An Interesting Modelling Problem – Magnetic Equilibria

I have just returned from a visit to the Max Planck Institute for Plasma Physics. The purpose of my visit was to explore the possibility of using some recent advances in nonlinear dimensionality reduction and related modelling to the problem of regime identification in tokamaks.

The core issue is that of using magnetic measurements at various points on the vacuum vessel and some additional measurements, such as from Motional Stark Effect, inside the vacuum vessel to identify the plasma’s flux profile, boundary and essentially the equilibrium configuration it is close to. All of this is defined by an analytical model called the Grad Shafranov PDE but accurately solving it is quite intensive and requires a distibuted computing environment, i.e., a reasonably large cluster. So, what the diagnostics and control people actually do is collect a database of equilibria from off-line computations, fit a simpler statistical model and then use that in the experimental system. Currently, this statistical model is a regression model that is something like a ‘quadratic’ PCA model (called function parameterization).

I got interested in this issue during my final few months at National Instruments, during a visit by one of the ASDEX Upgrade scientists involved with diagnostics. He is currently involved in setting up a closed loop system that will use all these methods to identify the regime, locate possible instabilities (e.g., turbulence that cools down the plasma) and ‘shoot them down’ using steerable mirrors that focus a beam that locally heats up the plasma – as in this figure:

My hypothesis is that NLDR algorithms may enable models that achieve at least the current levels of accuracy with reduced levels of computation, and perhaps even pick up on ‘quirks’ that matter for real time control. However, all this is still a bit speculative. For one thing, it may turn out that the function parameterization statistical procedure currently in use for the ASDEX Upgrade is capturing all there is to capture. Moreover, over time, it will surely become possible to solve the PDEs in real time and this extra step becomes redundant.

However, despite all this, the thing that intrigues me about this problem is this (somewhat orthogonal to the practical goal, but reason enough for me to spend a bit of time on it) – do techniques like NLDR actually recover abstractions that correspond directly to the essence of the problem in the way that the physical models are understood, e.g., if we recovered principal directions on a submanifold describing all magnetic equilibria in the database and built models using them then does the result directly correspond to analytical solutions of the Grad Shafranov PDE? If this happens, I would be very pleased indeed and it would say something important about the properties of these learning algorithms!

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