In machine learning, a neural differential equation is a differential equation whose right-hand side is parametrized by the weights θ of a neural network.[1] In particular, a neural ordinary differential equation (neural ODE) is an ordinary differential equation of the form
In classical neural networks, layers are arranged in a sequence indexed by natural numbers. In neural ODEs, however, layers form a continuous family indexed by positive real numbers. Specifically, the function maps each positive index t to a real value, representing the state of the neural network at that layer.
Neural ODEs can be understood as continuous-time control systems, where their ability to interpolate data can be interpreted in terms of controllability.[2]
Connection with residual neural networks
editNeural ODEs can be interpreted as a residual neural network with a continuum of layers rather than a discrete number of layers.[1] Applying the Euler method with a unit time step to a neural ODE yields the forward propagation equation of a residual neural network:
with ℓ being the ℓ-th layer of this residual neural network. While the forward propagation of a residual neural network is done by applying a sequence of transformations starting at the input layer, the forward propagation computation of a neural ODE is done by solving a differential equation. More precisely, the output associated to the input of the neural ODE is obtained by solving the initial value problem
and assigning the value to .
Universal differential equations
editIn physics-informed contexts where additional information is known, neural ODEs can be combined with an existing first-principles model to build a physics-informed neural network model called universal differential equations (UDE).[3][4][5][6] For instance, an UDE version of the Lotka-Volterra model can be written as[7]
where the terms and are correction terms parametrized by neural networks.
References
edit- ^ a b Chen, Ricky T. Q.; Rubanova, Yulia; Bettencourt, Jesse; Duvenaud, David K. (2018). "Neural Ordinary Differential Equations" (PDF). In Bengio, S.; Wallach, H.; Larochelle, H.; Grauman, K.; Cesa-Bianchi, N.; Garnett, R. (eds.). Advances in Neural Information Processing Systems. Vol. 31. Curran Associates, Inc. arXiv:1806.07366.
- ^ Ruiz-Balet, Domènec; Zuazua, Enrique (2023). "Neural ODE Control for Classification, Approximation, and Transport". SIAM Review. 65 (3): 735–773. arXiv:2104.05278. doi:10.1137/21M1411433. ISSN 0036-1445.
- ^ Christopher Rackauckas; Yingbo Ma; Julius Martensen; Collin Warner; Kirill Zubov; Rohit Supekar; Dominic Skinner; Ali Ramadhan; Alan Edelman (2024). "Universal Differential Equations for Scientific Machine Learning". arXiv:2001.04385.
- ^ Xiao, Tianbai; Frank, Martin (2023). "RelaxNet: A structure-preserving neural network to approximate the Boltzmann collision operator". Journal of Computational Physics. 490: 112317. arXiv:2211.08149. doi:10.1016/j.jcp.2023.112317.
- ^ Silvestri, Mattia; Baldo, Federico; Misino, Eleonora; Lombardi, Michele (2023), Mikyška, Jiří; de Mulatier, Clélia; Paszynski, Maciej; Krzhizhanovskaya, Valeria V. (eds.), "An Analysis of Universal Differential Equations for Data-Driven Discovery of Ordinary Differential Equations", Computational Science – ICCS 2023, vol. 10476, Cham: Springer Nature Switzerland, pp. 353–366, doi:10.1007/978-3-031-36027-5_27, ISBN 978-3-031-36026-8, retrieved 2024-08-18
- ^ Christoph Plate; Carl Julius Martensen; Sebastian Sager (2024). "Optimal Experimental Design for Universal Differential Equations". arXiv:2408.07143.
- ^ Patrick Kidger (2021). On Neural Differential Equations (PhD). Oxford, United Kingdom: University of Oxford, Mathematical Institute.