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\centerline{\large Bayesian Neural Network Model for}
\centerline{\large Austenite Formation in Steels}
\bigskip \singlespace
\centerline{L. Gavard$^{\dag}$, H. K. D. H. Bhadeshia$^{\ddagger}$, D. J.
C. MacKay$^{\ddagger}$ and S. Suzuki$^{\star}$}
\medskip
\centerline{$^{\dag}$University de Lille I {\bf (EUDIL)}, Lille, France}
\centerline{$^{\ddagger}$Mathematics \&\ Physical Sciences Group,
Darwin College, Cambridge}
\centerline{$^{\star}$NKK Corporation, Japan}
\bigskip \doublespace

Published in Materials Science and Technology, Vol. 12 (1996)
453--463
\sec{ABSTRACT}

{\parindent=20 pt  \narrower \medskip
\x The formation of austenite during the continuous heating of
steels has been investigated using neural network analysis with a
Bayesian framework. An extensive database  consisting of
the detailed chemical composition, $Ac_1$ and $Ac_3$ temperatures,
and the heating rate was compiled for this purpose, using data
from the  published literature. This has 
been assessed using a neural network, with the aim of modelling the
austenite--start and finish temperatures. The results from the neural
network analysis are found to be consistent with what might be
expected from phase transformation theory. 

\medskip
}

\sec{INTRODUCTION}

\x Most commercial processes rely to some extent on heat 
treatments which cause the steel to revert into the 
austenitic condition. This includes the processes involved 
in the manufacture of wrought steels, and in the fabrication 
of steel components by welding. It is useful, therefore, to
be able to  model quantitatively the transformation of an ambient
temperature steel microstructure into austenite [1].

 The formation of austenite during heating  differs in
many ways from those transformations that occur during the
cooling of austenite. For cooling transformations, the kinetics of
decomposition  follow the classical C-curve behaviour, in which
the rate goes through a maximum as a function of the undercooling
below the equilibrium transformation temperature. This is because 
diffusion becomes sluggish with decreasing temperature, but 
the driving force for transformation increases with the
undercooling. On the other hand, both the diffusion coefficient and the
driving  force increase with the extent of superheat above the
equilibrium temperature, so that the  rate  of austenite formation
must increase indefinitely  with temperature, \fagg.

\picture{ttt}{187}{134}{500}{\tabtit{\figg : }{ The
time--temperature--transformation  curves for the
$\gamma\rightarrow\alpha$ reaction, and for the  reverse
$\alpha\rightarrow\alpha$ transformation. $\alpha$ and $\gamma$
represent ferrite and austenite, respectively. $\Delta G$ represents
the chemical driving force for transformation; $D$
is the rate--controlling diffusion coefficient.}}


There is another important difference between the transformation
of austenite, and the transformation to austenite. In the former
case,  the kinetics of transformation can be
described completely in terms of the alloy composition and the
austenite grain size.  By contrast, the microstructure from which
austenite may grow can be infinitely varied. Many more variables are
therefore needed  to describe the kinetics of austenite
formation.  The  extent to which the starting  microstructure has to be
specified remains to be determined, but factors such as  particle size,
the distribution and chemistry of individual  phases, homogeneity, the
presence of  nonmetallic inclusions, \etc should all be important.

This discussion highlights the complexity of the problem.  A
fundamental attempt at modelling the formation of austenite is
therefore unlikely to be of general value, except at slow heating rates
consistent with the achievement of equilibrium. Some aspects of the
difficulties involved  have been reviewed recently for a variety of
starting microstructures [2].  Models of specific
metallurgical approaches exist for austenite formation
from a mixture of cementite and ferrite [3], from bainite [4],  and
from ferrite [5]. However, none of these are of
general applicability for the reasons described earlier.

In this work we adopt a different approach involving the use of an
artificial neural network to \lqq blindly" model a large set of
published experimental data on austenite formation in steels. The
results are then compared extensively against what might be
expected on the basis of metallurgical theory.

\sec{THE TECHNIQUE}

\x   Neural networks are parameterized non--linear models used for
 empirical regression and classification modelling. Their flexibility
 makes them able to discover more complex relationships in data than
 traditional linear statistical models.

 A neural network is `trained' on a set of examples of 
 input and output data. The outcome of training is a set of 
 coefficients (called weights) and a specification of the 
 functions which in combination with the weights relate 
 the input to the output. The training process involves a 
 search for the optimum non--linear relationship between 
 the inputs and the outputs and is computer intensive. 
 Once the network is trained, estimation of the outputs 
 for any given inputs is very rapid. 

 One of the difficulties with blind data modelling is that 
 of `overfitting', in which spurious details and noise in the 
 training data are overfitted by the model. This gives rise to 
 solutions that generalize poorly. MacKay [6--11] has developed 
 a Bayesian framework for neural networks in which the appropriate 
 model complexity is inferred from the data. 

 The Bayesian framework for neural networks has two further
advantages. First, the significance of the input variables is
automatically  quantified. Consequently the model--perceived
significance of each input variable can be compared against
metallurgical theory.Second, the  network's predictions are
accompanied by error bars which depend on the specific position in
input space. These quantify the model's certainty about its
predictions. 

