Diffusion Tempering Improves Parameter Estimation with Probabilistic
  Integrators for Ordinary Differential Equations
Jonas Beck Nathanael Bosch Michael Deistler Kyra L. Kadhim Jakob H. Macke Philipp Hennig Philipp Berens
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.