Solvers for ordinary differential equations (ODEs) belong to the best-studied algorithms of numerical mathematics. An ODE is an implicit statement about the relationship of a curve $x:\mathbb{R}_{\geq 0}\to\mathbb{R}^N$ to its derivative, in the form $x'(t) = f(x(t),t)$, where $x'$ is the derivative of curve, and $f$ is some function. To identify a unique solution of a particular ODE, it is typically also necessary to provide additional statements about the curve, such as its *initial value* $x(t_0)=x_0$. An ODE solver is a mathematical rule that maps function and initial value, $(f,x_0)$ to an estimate $x(t)$ for the solution curve. *Good* solvers have certain analytical guarantees about this estimate, such as the fact that its deviation from the true solution is of a high polynomial order in the step size used by the algorithms to discretize the ODE.

One of the main theoretical contributions of the group is the development of *probabilistic* versions of these solvers. In several works, we established a class of solvers for initial value problems that generalize classic solvers by taking as inputs Gaussian distributions $\mathcal{N}(x(t_0);x_0,\Psi)$, $\mathcal{GP}(f;\hat{f},\Sigma)$ over the initial value and vector field, and return a Gaussian process posterior $\mathcal{GP}(x;m,k)$ over the solution. We were able to show that these methods

- have the same (linear) computational computational complexity in solver's step-size $h$ as classic methods [ ] (they are Bayesian filters)
- can inherit the famous local and global polynomial convergence rates of classic solvers [ ] (i.e. $\|m-x\|\leq Ch^q$ for $q\geq 1$)
- produce posterior variance estimates that are calibrated worst-case error estimates [ ] (i.e. $\|m-x\|^2\leq Ck$). In Short, they produce meaningful uncertainty
- are in fact a generalization of certain famous classic ODE solvers (namely they reduce to explicit single-step Runge Kutta methods and multi-step Nordsieck methods in the limit of uninformative prior and steady-state operation, respectively. In practical operation, they offer a third, novel type of solver) [ ]
- they can be generalized to produce non-Gaussian, nonparametric output while retaining many of the above properties [ ].

Together, these results provide a rich and reliable new theory for probabilistic simulation that current ongoing research projects are seeking to leverage to speed up structured simulation problems inside of machine learning algorithms.

6 results

**Probabilistic Solutions To Ordinary Differential Equations As Non-Linear Bayesian Filtering: A New Perspective**
*ArXiv preprint 2018*, arXiv:1810.03440 [stat.ME], October 2018 (article)

**Convergence Rates of Gaussian ODE Filters**
*arXiv preprint 2018*, arXiv:1807.09737 [math.NA], July 2018 (article)

**A probabilistic model for the numerical solution of initial value problems**
*Statistics and Computing*, Springer US, 2018 (article)

**Active Uncertainty Calibration in Bayesian ODE Solvers**
*Proceedings of the 32nd Conference on Uncertainty in Artificial Intelligence (UAI)*, pages: 309-318, (Editors: Ihler, A. and Janzing, D.), AUAI Press, June 2016 (conference)

**Probabilistic ODE Solvers with Runge-Kutta Means**
In *Advances in Neural Information Processing Systems 27*, pages: 739-747, (Editors: Z. Ghahramani, M. Welling, C. Cortes, N.D. Lawrence and K.Q. Weinberger), Curran Associates, Inc., 28th Annual Conference on Neural Information Processing Systems (NIPS), 2014 (inproceedings)

**Probabilistic Shortest Path Tractography in DTI Using Gaussian Process ODE Solvers**
In *Medical Image Computing and Computer-Assisted Intervention – MICCAI 2014, Lecture Notes in Computer Science Vol. 8675*, pages: 265-272, (Editors: P. Golland, N. Hata, C. Barillot, J. Hornegger and R. Howe), Springer, Heidelberg, MICCAI, 2014 (inproceedings)