 

\sec{THE DATABASE}

\x The dataset was constructed using information from the published
literature, particularly time--temperature--transformation atlases
[12--16]. The variables accessed are listed in \tablaa. The dataset
therefore consisted of 22 input variables, and two output variables,
the $Ac_1$ and $Ac_3$ temperatures, which describe the onset and
completion (respectively) of austenite formation during continuous
heating beginning from ambient temperature. There were 788 cases
included in the analysis.

It is expected that the measured $Ac_3$ temperature should
always be larger than the temperature at which the austenite
should become fully austenitic under equilibrium conditions (\ie
the $Ae_3$ temperature). To demonstrate this, the measured
$Ac_3$ temperatures for each of the  low--alloy steels were
compared against the corresponding calculated paraequilibrium
$Ae_3'$ temperatures. The method for calculation has been described
elsewhere (Bhadeshia, 1985).
\footnote\dag{\medtype The paraequilibrium $Ae_3'$ temperature is
calculated in the same way as the equilibrium $Ae_3$ temperature,
but whilst maintaining the iron to substitutional solute atom ratio
constant everywhere. This means that substitutional solute do not
partition between the phases. The paraequilibrium temperature is
easier to calculate but always represents an underestimate of the
$Ae_3$ temperature.} \fagg\ shows that the measured $Ac_3$
temperatures are almost always larger than the corresponding
$Ae_3$  temperatures, often by several hundred \degg. Thus, the
$Ac_3$ temperatures in the dataset are dominated by kinetic effects
so that it is important to include the heating rate as a variable.

\picture{para}{149}{128}{500}{\tabtit{\figg : } { Comparison of the
kinetic $Ac_3$ temperatures against the calculated paraequilibrium
$Ac_3'$ temperatures for the low--alloy steels included in the
dataset.}}

\midinsert \thicksize=1pt \thinsize=0.8pt 
\tablewidth=5.2truein \begintable 

Variable | Range & Mean & Standard \nr
               |                 &          &  Deviation \cr
C | 0--0.96 & 0.30 & 0.02 \nr
Si | 0--2.13 & 0.39 & 0.06 \nr
Mn | 0--3.06 & 0.82 & 0.08 \nr
S | 0--0.09 & 0.007 & 0.003 \nr
P | 0--0.12 & 0.008 & 0.004 \nr
Cu | 0--2.01 & 0.046 & 0.07 \nr
Ni | 0--9.12 & 1.01 & 0.29 \nr
Cr | 0--17.98 & 1.23 & 0.60 \nr
Mo | 0--4.80 & 0.32 & 0.16 \nr
Nb | 0--0.17 & 0.003 & 0.006 \nr
V | 0--2.45 & 0.05 & 0.09 \nr
Ti | 0--0.18 & 0.0014 & 0.006 \nr
Al | 0--1.26 & 0.006 & 0.05 \nr
B | 0--0.05 & 0.0005 & 0.002 \nr
W | 0--8.59 & 0.06 & 0.31 \nr
As | 0--0.02 & 0.0001 & 0.001 \nr
Sn | 0--0.008 & 0.0001 & 0.0003 \nr
Zr | 0--0.09 & 0.0001 & 0.003 \nr
Co | 0--4.07 & 0.06 & 0.14 \nr
N |  0--0.06 & 0.003 & 0.002 \nr
O | 0--0.005 & 0.0001 & 0.0002 \nr
Heating Rate / ${\rm K\,s^{-1}}$ | 0.03--50 & 1.0 & 1.7 \cr
Ac1 / \degg\ | 530--921 & 724 & 7 \nr
Ac3 / \degg\ | 651--1060 & 819 & 9   
\endtable 
\tabtit {\tablee : }{  The variables. The concentrations are all in wt\%.
 }
\endinsert 
\sec{THE ANALYSIS}

\x The aim of the work was to predict the austenite formation
temperatures for the steels as a function of the variables listed in
\tablee. Both the input and output variables
were first normalized within the range $\pm 0.5$ as follows:
$$ x_{N} = {{x - x_{min}}\over{x_{max}-x_{min}}} - 0.5
\numeqn
$$ where $x_{N}$ is the normalized value of $x$ which has
maximum and minimum values given by $x_{max}$ and $x_{min}$
respectively.

The network consisted of 22 input nodes, a number of hidden
nodes and an output node representing either the $Ac_1$ or
the $Ac_3$ temperature (\fagg). The network was trained using a
randomly chosen 394 of the examples from a total of 788 available,
the remaining 394 examples being used as \lq new\rq\ experiments to
test the trained network.

