A recurring movement within deep learning research which tries to render the learning of prediction functions tractable by considering them as dynamical systems, and using the theory of stability in the context of Hamiltonians,r optimal control and/or ODE solvers, to make it all work.

I’ve been interested by this ever since seeing the Haber and Ruthotto paper, but it’s got a real kick recently since the Vector Institute team’s paper won the prize at NeurIPS for learning the ODEs themselves.

# Stability of training

Related, but not quite the same, notion of stability, as in data-stability in learning. Arguing that neural networks are in the limit approximants to quadrature solutions of certain ODES, work and gain insights and new tricks into neural nets by using ODE tricks.. This is mostly what Haber and Rhutthoto et al do. ([Haber, Ruthotto, Holtham, & Jun, 2017][#HRHJ17], [Haber, Lucka, & Ruthotto, 2018][#HaLR18], [Ruthotto & Haber, 2018][#RuHa18])

## Can it work on time series?

Good question; It looks like it should, since there is an implicit time series the ODE-solver. But these problems so far have use non-time-series data.

## Neural ODE regression

By which I mean *learning an ODE whose solution is the regression problem*. This is what the Vector Institute paper did. There are various laypersons’ introductions to this, including the simple and practical magical take in julia.

There are some syntheses of these approaches that try to do everything with ODEs, all the time. [Niu, Horesh, & Chuang, 2019][#NiHC19], [Rackauckas et al., 2018][#RMDG18], and even some tutorial implementations by the inexhaustible Chris Rackauckas.

## Random stuff

My question: How can this be made Bayesian? Priors on dynamics, posterior uncertainties etc.

TBC. Lyapunov analysis, Hamiltonian dynamics.

## Refs

- RMDG18: Christopher Rackauckas, Yingbo Ma, Vaibhav Dixit, Xingjian Guo, Mike Innes, Jarrett Revels, … Vijay Ivaturi (2018) A Comparison of Automatic Differentiation and Continuous Sensitivity Analysis for Derivatives of Differential Equation Solutions.
*ArXiv:1812.01892 [Cs]*. - WiBö15: Thomas Wiatowski, Helmut Bölcskei (2015) A Mathematical Theory of Deep Convolutional Neural Networks for Feature Extraction. In Proceedings of IEEE International Symposium on Information Theory.
- EHL18: Weinan E, Jiequn Han, Qianxiao Li (2018) A Mean-Field Optimal Control Formulation of Deep Learning.
*ArXiv:1807.01083 [Cs, Math]*. - E17: Weinan E (2017) A Proposal on Machine Learning via Dynamical Systems.
*Communications in Mathematics and Statistics*, 5(1), 1–11. DOI - GhKB19: Amir Gholami, Kurt Keutzer, George Biros (2019) ANODE: Unconditionally Accurate Memory-Efficient Gradients for Neural ODEs.
*ArXiv:1902.10298 [Cs]*. - CCHC19: Bo Chang, Minmin Chen, Eldad Haber, Ed H. Chi (2019) AntisymmetricRNN: A Dynamical System View on Recurrent Neural Networks. In Proceedings of ICLR.
- Haro08: A. Haro (2008) Automatic differentiation methods in computational dynamical systems: Invariant manifolds and normal forms of vector fields at fixed points.
*IMA Note*. - LiSY18: Hanxiao Liu, Karen Simonyan, Yiming Yang (2018) DARTS: Differentiable Architecture Search.
*ArXiv:1806.09055 [Cs, Stat]*. - RuHa18: Lars Ruthotto, Eldad Haber (2018) Deep Neural Networks motivated by Partial Differential Equations.
*ArXiv:1804.04272 [Cs, Math, Stat]*. - MHRB17: Zakaria Mhammedi, Andrew Hellicar, Ashfaqur Rahman, James Bailey (2017) Efficient Orthogonal Parametrisation of Recurrent Neural Networks Using Householder Reflections. In PMLR (pp. 2401–2409).
- WiGB17: Thomas Wiatowski, Philipp Grohs, Helmut Bölcskei (2017) Energy Propagation in Deep Convolutional Neural Networks.
*IEEE Transactions on Information Theory*, 1–1. DOI - MWCW16: Qi Meng, Yue Wang, Wei Chen, Taifeng Wang, Zhi-Ming Ma, Tie-Yan Liu (2016) Generalization Error Bounds for Optimization Algorithms via Stability. In arXiv:1609.08397 [stat] (Vol. 10, pp. 441–474).
- HSNB19: Junxian He, Daniel Spokoyny, Graham Neubig, Taylor Berg-Kirkpatrick (2019) Lagging Inference Networks and Posterior Collapse in Variational Autoencoders. In PRoceedings of ICLR.
- HRHJ17: Eldad Haber, Lars Ruthotto, Elliot Holtham, Seong-Hwan Jun (2017) Learning across scales - A multiscale method for Convolution Neural Networks.
*ArXiv:1703.02009 [Cs]*. - CMHT18: Bo Chang, Lili Meng, Eldad Haber, Frederick Tung, David Begert (2018) Multi-level Residual Networks from Dynamical Systems View. In PRoceedings of ICLR.
- ChGS15: Tianqi Chen, Ian Goodfellow, Jonathon Shlens (2015) Net2Net: Accelerating Learning via Knowledge Transfer.
*ArXiv:1511.05641 [Cs]*. - CRBD18: Tian Qi Chen, Yulia Rubanova, Jesse Bettencourt, David K Duvenaud (2018) Neural Ordinary Differential Equations. In Advances in Neural Information Processing Systems 31 (pp. 6572–6583). Curran Associates, Inc.
- HaLR18: Eldad Haber, Felix Lucka, Lars Ruthotto (2018) Never look back - A modified EnKF method and its application to the training of neural networks without back propagation.
*ArXiv:1805.08034 [Cs, Math]*. - NiHC19: Murphy Yuezhen Niu, Lior Horesh, Isaac Chuang (2019) Recurrent Neural Networks in the Eye of Differential Equations.
*ArXiv:1904.12933 [Quant-Ph, Stat]*. - CMHR18: Bo Chang, Lili Meng, Eldad Haber, Lars Ruthotto, David Begert, Elliot Holtham (2018) Reversible Architectures for Arbitrarily Deep Residual Neural Networks. In arXiv:1709.03698 [cs, stat].
- AnLG18: Cem Anil, James Lucas, Roger Grosse (2018) Sorting out Lipschitz function approximation.
- HaRu18: Eldad Haber, Lars Ruthotto (2018) Stable architectures for deep neural networks.
*Inverse Problems*, 34(1), 014004. DOI - BaKS14: Ann C. Babtie, Paul Kirk, Michael P. H. Stumpf (2014) Topological sensitivity analysis for systems biology.
*Proceedings of the National Academy of Sciences*, 111(52), 18507–18512. DOI - HaRS15: Moritz Hardt, Benjamin Recht, Yoram Singer (2015) Train faster, generalize better: Stability of stochastic gradient descent.
*ArXiv:1509.01240 [Cs, Math, Stat]*. - JSDP17: Li Jing, Yichen Shen, Tena Dubcek, John Peurifoy, Scott Skirlo, Yann LeCun, … Marin Soljačić (2017) Tunable Efficient Unitary Neural Networks (EUNN) and their application to RNNs. In PMLR (pp. 1733–1741).
- ArSB16: Martin Arjovsky, Amar Shah, Yoshua Bengio (2016) Unitary Evolution Recurrent Neural Networks. In Proceedings of the 33rd International Conference on International Conference on Machine Learning - Volume 48 (pp. 1120–1128). New York, NY, USA: JMLR.org