 Hi, my name is Katharina Ott and today I would like to talk to you about differential equations. So why should you even care about differential equations? Differential equations are everywhere, so let's look at a few examples. One example are physical systems. For example, the well-known equations of motion, here depicted by this pendulum. Another prominent example are fluid dynamics, where differential equations also play a crucial role. In the image, you can see the result of a simulation of a shock wave passing through two fluids with different densities. In biochemical systems, differential equations are for example needed for mechanical reaction networks. And in the image on the lower right, you can see the result of a simulation of an infectious disease using a compartmental model, as used in epidemiology. There are of course many other examples not shown on the slide, like climate modeling or economics. But let's take a look at a concrete example. The SIRD model used for modeling an infectious disease. So the SIRD model describes the number of individuals susceptible, infectious, recovered and diseased. And the differential equations shown on the left side describe this model. You don't need to care about the exact form of these differential equations right now. So the parameter beta in these equations describes the rate of infection. And one way of changing this parameter beta is by social distancing, which leads to a decrease in the value of beta. So if we are lowering the value of beta, then we can observe the effect shown in the animation, which is also known as flattening the curve. We also observe that there is a non-linear dependence between the parameter beta and the model. So if we are given some data and we want to inferior the value of beta, then this becomes a non-revealed task. Another question a machine learner might ask is how to use this model to make predictions into the future? I hope I was able to convince you with the simple example that differential equations can pose some really interesting machine learning challenges. And in the next videos, we'll give you some tools to tackle such challenges.