\picture{network} {114.0} {67} {700} {\tabtit{\figg :}{
A typical network used in the analysis. Only the connections
originating from one input unit are illustrated, and the two
bias units are not illustrated.}}

Linear functions of the inputs $x_j$ are operated on by a
hyperbolic tangent transfer function:
$$ h_i = \tanh\bl(\sum_j w^{(1)}_{ij}x_j + \theta^{(1)}_i\br)
 \numeqn $$
so that each input contributes to every hidden unit. The
bias is designated $\theta_i$ and is analogous to the
constant that appears in linear regression. The strength of
the transfer function is in each case determined by the
weight $w_{ij}$. The transfer to the output $y$ is linear:
$$y=\sum_i w^{(2)}_{ij}h_i + \theta^{(2)}\numeqn $$
The specification of the network structure, together with
the set of weights is a complete description of the formula
relating the inputs to the output. The weights are
determined by training the network; the details are
described elsewhere (MacKay, 1992a--b, 1993, 1995, 1995; Bhadeshia
\et, 1995). The training involves a minimisation of the regularised sum
of squared errors. The term $\sigma_\nu$ used below is the
framework estimate of the noise level of the data.

The complexity of the model is controlled by the
number of hidden units (\fagg), and the values of the
24 regularisation constants ($\sigma_w$), one associated with
each of the 22 inputs, one for biases and one for all weights
connected to the output. 

\picture{sigmanu} {199.0} {95} {700} {\tabtit{\figg : }{ 
Variation in $\sigma_\nu$ as a function of the number of
hidden units. Several values are presented for  each set of
hidden units because the training for each network was started
with a variety of random seeds. (a) Analysis for the $Ac_1$
temperature. (b) Analysis for the $Ac_3$ temperature.}}

\figg\  shows that for both cases, the inferred noise level decreases
as the number of hidden units increases. However,
the complexity of the model also increases with the number of
hidden units. A high degree of complexity may not be justified, and
in an extreme case, the model may in a meaningless way attempt to
fit the noise in the experimental data. MacKay has made a
detailed study of this problem and has defined a quantity (the
{\it evidence}) which comments on the probability of a model. In
circumstances where two models give similar results over the
known dataset, the more probable model would be predicted to be
that which is simpler; this simple model would have a higher value
of \lq evidence'.  The  evidence framework was used to
control the regularisation constants and $\sigma_\nu$. The
number of hidden units was set by examining performance on test
data (\fagg). A combination of Bayesian and pragmatic statistical
techniques were therefore used to control the model complexity. Four
hidden units were found to give a reasonable level of complexity to
represent the variations in the $Ac_1$ temperature as a function of
the input variables. A less complex model (two hidden units) was
necessary for the $Ac_3$ temperature, presumably because this
should be less dependent on the starting microstructure. 

\picture{testen} {199.0} {79.0} {700} {\tabtit{\figg : }{ The
 test error as a function of the number of hidden units. Several values are presented for  each set of
hidden units because the training for each network was started
with a variety of random seeds. (a) Analysis
of the $Ac_1$ data. (b) Analysis of the $Ac_3$ data.}}

Once the optimum number of hidden units was established for each
analysis, the data were retrained to give a more accurate model. This
time, all 788 of the cases were included in the training process.

The weights for the selected networks are presented in
the appendix; these listings are sufficient to  reproduce
the predictions described, though not the error bars. The levels of
agreement for the training datasets are illustrated in
\fagg, which shows good prediction. It should be
emphasized that all data were included in deriving the
weights given in the appendix.
The good fit is established to work well over the range of data
included in the analysis.


\picture{traintest} {197.0} {184.0} {700} {\tabtit{\figg : }{  Plot
of the estimated versus measured temperatures, both for the
training and test datasets for the $Ac_1$ and $Ac_3$ temperatures,
using four and two hidden units respectively.}}


\sec{USE OF THE MODEL}

\x We now examine the metallurgical significance of the results.
\fagg\ illustrates the significance ($\sigma_w$) of each of the input
variables, as perceived by the neural network, in influencing the
austenite transformation temperatures, within the limitations of
the dataset. A high value of $\sigma_w$ implies that the
input parameter concerned explains a relatively large amount of the
variation in transformation temperature in the dataset (rather like a
partial correlation coefficient in multiple regression analysis). It
follows  that
$\sigma_w$ is not necessarily an indication of the sensitivity of  the
transformation temperature to that input parameter. The
interpretation of $\sigma_w$ is therefore best considered alongside
the predictions of transformation temperatures presented below.



\picture{importance} {141.0} {153} {700} {\tabtit{\figg : }{
Bar charts showing a measure of the model--perceived significance
of each of the input variables in influencing either the $Ac_1$ or the
$Ac_3$ temperatures (for models with four and two hidden units for
the $Ac_1$ and $Ac_3$ temperatures respectively). Note that the
magnitude of
$\sigma_W$ for a particular input does not necessarily reflect the
sensitivity of the transformation temperature to that variable; it
should be interpreted like a partial correlation coefficient. It reflects
the degree of variation in temperature, explained by the variable
within the scope of the dataset. The term $HR$ stands for heating
rate.}}
 

\x \fagg\ shows the predicted effects of the  carbon
concentration and heating rate on the $Ac_1$ and $Ac_3$
temperatures of a plain carbon steel.  The data presented as a
function of carbon are calculated for a heating rate of
$1\,{\rm ^{\circ}C\,s^{-1}}$. As might be expected, the $Ac_1$
temperature decreases with carbon concentration reaching a limiting
value which is very close to the eutectoid temperature of about 723
\degg. This latter limit is expected because of the slow heating rate
and the fact that the test steel does not contain any substitutional
solutes. Note that there is a slight underprediction of the $Ac_1$
temperature for pure iron, although the expected temperature of
about 910 \degg\ is within the 95\%\ confidence limits of the
prediction (twice the width of the error limits illustrated in \figg).

