 Hi everyone, I'm Ding Ling. Last time we learned that differential equations are useful from many perspectives. Today we will look into what is actually an ordinary differential equation and how do classic ODE servers work. An ODE is an equation involved with some ordinary derivatives, given a particular initial value, but not. The derivative function is called the vector field F. Mostly our goal is to find some function yt, which satisfies the ODE and the initial value at the same time. This is so-called the solution to this initial value problem. One of the most common ODEs is the logistic equation, which is often used to model initial growth of the population size in a particular species in an environment. The derivative function is co-determined by the population size pt and the growth rate r and the current capacity k. Given these parameters and the initial value, this figure shows the solution to this logistic equation. The gray arrows depicts the vector field. But there remains a question, namely how can we solve ODE's more generally? Well, one could come up with some smart transformation trick and hope the chain rule will do its work. But for more general ODE's, guessing almost never works. A simple but generally working method is the Euler's method. Starting from the initial value, it goes iteratively forward in the time dimension. To take a closer look, it works actually like a learning agent. Firstly, it observes the current vector field value f and then it predicts the next value based on the current value pt and shifts it by the product of step size h and the vector field, namely our observation f. Well, as you may have noticed, there is a problem. Namely, we have some fixed step size, which means there are only finite steps available. At the result, there will be some numerical uncertainties, which are unfortunately completely ignored by the classic ODE servers. To conclude, classic ODE servers are like learning machines. However, there are important details missing. That's exactly why we need a probabilistic version of that, the probabilistic ODE servers. If you are interested, please do not forget to follow up. And in the next video, Nico will give an elaborate explanation to this topic. Thanks for watching. Bye.