. However, existing methods fall short in capturing the temporal dynamics comprehensively, as they typically focus solely on next-step predictions. The paper "Equivariant Graph Neural Operator for Modeling 3D Dynamics" addresses this limitation by proposing a novel approach: the Equivariant Graph Neural Operator (EGNO).

Methodological Contributions

  1. Trajectory-Based Modeling: Unlike traditional methods that predict one time-step ahead, EGNO models the dynamics as trajectories. This means the method captures the entire evolution of the system over a time window, rather than just the subsequent state. This approach naturally incorporates temporal dependencies that are essential for understanding the system's behavior over time.
  2. SE(3)-Equivariance: A key feature of EGNO is its equivariance with respect to the Special Euclidean group SE(3), which includes rotations and translations. This ensures that the predictions are physically meaningful and consistent with the inherent symmetries of 3D spaces. The equivariance is achieved through innovative use of equivariant temporal convolutions, parameterized in the Fourier domain, maintaining consistency across spatial and temporal transformations.
  3. Fourier-Based Temporal Convolution: EGNO integrates temporal convolutions in Fourier space, leveraging the intrinsic properties of Fourier transforms to capture temporal correlations while preserving spatial symmetries. This is a significant advancement over traditional GNNs, which typically operate in the spatial domain without adequate attention to temporal equilateralism.
  4. Composability with Existing Architectures: The design of the equivariant temporal convolutional layers is general enough to be employed alongside various existing equivariant GNN layers. This makes EGNO a versatile enhancement to other models tasked with modeling physical dynamics.

Experimental Validation

EGNO has been evaluated across multiple domains including particle simulations, human motion capture, and molecular dynamics. The experiments demonstrate clear improvements over state-of-the-art methods:

  • Particle Simulations: EGNO reduces the final mean squared error (F-MSE) in predicting particle dynamics compared to existing models, showcasing its strength in simulating physical forces over time.
  • Motion Capture: In modeling human motion, EGNO achieves significant improvements over competitors by accurately capturing complex kinematic pathways, suggesting its efficacy in applications involving biomechanics.
  • Molecular Dynamics: EGNO excels in predicting the trajectories of small molecules and complex proteins, highlighting its potential utility in computational chemistry and drug discovery.

Implications and Future Directions

The proposed EGNO framework opens up several avenues for future research and application:

  • Scalability and Efficiency: By enabling parallel state predictions within a time window, EGNO presents an opportunity to scale to even more complex systems, such as fluid dynamics or large-scale biological simulations.
  • Interdisciplinary Applications: The robustness of EGNO makes it suitable for integration into various interdisciplinary domains, from astrophysics simulations to the modeling of biological organisms.
  • Improved Accuracy in Physical Simulations: The retention of SE(3) symmetry means that EGNO could enable more accurate physical simulations where rotational and translational invariances are critical.

In summary, the EGNO model represents a significant step forward in the modeling of dynamic systems. By directly addressing the limitations of temporal prediction in GNNs and integrating symmetry considerations, it sets a new standard for trajectory-based modeling in 3D dynamics, offering enhanced accuracy and computational efficiency. The results suggest considerable practical and theoretical implications, paving the way for further innovation in machine learning for the natural sciences.

File Document Download Save Streamline Icon: https://streamlinehq.com PDF File Document Download Save Streamline Icon: https://streamlinehq.com Markdown