Following up on things I learned at the recent IMA workshop, I have become quite impressed by the “equation-free” modeling and simulation approach of teams such as Kevrekidis et al. This is an interesting application of a cutting edge machine learning method, although in a domain that may be somewhat unfamiliar to the typical AI person.

One of the motivating issues is this – with Laplace, one could argue that if we know the structure of a system such as a protein or other macromolecular complex, we could write down the equations of motion, solve them or simulate them on a large computer and make ‘exact’ predictions about dynamic behavior. In fact, what we would be simulating is a multiscale entity that is evolving at timescales ranging from femtoseconds (bonds forming and breaking) to microseconds (large scale evolution of the backbone shape). So, in fact, we have a large collection of atoms that are dancing around according to the laws of (quantum) statistical mechanics whose slow scale behavior yields the near-deterministic evolution of geometry, which determines function and eventually keeps us laughing and kicking. If we simulate the system at this femtosecond level of gory detail then even our best supercomputers take too long to allow us to make meaningful high level statements. More importantly, this seems to imply that the only systems (and families of systems) we can simulate over long time-scales are the ones that we understand well enough to replace the tedious computation with meaningful simplifications – in which case open ended exploration by simulation seems hard to arrive at.

This is where the dimensionality reduction comes in. To the extent that the fast dynamics (that can be simulated knowledge-free over short horizons) is that of an equilibriating ensemble, the configurations settle into some arbitrary distribution that can be captured by a low-dimensional continuum model in suitable reaction coordinates. Machine learning provides numerous algorithms (Isomap, diffusion maps, etc.) for ‘automatically’ discovering these reaction coordinates from the fast time-scale simulation data. Then, instead of the brute -force marching at this time scale, we could leap ahead using the low-dimensional continuum model and restart somewhere else. There is a lot of detail to be handled (e.g., see here). However, when all this comes together, then this brings us closer to the vision of ‘just watching’ the system in order to understand the higher-level behavior. This is exciting!

This doesn’t just apply to molecular dynamics. This notion of automated coarse-grained modeling seems relevant to a variety of applications ranging from neuroscience and population biology to socio-economic systems.

Ram,

There is a detail which was not clear to me: the automated discovery of the low-dimensional spaces does require a lot of data from the smallest time scales, doesn’t it? I was not able to follow Kevrekidis presentation in this regard. My own particular view of this has to do with the two-time scale models of nonlinear dynamical systems. There, the slower time-scale can be solved statically as a equilibrium equation (“x* st. g(x^*) = 0”), and the faster time scale is then solved around x^* (dot{(x-x^*)) = f(x-x^*)) . I guess it is all the same idea, except there the model is given by f and g.

One nice question might be, given data from such a model “f,g”, can you learn g? The correct equations can be found in Shastry’s book. By the way, the blog is great!

Ram

Yes, it is essentially the time scale separation idea and they are precisely interested in your last question. Specifically, if you are simulating a new molecule or system and you want to quickly see the big picture, can you bootstrap your way from the molecular dynamics simulations.

However, remember that there is the dimensionality aspect in all this – the fast dynamics is that of a million atoms in their statistical mechanical glory while the slow dynamics is (hopefully) that of a much simpler and lower-dimensional continuum model (master equation) that captures the relevant information. Then, instead of chugging along at femtoseconds, one could jump at the larger timestep.