Method

A versatile method for treating empirical data is the neural network in a Bayesian framework. The theory behind practical Bayesian networks has been described in [14,13] and the background information theory is available in a seminal textbook on the subject [15]. In addition, this method been reviewed thoroughly [11], as have been its applications [16]. Indeed, there have been diverse applications which lead to useful and verifiable predictions in the context of low-cycle fatigue [17], the estimation of bainite plate thickness [18], the calculation of ferrite number in stainless steels [19], the estimation of tensile strength [20,27], impact strength [26,21], the effect of processing parameters on marageing steels [22], the modelling of strain induced martensitic transformation [23], and the reduction in steel varieties [28], to name but a few. There has even been an assessment of procedures needed to design networks which are well-assessed in their performance [24]. Given this plethora of literature, only specific points of relevance are introduced here.

With neural networks, the input data $x_j$ are multiplied by weights, but the sum of all these products forms the argument of a flexible mathematical function (known as the transfer function), here a hyperbolic tangent. The output $y$ is therefore a non-linear function of $x_j$. The exact shape of the hyperbolic tangent can be varied by altering the weights. Further degrees of non-linearity can be introduced by combining several of these hyperbolic tangents, so that the network is able to model highly non-linear relationships. The nature of the transfer functions and the weights define a reproducible mathematical function which represents the empirical data.

The network just described is essentially a non-linear regression method which, because of its flexibility, is able to capture complicated data, whilst at the same time avoiding overfitting. There are a number of interesting outputs other than the coefficients which help recognise the significance of each input. First, there is the noise in the output, associated with the fact the input set is unlikely to be comprehensive - i.e., a different result is obtained from identical experiments. Secondly, there is the uncertainty of modelling because many mathematical functions may be able to adequately represent known data but which behave differently when extrapolated. A knowledge of this uncertainty helps make the method less risky in extrapolation. This uncertainty can be expected to be large in regions of the input domain where data are sparse or exceptionally noisy.