It is has been quite a while since I last posted. I was reminded to write something by a nice remark about the blog from a colleague at Microsoft who I met through an Engineering and Physical Sciences Research Council event. He also pointed me to a fascinating blog by Neil Dalchau at Microsoft on how to use machine learning to pin down parameters in models that can only be solved numerically. I was very excited by this because, as it happens, this has become a focus of our own research.
Before I discuss that in a future blog though, I want to share what I think is an exciting example of where machine learning can help us learn more from existing data than we previously thought possible.
First, recall that I’m fascinated by the physics behind structural evolution that leads to images like these:
Ultimately we want to know how a structure relates to some property of the material, which might be its strength or how it absorbs light, depending on the intended application. Ideally, we want a succinct way to characterise these structures. A very powerful experimental approach involves shining a beam of some sort (neutrons are particularly good if the mixture is different polymers, but X-rays or light can also be used), and observing how the beam is affected as it exits the material. A significant piece of information we obtain is a length scale, which tells us something about the size of the structures inside our material. The problem is that the length scale is not unique to a given structure. As an example, the two structures above have the same length scale according to a scattering experiment, as shown in the figure below, but just by looking at images above we can see that they are different. On the left, the structure corresponds to droplets of one type of polymer in a matrix of another, whilst we refer to the structure on the right as co-continuous.
By borrowing tools from cosmology, scientists in the 1990s proposed some extra measures that can characterise the structure. These characteristics, known as Minkowski functionals, are volume, surface area, curvature and connectivity. Volume represents the relative volume occupied by each of the two phases. Surface area is the total contact area between the two structures. Curvature is a measure of how curved the structures are – small spheres have a much higher curvature than large spheres. Connectivity, also called the Euler measure, tells us how easy it is to trace a path from one part of the structure that is occupied by one of the polymers to another that is occupied by the same polymer without having to pass through regions occupied by the other polymer. A droplet structure has low connectivity – if you are inside one droplet you cannot trace e a path to another droplet without passing through the matrix of the other polymer. An ideal co-continuous structure will have perfect connectivity – you can reach anywhere that is also occupied by the same polymer without having to pass through a region occupied by the other polymer.
The challenge with experimentally determining Minkowski functionals is that it remains difficult to obtain three dimensional images of internal structures. Confocal microscopy is one tool for doing this, but it is limited to structures of the order of microns, whilst many mixtures have structures that are much smaller. There have been amazing developments in X-ray and neutron tomography which do enable the images to be obtained, but it is still much more time consuming and costly compared to scattering experiments.
It would be really nice if we could learn more from the scattering than just the length scale. The question we asked ourselves was is there a relation between the scattering and Minkowski functionals? A strong motivation for this is that we know that the volume can be obtained from the scattering from a reasonably simple mathematical relation, so perhaps it is also the case that the other functionals are also hidden in the scattering data.
To answer this question we trained some machine learning algorithms on pairs of scattering data and Minkowski functionals both calculated from the same microstructure image. We did this for approximately 200 different pairs with different microstructures. We then tested the algorithm to predict the Minkowski functionals given 200 different scattering data that it had not previously seen and compared the ML predictions with the direct calculations. The results were considerably more impressive than I anticipated!
We’ve since refined our algorithm further and the early indications are that we can do even better at the extreme values, which is a reminder that whilst machine learning is a black box technique, how the question is framed and how the data is prepared remain crucial to success.