Physics-Informed Symplectic Operator Network

Overview

PI-SONet is a structure-preserving operator learning framework for parameterized families of optimal control problems. It combines a latent Hamiltonian solver with a conditional symplectic operator to approximate PMP solution operators while preserving symplectic structure by construction.

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Animations

The animations/ folder contains animations corresponding to the figures in the paper.

Animation	Description
animations/figure2/free.gif	Multi-agent collision avoidance in free space (Figure 2a).
animations/figure2/obstacle.gif	Multi-agent navigation around a circular obstacle (Figure 2b).
animations/figure2/maze.gif	Multi-agent navigation in a maze-like environment (Figure 2c).
animations/figure3/small.gif	Generalization to a small circular obstacle (Figure 3a).
animations/figure3/intermediate.gif	Generalization to an intermediate circular obstacle (Figure 3b).
animations/figure3/large.gif	Generalization to a large circular obstacle (Figure 3c).
animations/figure4/two_dim.gif	Planar navigation with heterogeneous agents around a circular obstacle (Figure 4a).
animations/figure4/three_dim.gif	Three-dimensional navigation with heterogeneous agents in a cluttered environment (Figure 4b).
animations/figure4/three_dim.html	Interactive 3D visualization of Figure 4b. Download and open in any browser to rotate, zoom, and inspect the three-dimensional trajectories freely.

In all animations, colored curves denote agent trajectories, blue arrows denote velocity vectors, and red arrows denote control inputs.

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Code

🔒 The full codebase will be released upon acceptance of the paper. Stay tuned!

In the meantime, feel free to reach out with any questions.

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Citation

If you find this work useful, please consider citing:

@misc{varghese2026pisonetphysicsinformedsymplecticoperator,
      title={PI-SONet: A Physics-Informed Symplectic Operator Network for Real-Time Optimal Control of Multi-Agent Systems}, 
      author={Alan John Varghese and Shanqing Liu and Paula Chen and Yaochen Zhu and Jérôme Darbon and George Em Karniadakis},
      year={2026},
      eprint={2605.14332},
      archivePrefix={arXiv},
      primaryClass={math.OC},
      url={https://arxiv.org/abs/2605.14332}, 
}

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Contact

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