# Solving partial differential equations with generative modeling

Another field in which deep learning in general, and generative learning in particular, have led to recent breakthroughs is the solution of **partial differential equations** (**PDEs**), a kind of mathematical model used for diverse applications including fluid dynamics, weather prediction, and understanding the behavior of physical systems. More formally, a PDE imposes some condition on the partial derivatives of a function, and the problem is to find a function that fulfills this condition. Usually some set of initial or boundary conditions is placed on the function to limit the search space within a particular grid. As an example, consider Burger's equation,^{8} which governs phenomena such as the speed of a fluid at a given position and time (*Figure 13.8*):

Where *u* is speed, *t* is time, *x* is a positional coordinate, and is the viscosity ("oiliness") of the fluid. If the viscosity is 0, this simplifies...