By contrast, the $Ac_3$ temperature appears to go through
a minimum at about the eutectoid carbon concentration. This is also
expected because the $Ae_3$ temperature also goes through a
minimum at the eutectoid composition. Furthermore, unlike the
$Ac_1$ temperature, the minimum value of the calculated $Ac_3$
never reaches the eutectoid temperature; even at the slow heating
rate it is expected that the austenite transformation finishes at
some superheat above the eutectoid temperature, the superheat
being reasonably predicted to be about 25~\degg.

At slow heating rates, the predicted $Ac_1$ and $Ac_3$ temperatures
are in fact very close to the equilibrium $Ae_1$ and $Ae_3$
temperatures and insensitive to the rate of heating. As expected, they
both increase more rapidly when the heating rate rises exceeds about
$10\,{\rm ^{\circ}C\,s^{-1}}$.  Since the the mean heating
rate in the experimental dataset is only $1\,{\rm^{\circ}C\,s^{-1}}$
(\tablee), it is not surprising that the
$\sigma_w$ values associated with heating rate are relatively small
in Fig.~7.

\picture{HR,C}{194}{193}{700}{\tabtit{\figg : }{(a,b) The predicted
variation in the $Ac_1$ and $Ac_3$ temperatures of plain carbon
steels as a function of the carbon concentration at a heating rate of
$1\,{\rm ^{\circ}C\,s^{-1}}$.  (c,d) The predicted
variation in the $Ac_1$ and $Ac_3$ temperatures of an
Fe--0.2~wt\%\ alloy as a function of the heating rate. In all of these
diagrams, the lines represent the $\pm 1\sigma$ error bars about
the calculated points. All the results presented here are based on  
models with four and two hidden units for the $Ac_1$ and $Ac_3$
temperatures respectively.}}
 
It is not at first sight expected that the
transformation temperatures should go through a maximum as a
function of the heating rate. This maximum appears to be significant
even when the error bars are taken into account (note that the
predicted errors become very large at rates well outside the range
of the experimental database). The occurrence of a maximum is
possible if retained austenite is present in the microstructure, as
might be the case for many of the steels for which high heating rate
experiments have been conducted. This is because at high heating
rates, the retained austenite simply grows as soon as the
equilibrium $Ae_1$ temperature is exceeded. However, slow heating
allows the austenite to first decompose into an equilibrium mixture
of ferrite and carbides, thereby making it necessary for new
austenite to nucleate when equilibrium permits. This is illustrated in
\fagg. Finally, it is worth noting that at heating rates well outside of
the range of experimental data, there are excessive error bars
associated with the predicted transformation temperatures.

\picture{HR}{177}{210}{400}{\tabtit{\figg : }{ Schematic illustration of
why when some critical heating rate is exceeded,  a smaller
superheat is necessary to grow austenite. A very large heating rate
can involve the growth of existing retained austenite, thereby
avoiding the need to nucleate new austenite, and hence leading to a
decrease in the superheat needed to begin transformation (Yang and
Bhadeshia, 1989).}}


Fig.~7 shows that the $\sigma_w$ value for Mn is larger for the
$Ac_1$ data than for the corresponding $Ac_3$ data. Noting also that
the variation of manganese in the dataset is large, and that the data
are likely to be reliable since it is a key alloying element in steel, it
is likely that the difference in the two $\sigma_w$ values is
important. This is confirmed by the predicted effect of Mn on the
transformation temperatures (\fagg a), since the $Ac_1$ temperature
is indeed found to be more sensitive to Mn than the $Ac_3$
temperature. This behaviour is also expected from the Fe--C--Mn
phase diagram. Whereas the effect of Mn on the $Ac_3$ temperature
is simply to lower the $(\alpha + \gamma)/\gamma$ phase boundary
on the temperature scale (via the thermodynamic effect of Mn on
austenite stability), the influence on $Ac_1$ is much larger since
a three phase $\alpha+\gamma+M_3C$ field develops. This is
confirmed by the phase diagram calculations carried out using the
MTDATA (1995) thermodynamic package. Thus, the $Ae_1$ and $Ae_3$
temperatures for Fe--1Mn--0.2C wt\%\ alloy were
predicted to be 696 and 823 \deg\ respectively. This compares with
the corresponding equilibrium temperatures for a plain carbon
Fe--0.2 wt.\%\ steel at 723 and 839 \degg. Thus, manganese should
indeed influence $Ac_1$ more than  $Ac_3$ in the
present context.  The calculations allowed the existence of ferrite,
austenite and $M_3C$, where $M$ refers to metal atoms. 



\figg b shows that copper, over the concentration range considered
has a negligible influence on the formation of austenite. The
difference between the $Ac_1$ and $Ac_3$ temperatures decreases
with increasing cobalt concentration (\figg c).

