Modeling Energy Confinement in Plasma Devices by Neural Networks

Energy confinement data of large fusion devices are analyzed in terms of reduced variables which consist of certain combinations of the machine variables. The goal is to predict the single variable behaviour from a data set with entries which differ in more than one variable setting from each other. Bayesian neural networks are used to model the hyper-surface evolving for the energy confinement as a function of the reduced variables. In order to tell which neural net is best and to provide expectation values the multi-dimensional multi-modal marginalization integrals are calculated employing a Monte Carlo method called perfect tempering. INTRODUCTION In the 1998 conference of this series we presented a paper [1] which analyzed energy confinement data of the stellarator W7-AS in terms of dimensionless form free scaling functions. Since no first principles theory exists to describe the energy confinement in such a device empirical scaling laws are employed to characterize the plasma machine properties. These are for W7-AS the particle density n, magnetic field B, heating power P and the minor radius r, apart from other – dimensionless – quantities like the rotational transform . Though the functional form of the scaling law is unknown one can use basic plasma models in order to create a set of reduced variables with which one composes the scaling function. This was achieved by Connor and Taylor [2] in requiring that the invariances of plasma models under similarity transformations should be the same as those of the scaling function belonging to the respective model. Four kinetic models (see table 1) were obtained in considering the Boltzmann equation of motion (describing a collisional plasma) or the Vlasov equation (collisionless plasma) and discriminating further between the electrostatic limit (lowcase) or a self-consistent calculation of the electro-magnetic fields from the Maxwell equations (high). By the invariance requirement the plasma variables are combined in one to three terms xj with scaling exponents j according to the basic plasma model (see table 1). W theo nr4B2 / 1 P nr4B3 1 2 r3B4 n ! 2 3 1 nr2 3 (1) = f (x; ) : (2) The constant in (2) carries fundamental physical constants and units and contains the product 1 1 2 2 3 3 . The scaling terms are represented by a dimensionless function f(x; )