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.

翻译：常微分方程（ODE）被广泛应用于描述科学中的动态系统，但确定解释实验测量的参数仍具挑战性。特别地，尽管ODE具有可微性且支持基于梯度的参数优化，其非线性动力学常导致大量局部极小值并使得初始条件敏感性极强。为此，我们提出扩散退火——一种面向概率数值方法的新型正则化技术，可改善ODE中基于梯度的参数优化收敛性。通过逐步降低概率积分器的噪声参数，所提方法能够更可靠地收敛至真实参数。我们证明该方法对具有不同复杂度的动态系统均有效，并展示其在霍奇金-赫胥黎模型中能够针对具有实际相关参数数量的情形获得可靠参数估计。