 Boron has consistently
small values of $\sigma_w$ for both the $Ac_1$ and $Ac_3$
temperatures (Fig.~7). Table~1 shows that there is a significant
concentration range of boron included in the dataset, the maximum
boron content being  500 parts per million by weight. The
concentration of boron can be precisely controlled in steel to an
accuracy of $\pm 5$ ppmw, and the accuracy of the chemical
analysis is generally better than $\pm 3$.  Consequently a low value
of $\sigma_w$ truly indicates that the austenite formation
temperatures are insensitive to the boron concentration, and this is
confirmed the predicted effect of boron in \figg d.

Vanadium is an extremely strong carbide--forming element, with
limited solubility even in austenite. Hence, it is not surprising that
both the $Ac_1$ and $Ac_3$ temperatures are very high for the
Fe--0.2C--1V~wt\%\ alloy. Austenite growth would be retarded if
the carbon is tied up in the form of vanadium carbide. 

\picture{MN}{167}{233}{700}{\tabtit{\figg : } { The
predicted variation in the $Ac_1$ and $Ac_3$ temperatures for a
variety of Fe--0.2 wt\%\  steels  at a
heating rate of $1\,{\rm ^{\circ}C\,s^{-1}}$.  In all of these diagrams,
the lines represent the $\pm 1\sigma$ error bars about the calculated
points.}}

When compared with vanadium,  molybdenum is a less potent
carbide--forming element. Hence, at comparable concentrations, the
transformation temperatures are found to be  less sensitive to
the Mo content. However, at even larger molybdenum contents the
gap $Ac_3-Ac_1$ increases greatly. This is because the high
temperature
$\delta-$ferrite and low temperature $\alpha-$ferrite phase fields
tend to join up in the low--carbon end of the phase diagram, thereby
raising the temperature at which austenite formation is completed
(\fagg). These results are verified by the phase diagram calculations
presented in \tablaa .


\midinsert \thicksize=1pt \thinsize=0.8pt 
\tablewidth=5.2truein \begintable 
Alloy Composition / wt\% | $Ae_1$ / \deg & $Ae_3$  / \deg \cr
Fe--1Mo--0.2C | 727 & 845 \nr
Fe--4Mo--0.2C | 727 & 897 \nr
Fe--6Mo--0.2C | 715 & 955
\endtable 
\tabtit { \tablee : }{  Phase equilibrium calculations conducted for a
series of Fe--0.2C--Mo wt\%\ alloys, using MTDATA (1995). The
calculations allowed the existence of ferrite, austenite, $M_3C$
and $M_2C$ carbides, where $M$ refers to metal atoms. They show
that the
$Ae_3$ temperature increases sharply beyond a certain molybdenum
concentration, whereas $Ae_1$ decreases slightly. These results are
consistent with the predictions of the neural network model, as
discussed in the text. }
\endinsert 

\picture{MOPHASE}{116}{82}{600}{\tabtit{\figg : }{An illustration of
the way in which molybdenum modifies the phase diagram of a steel.
At low concentrations, the $\alpha$ and $\delta-$ferrite phase fields
are separated. These fields connect when the molybdenum
concentration is increased. This in turn leads to a large change in the
$Ac_3$ temperature, though not the $Ac_1$ temperature, as
indicated by the pairs of points marked on each diagram.}}


Predicted data for the effects of nickel and chromium are illustrated
in \fagg. Nickel is an austenite stabilizer and judging from the phase
diagram, both the $Ac_1$ and $Ac_3$ temperatures should decrease
with increasing nickel concentration. This clearly is not the case with
the predicted $Ac_1$ temperature, which seems to exceed the
predicted $Ac_3$ temperature for nickel concentrations greater
than about 1 wt\%\ (although the data beyond about 6~wt\%\ are
not reliable, with large error bars).  This behaviour reflects
the poor quality of the $Ac_1$ data for the nickel steels.

The predicted data for chromium are more interesting. The $Ac_3$
temperature appears to go through a minimum with increasing
chromium concentration, a behaviour to some extent replicated by the
calculated $Ae_3$ temperatures for the same alloy system. The trend
for $Ac_1$ is different, but follows what is expected from the
calculated phase diagram. At about Fe--0.2C--7.5Cr  wt\%, the
$Ae_1$ temperature becomes virtually identical to the $Ae_3$
temperature. This is because of the existence of a $\gamma-$loop in
the phase diagram, as illustrated schematically in \figg e.  The
results for chromium are therefore consistent with what is expected
from the phase diagram: there are some results where the
calculated $Ac_1$ exceeds $Ac_3$, but this is not significant when
the error bars are taken into account. The work emphasizes that
there will be difficulties at high chromium concentrations since the
difference between the two transformation temperatures becomes
rather small.

