# Diffusion Tempering Improves Parameter Estimation with Probabilistic Integrators for Ordinary Differential Equations

arXiv: https://arxiv.org/abs/2402.12231 · HTML: https://mlsanity.com/paper/2402.12231/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- Builds on 14 of its 44 references that are in the map.
- First posted on arXiv 19 Feb 2024.

## Builds on (most cited first, 14 of 14)

- [Neural Ordinary Differential Equations](https://mlsanity.com/paper/1806.07366/) (2018)
- [The frontier of simulation-based inference](https://mlsanity.com/paper/1911.01429/) (2019)
- [Flexible statistical inference for mechanistic models of neural dynamics](https://mlsanity.com/paper/1711.01861/) (2017)
- [Adding Gradient Noise Improves Learning for Very Deep Networks](https://mlsanity.com/paper/1511.06807/) (2015)
- [Bayesian Probabilistic Numerical Methods](https://mlsanity.com/paper/1702.03673/) (2017)
- [A probabilistic model for the numerical solution of initial value problems](https://mlsanity.com/paper/1610.05261/) (2016)
- [Probabilistic Solutions To Ordinary Differential Equations As Non-Linear Bayesian Filtering: A New Perspective](https://mlsanity.com/paper/1810.03440/) (2018)
- [Bayesian ODE Solvers: The Maximum A Posteriori Estimate](https://mlsanity.com/paper/2004.00623/) (2020)
- [Stable Implementation of Probabilistic ODE Solvers](https://mlsanity.com/paper/2012.10106/) (2020)
- [Convergence Rates of Gaussian ODE Filters](https://mlsanity.com/paper/1807.09737/) (2018)
- [Probabilistic ODE Solutions in Millions of Dimensions](https://mlsanity.com/paper/2110.11812/) (2021)
- [Fenrir: Physics-Enhanced Regression for Initial Value Problems](https://mlsanity.com/paper/2202.01287/) (2022)
- [Laplace based approximate posterior inference for differential equation models](https://mlsanity.com/paper/1607.07203/) (2016)
- [Evolutionary algorithms as an alternative to backpropagation for supervised training of Biophysical Neural Networks and Neural ODEs](https://mlsanity.com/paper/2311.10869/) (2023)

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

- [Adaptive Probabilistic ODE Solvers Without Adaptive Memory Requirements](https://mlsanity.com/paper/2410.10530/) (2024)

## Abstract (arXiv)

Ordinary differential equations (ODEs) are widely used to describe dynamical systems in science, but identifying parameters that explain experimental measurements is challenging. In particular, although ODEs are differentiable and would allow for gradient-based parameter optimization, the nonlinear dynamics of ODEs often lead to many local minima and extreme sensitivity to initial conditions. We therefore propose diffusion tempering, a novel regularization technique for probabilistic numerical methods which improves convergence of gradient-based parameter optimization in ODEs. By iteratively reducing a noise parameter of the probabilistic integrator, the proposed method converges more reliably to the true parameters. We demonstrate that our method is effective for dynamical systems of different complexity and show that it obtains reliable parameter estimates for a Hodgkin-Huxley model with a practically relevant number of parameters.