Joe's Notes

    Other architectures

    There is a wide array of literature on architectures and methods that try to embed ideas from classical mechanics to target nicher, more specific problems with the tools of physics-informed neural networks. We will discuss in what follows some of those contributions.

    In the study of Liouville integrable systems, systems with "enough" conserved quantities, Daigavane et al. introduce Action-Angle Networks for temporally evolving these systems using the concept of Action-Angle coordinates1. Also, Ishikawa et al. introduce a data-driven approach to learn the Hamiltonian of integrable systems (tested in the paper on the Toda lattice problem)2.

    Another interesting method is Generating Function Neural Network, introduced by Chen and Tao3.

    Karniadakis's group has published two papers on the use of the Hamilton-Jacobi equation in neural networks in application to optimal control problems4 and 5. See The Hamilton-Jacobi Equation in Optimal Control Setting.

    1

    Daigavane, A., Kosmala, A., Cranmer, M., Smidt, T., & Ho, S. (2022). Learning Integrable Dynamics with Action-Angle Networks. https://arxiv.org/abs/2211.15338

    2

    Ishikawa, F., Suwa, H., & Todo, S. (2021). Neural Network Approach to Construction of Classical Integrable Systems. Journal of the Physical Society of Japan, 90(9), 093001. https://doi.org/10.7566/jpsj.90.093001

    3

    Chen, R., & Tao, M. (2021). Data-driven Prediction of General Hamiltonian Dynamics via Learning Exactly-Symplectic Maps. https://arxiv.org/abs/2103.05632

    4

    Meng, T., Zhang, Z., Darbon, J., & Karniadakis, G. E. (2022). SympOCnet: Solving optimal control problems with applications to high-dimensional multi-agent path planning problems. https://arxiv.org/abs/2201.05475

    5

    Zhang, Z., Wang, C., Liu, S., Darbon, J., & Karniadakis, G. (2024). A time-dependent symplectic network for non-convex path planning problems with linear and nonlinear dynamics. https://arxiv.org/abs/2408.03785