\picture{CR,NI}{193}{270}{500}{\tabtit{\figg : } {(a,b) The predicted
variation in the $Ac_1$ and $Ac_3$ temperatures of Fe--0.2 wt\%\
steel as a function of the nickel concentration at a heating rate
of $1\,{\rm ^{\circ}C\,s^{-1}}$.  (c,d) The predicted
variation in the $Ac_1$ and $Ac_3$ temperatures of an
Fe--0.2~wt\%\ alloy as a function of the chromium concentration at
a heating rate of $1\,{\rm ^{\circ}C\,s^{-1}}$. In all of these diagrams,
the lines represent the
$\pm 1\sigma$ error bars about the calculated points. (e) An
illustration of the $\gamma-$loop. The portions $ab$, $bc$ and $cd$
represent the $\alpha+\gamma/\gamma$,
$(\gamma+M_{23}C_6)/\gamma$ and $\gamma/(\gamma+\alpha)$
phase boundaries respectively.}}


Disregarding the effect of silicon on the $Ac_1$ temperature (due to
the unreliability of the predictions with large error bars), its
influence on the $ac_3$ temperature is consistent with the fact that
it is a ferrite stabilizing element (\fagg).  Titanium raises the
transformation temperatures, presumably because it is combined
with carbon in the starting microstructure. The effect of niobium is
similar to that of molybdenum. Nitrogen and phosphorus,
at the concentrations studied, have no significant effects on
austenite formation, consistent with their small $\sigma_w$ values
(Fig.~7).



\picture{OTHER2}{170}{225}{500}{\tabtit{\figg : } { The
predicted variation in the $Ac_1$ and $Ac_3$ temperatures for a
variety of Fe--0.2 wt\%\  steels  at a
heating rate of $1\,{\rm ^{\circ}C\,s^{-1}}$.  In all of these diagrams,
the lines represent the $\pm 1\sigma$ error bars about the calculated
points.}}


\sec{FURTHER TESTS OF THE MODEL}

\x Substitutional solutes affect transformations in steels by two
mechanisms:

{\parindent=14pt\narrower\medskip
\zz{(a) } A solute can alter the relative thermodynamic stabilities of
the parent and product phases. For example, solutes like nickel
which stabilize austenite might be expected to lower the
$Ae_3$ temperature, which is an {\it equilibrium} transformation
temperature.

\zz{(b) } The solute is likely to have different solubilities in the
parent and product phases. When transformation is
diffusion--controlled, the necessity for the solute to partition is
expected to reduce the rate of transformation. This {\it kinetic}
effect is  independent of the thermodynamic effect
emphasized in (a).
\medskip} 

It is possible, using the trained neural network model, to examine
both of these issues, and hence test whether it behaves correctly
from a metallurgical point of view. \fagg\ contains comparisons
between the kinetic $Ac_3$ temperatures and corresponding
thermodynamically calculated $Ae_3$ equilibrium transformation
temperatures.  Any difference between these represents some
kinetic hindrance to transformation. 

\figg a shows clearly that the
Fe--C alloys transform  easily to austenite, at temperatures which
are not very different from equilibrium. The superheat needed for
the higher carbon Fe--C alloy is larger because the
extra carbon depresses the transformation
temperature, leading to a reduction in the mobility of iron atoms. 
The addition of manganese clearly leads to much larger deviations
from equilibrium, even when the transformation occurs at
temperatures higher than for the Fe--3C~at\%\ steel. Furthermore,
the deviation increases disproportionately with the concentration of
manganese. This confirms the fact that the presence of a
substitutional solute greatly retards the transformation to austenite
because it is necessary for the solute to diffuse during
transformation.

To show that it is the diffusion of solute which retards the formation
of austenite, \figg b shows cases where the addition of Mo or V
raises the transformation temperature, but nevertheless increases
the deviation from equilibrium.


\picture{OTHER3}{195}{76.5}{700}{\tabtit{\figg : } { A comparison of
calculated $Ac_3$ temperatures against the calculated equilibrium
$Ae_3$ temperature. The solute concentrations are in atomic
percent. The largest deviation of $Ac_3$ from $Ac_1$ is in each case
highlighted using a horizontal arrow.}}

\sec{APPLICATION TO STEEL T91}

\x Steel T91 (\tablaa) is an alloy designed for power plant applications
and for use in the nuclear industry. It is of particular interest to us
and a number of experiments have been conducted to measure the
austenite formation temperatures as a function of the starting
microstructure (Brachet and Gavard, 1995). The latter has not been
included as a variable in the neural network model, because of the
absence of data. The following results therefore illustrate the level of
error in prediction, which can be attributed to microstructural effects.

Martensite was obtained by directly quenching the alloy after
austenitising at 1050\degg\ for 30~min, to room temperature. A
number of tempered martensite microstructures were also
generated from the quenched samples. In addition,  
microstructure consisting of allotriomorphic ferrite and
pearlite--like alloy carbides was obtained by isothermal
transformation of austenite at 725\deg for 6~h
before cooling to ambient temperature. 

The calculated $Ac_1$ temperature is compared against the value
measured for the untempered martensite, as a function of the
heating rate, in \fagg a. The calculation overestimates the
transformation temperature by 15--50 \degg, which is within the
error limits of the neural network model. It is good that
consistent with the experimental data, the model predicts that the
transformation temperature does not increase with the heating
rate. If anything, there is a slight decrease predicted.

