Empirical studies of the scaling of Tokamak energy confinement times with machine parameters constitute a useful point of contact with physics-based transport theories. They also form the basis for the design of next-step and reactor grade Tokamaks. In most cases a simple power law (or sometimes offset linear) functional form is fitted to the data. Such linear regression techniques have the advantage of computational simplicity, but otherwise have little a-priori justification. Neural networks provide a powerful general-purpose technique for nonlinear regression which exhibits no essential limitations on the functional form which can be fitted. In this paper we apply neural networks to the problem of energy confinement scaling in Tokamaks, and we illustrate the technique using data from the JET (Joint European Torus) Tokamak. The results show that the neural network approach leads to a substantial improvement in the ability to predict the energy confinement time as compared with linear regression. The significance of this result is discussed.