# Path differentiability of ODE flows

arXiv: https://arxiv.org/abs/2201.03819 · HTML: https://mlsanity.com/paper/2201.03819/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 39 references that are in the map.
- First posted on arXiv 11 Jan 2022.

## Builds on (most cited first, 8 of 8)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [Optimization Methods for Large-Scale Machine Learning](https://mlsanity.com/paper/1606.04838/) (2016)
- [Augmented Neural ODEs](https://mlsanity.com/paper/1904.01681/) (2019)
- [Conservative set valued fields, automatic differentiation, stochastic gradient method and deep learning](https://mlsanity.com/paper/1909.10300/) (2019)
- [A mathematical model for automatic differentiation in machine learning](https://mlsanity.com/paper/2006.02080/) (2020)
- [Differential Inclusions in Wasserstein Spaces: The Cauchy-Lipschitz Framework](https://mlsanity.com/paper/2007.08906/) (2020)
- [Conservative and semismooth derivatives are equivalent for semialgebraic maps](https://mlsanity.com/paper/2102.08484/) (2021)
- [Examples of pathological dynamics of the subgradient method for Lipschitz path-differentiable functions](https://mlsanity.com/paper/2007.11699/) (2020)

## Built on by (most cited first, 1 of 1)

- [Automatic differentiation of nonsmooth iterative algorithms](https://mlsanity.com/paper/2206.00457/) (2022)

## Abstract (arXiv)

We consider flows of ordinary differential equations (ODEs) driven by path differentiable vector fields. Path differentiable functions constitute a proper subclass of Lipschitz functions which admit conservative gradients, a notion of generalized derivative compatible with basic calculus rules. Our main result states that such flows inherit the path differentiability property of the driving vector field. We show indeed that forward propagation of derivatives given by the sensitivity differential inclusions provide a conservative Jacobian for the flow. This allows to propose a nonsmooth version of the adjoint method, which can be applied to integral costs under an ODE constraint. This result constitutes a theoretical ground to the application of small step first order methods to solve a broad class of nonsmooth optimization problems with parametrized ODE constraints. This is illustrated with the convergence of small step first order methods based on the proposed nonsmooth adjoint.