\figg b shows the variation in transformation temperatures with the
starting microstructure. It is evident that the variation can be as
large as 75 \degg, a value comparable with the overall error implied
by the $\sigma_{\nu}$ noise for the optimized neural network
models (Fig.~4).

\picture{OTHER4}{175}{180}{500}{\tabtit{\figg : } {Some calculations
for steel T91. (a) Variation in the calculated and measured $Ac_1$
temperatures. The measured temperatures refer to samples with an
untempered martensitic microstructure. (b) Illustration of the
variation in the austenitisation behaviour as a function of the
starting microstructure. The tempered martensite is identified by
the tempering temperature, the tempering time being 1~h in each
case. A heating rate of 50 \degg${\rm\,^s{-1}}$ was used to
measure the $Ac_1$ and $Ac_3$ temperatures.}}


\midinsert \thicksize=1pt \thinsize=0.8pt 
\tablewidth=5.2truein \begintable 

Element & Concentration / wt\% & Element & Concentration /
wt\%\cr C & 0.105 &N & 0.051 \nr
Cr & 8.20 & Mo & 0.97 \nr
V & 0.20 & Ni & 0.13 \nr
Si & 0.43 & Mn & 0.37 \nr
Nb & 0.075 & &  
\endtable 
\tabtit {\tablee : }{  The chemical composition of steel T91,  in wt\%. }
\endinsert 

\sec{CONCLUSIONS}

\x The temperature at which austenite first begins to form during
heating, and that at which the transformation to austenite is
completed, have been modelled as a function of the steel chemical
composition and heating rate. The model is based on a neural
network analysis of an experimental database compiled from
published data.

The model is found capable of estimating the transformation
temperatures to an accuracy of about $\pm 40$\degg\ (95\%\
confidence limits). The neural network technique used is
based on a Bayesian framework and hence is capable of
associating different error bars depending on the location in the
input parameter space. This has demonstrated that the predictability
of the 
$Ac_1$ temperature is in many cases, less reliable than is the case
for the $Ac_3$ temperature. This is probably a reflection of the fact
that in many cases, the $Ac_1$ temperature is more difficult to
measure experimentally. 

The predictions of the model have been assessed against
metallurgical theory and found to be reliable.

\sec{ACKNOWLEDGMENTS}

\x The authors are grateful to {\bf EUDIL} (University de Lille I) and
Professor Rolland Taillard for facilitating  Laurent Gavard's  work in
Cambridge, and to Professor Colin Humphreys of Cambridge
University for his support and encouragement of our work. We
sincerely thank Machiko Suzuki for her help in preparing some of the
data into a computer format.

\sec{REFERENCES}


{\singlespace\parindent=14pt \narrower \medskip

\def\ref#1#2#3#4#5#6{\parindent=-3truemm #1) #2,  {\it #4} {\bf
#5} (#3) #6\vskip 0.5truemm} 

\ref{1}{Bhadeshia, H. K. D. H. and Svensson, L.--E.}{1993}{Mathematical
Modelling of Weld Phenomena, {\rm eds. H. Cerjak and K. E.
Easterling, Institute of Materials, London,}}{}{109--182.}

\ref{2}{Bhadeshia, H. K. D. H.} {1992} {Bainite in Steels, } {} {1--458}


\ref{3}{Hillert, M., Nilsson, K.  and Torndahl, L-E.} {1971}  
{JISI, } {209{\rm,}} {49--66.} 


\ref{4}{Yang, J. R. and Bhadeshia, H. K. D. H}{1989}{Materials Science
and Engineering A, }{A118,}{155--170.}


\ref{5}{Atkinson, C., Akbay, T. and Reed, R. C.}{1995}{Acta Metallurgica
et Materialia, }{43,}{2013--2031.}

\ref{6}{MacKay, D. J. C.} {1992} {Neural Computation, } {4} 
{415.}

\ref{7}{MacKay, D. J. C.} {1992} {Neural Computation, } {4} {448.}


\ref{8}{MacKay, D. J. C.} {March 1993} {Darwin College Journal, } 
{} {81.}

\ref{9}{MacKay, D. J. C.} {1994} {ASHRAE (American Society of Heating, Refrigerating and Air--conditioning Engineers) Transactions, } {100, pt. 2} 
{1053--1062.}

\ref{10}{MacKay, D. J. C.} {1995}{Network: Computation in Neural
Systems, }{6}{in press.}



\ref{11}{Bhadeshia, H. K. D. H., MacKay, D. J. C. and Svensson,
L.--E} {1995} {Materials Science and Technology, } {} {in
press.}

\ref{12}{Abiko, K.}{January 1993}{Scientific American, {\rm Japanese
edition, }}{}{20--29.}

\ref{13}{Cool, T.}{1994}{Systematic design of welding alloys for power
plant steels, {\rm CPGS Thesis, University of Cambridge.}}{}{}

\ref{14}{Reed, R.}{1987}{Ph.D. Thesis, {\rm University of Cambridge}}{}{}

\ref{15}{Special Report 56}{1956}{Atlas of isothermal transformation
diagrams of B.S. En steels, {\rm 2nd edition, Iron and Steel Institute,
London.}}{}{}

