In this talk I will discuss how information about the long time dynamical behaviour of dynamical systems can be extracted from simulation trajectory data. I will start by considering the much more well-known case of learning low-dimensional geometrical structure from point cloud data in some metric space, where many established unsupervised machine learning methods exist. I will then advocate an approach of first constructing a ‘dynamical distance’ which incorporates information about the dynamical features one is interested in, and mapping the data into a space where the Euclidean distance equals the dynamical distance. This approach has the advantage that after the mapping is constructed, all the established geometrical learning methods can be used to find structure.
I will showcase this strategy on two examples. The first is the commute map, which defines distance based on the time it takes to travel to and forth between two regions. I will show how the commute distance can be estimated from Molecular Dynamics simulation data, and discuss the relationship with the diffusion distance. Then I will show how reduced order kinetic models can be computed by simply using geometric clustering approaches and the commute map embedding. The second example is a time-averaged diffusion map, which defines distance based on mutual closeness over a finite time interval. I will show that the time-averaged diffusion map together with spectral clustering leads to the uncovering of coherent structures, which are structures that are dynamically stable.