 Hi, I'm Nina. In the last videos you already heard a lot about probabilistic ODE solvers. All of those solvers were based on filter methods. And today I want to show you another way of solving ODE's probabilistically with sampling-based solvers. So first of all, credits to the people who came up with these methods, Giacomo Garegnani, Asi Abdulet and Patrick Conrad. What actually are sampling-based ODE solvers? Sampling-based ODE solvers are probabilistic ODE solvers that are based on classic iterative methods. For example, Runge-Kutter methods, and they for sure take an ODE into account. By introducing some noise-term psi, the deterministic solver becomes non-deterministic and probabilistic. How can this be done? Abdulet and Garegnani propose to put up the step psi. So instead of evaluating the ODE at the position t, it is evaluated at a slightly shifted position t plus some noise term. And Conrad et al propose to put up the original solution. So instead of taking the deterministic solution y of t, we add some scaled version of the error estimation to the output. How do we achieve a solution to the ODE? We solve the ODE not only once, but multiple times. And vice is actually useful. The solution, or what we call the solution, is actually just an approximation to the solution of the ODE. And those uncertainties of the ODE can also be non-gaussian. This is what the filter-based methods can't take into account. The error estimation is not in addition to the solution, but it can also be a part of the solution. And the step size is not just some parameter, but it directly influences the solution. So how does this look like in practice? In this case, you can see two samples of the Lorentz system solved by an ODE solver as proposed by Conrad et al. You can see that the trajectories are really similar in the beginning, and they start to go a little bit into different directions after a while. And if we wait even longer, they go into completely different directions. This is not an accident, but actually a property of the Lorentz system. The Lorentz system is symmetric. So we can see that sampling-based methods can give us an intuition for the uncertainty of the solution, but they can also show us some properties of the ODE. Or something like that.