\ref{16}{Vander Vroot, G. F. editor}{1991}{Atlas of
time--temperature--transformation diagrams for irons and steels,
{\rm ASM International, Ohio, U.S.A.}}{}{}

\ref{17}{MTDATA}{1995}{Metallurgical Thermochemical data, {\rm National
Physical Laboratory, Teddington, Essex, U. K.}}{}{}


\ref{18}{Brachet, J., Pradelles, R. and Gavard, L.}{1995}{Unpublished
research,}{}{CEA, Saclay, France.}



\medskip}
\vfill\eject

\sec{APPENDIX}

\midinsert \thicksize=1pt \thinsize=0.8pt
\tablewidth=6.2truein \begintable
-0.273272 & -0.584284 & 1.42725 & -0.675013 & -0.297617 & 0.149305 & -2.09248 & 0.558102 \nr
 1.16302 & 0.136662 & 0.00893805 & 0.0271082 & 0.0482528 & 0.285341 & 0.0135071 & 0.0350339 \nr
 -0.074152 & -0.0884704 & -0.0661309 & -0.133209 & -0.224594 & -0.102685 & 0.810902 & -3.00552 \nr
 -1.58987 & 0.515423 & -7.77932 & -1.22046 & -0.294707 & -0.160822 & 0.844392 & -1.54547 \nr
 -0.0628648 & 0.0217054 & 0.138556 & 0.0306007 & 0.124344 & 0.00430856 & 0.00473683 & 1.56578 \nr
 1.85985 & 1.57345  & 0.221986 & 0.179021 & 1.77703 & -0.646938 & 0.902249 & 0.677717 \nr
  -1.55427 & 0.638917  &  0.424401 & 0.232198 & 3.09303 & -0.533354 & 0.0072862 & -0.120021 \nr
 -0.0700783 & 0.415026  &  0.0173118 & -0.0489667 & 0.00607633 & -0.196499 & -0.476342 & -0.658654 \nr
 -0.475611 & 0.177044  &  0.44253 & -0.574891 & -0.266723 & -0.512698 & 0.303814 & -0.779683 \nr
 0.417492 & 0.0995294  & -0.542185 & 0.66351 & -0.846634 & -1.68699 & -0.146489 & 0.0761116 \nr
 -0.270882 & -0.0883608 & -0.413762 & 4.99285e-06 & 0.162749 & 0.215546 & 0.238404 & 0.221876 \nr
 0.0765538 & 0.0192461  &  0.227757 & -0.918695 & 0.416309 & 2.72348 & 1.33381 & 1.82862 \nr
 1.8775 & & & & & & &
\endtable \tabtit{\tablaa : }{The weights for the $Ac_1$ model. The data are arranged in a 
continuous horizontal sequence in the following order:
 $$ \eqalign{& \theta^{(1)}_1, w_{1,1}^{(1)} \ldots w_{1,22}^{(1)}, \cr
& \theta^{(1)}_2, w_{2,1}^{(1)} \ldots w_{2,22}^{(1)},\cr
& \theta^{(1)}_3, w_{3,1}^{(1)} \ldots w_{3,22}^{(1)},\cr
& \theta^{(1)}_4, w_{4,1}^{(1)} \ldots w_{1,22}^{(1)},\cr
& \theta^{(2)}, w_{1}^{(2)} \ldots w_{4}^{(2)}}$$}
 \endinsert

\midinsert \thicksize=1pt \thinsize=0.8pt
\tablewidth=6.2truein \begintable
-1.41522 & -0.811777 & 0.292749 & -0.212523 & -0.086742 & 6.22651e-05 & -0.0097287 & -0.224123 \nr
 -1.03976 & 0.466825 & 0.849595 & -0.250228 & 0.123227 & -0.56492 & 0.215049 & -0.938616 \nr
 0.309341 & 0.0687889 & 0.255502 & -0.129074 & 0.0229901 & 0.069863 & -0.479553 & 1.13112 \nr
 0.70752 & -0.241007 & 0.203091 & 0.129634 & 5.8642e-05 & -0.00661998 & 0.103754 & 1.16011 \nr
 -0.422398  & -1.07388 & 0.545212 & -0.131634 & 0.711098 & -0.287479 & 1.46768 & -0.639086\nr
 -0.142399 & -0.532336 & 0.171956 & 0.0100161 & -0.143582 & 0.661583 & 1.62705 & 4.13134 \nr
 2.53442 & & & & & & &
\endtable \tabtit{\tablaa : }{The weights for the $Ac_3$ model. The data are arranged in a 
continuous horizontal sequence in the following order:
 $$ \eqalign{& \theta^{(1)}_1, w_{1,1}^{(1)} \ldots w_{1,22}^{(1)}, \cr
& \theta^{(1)}_2, w_{2,1}^{(1)} \ldots w_{2,22}^{(1)},\cr
& \theta^{(2)}, w_{1}^{(2)}, w_{2}^{(2)}}$$}
 \endinsert

\vfill\eject\bye