Modeling the field of laser welding melt pool by
RBFNN
Anamarija Borštnik Bračič, Edvard Govekar, Igor Grabec
Faculty of Mechanical Engineering, University of Ljubljana,
Aškerčeva 6, POB 394, SI-1001 Ljubljana, Slovenia
Email: anamarija.bracic@fs.uni-lj.si
Abstract— Efficient control of a laser welding process requires
the reliable prediction of process behavior. A statistical method of
field modeling, based on normalized RBFNN, can be successfully
used to predict the spatiotemporal dynamics of surface optical
activity in the laser welding process. In this article we demon-
strate how to optimize RBFNN to maximize prediction quality.
Special attention is paid to the structure of sample vectors, which
represent the bridge between the field distributions in the past
and future.
I. INTRODUCTION
Laser systems are efficiently applied in welding pro-
cesses [1], where a laser beam is used to melt material. To
maintain high performance in a welding process, efficient
control should be established.
The crucial task in planning the control system is to deter-
mine representative variables which can effectively describe
the welding process. For this purpose, the intensity and spatial
distribution of reflected light, surface temperature values or
properties of the emitted electron plasma are usually chosen.
However, characteristic dynamic properties in space and time
can also be obtained by recording surface optical activity in
the heated zone, known as the melt pool [2].
After choosing the representative variables, extraction of
evolution laws from temporal data becomes a crucial problem.
To date, this problem has been extensively studied in relation
to chaotic time series prediction [3], [4], [5], [6], [7]. The
basis of these methods is to reconstruct a state-space from a
recorded scalar time series by using an embedded technique,
and then to estimate deterministic dynamic evolution from
the reconstructed trajectory using statistical average estima-
tors. We present a generalization of this approach, where
the modeling of dynamic laws is extended from one dimen-
sion (time) to multiple dimensions in spatiotemporal space.
This generalization requires a new embedding method, which
makes feasible a reconstruction of trajectory in the state-space
from spatiotemporal data. The embedded technique, which
was initially developed for time series analysis, can be simply
generalized to spatially related data [8], [9], [10], [11], [12] and
results in a good agreement between predicted and original
chaotic fields over short time scales. Since, in a properly
reconstructed state-space, the modeled dynamics must have
similar statistical properties to the actual dynamics, we use a
new state-space reconstruction method which also considers
Manuscript received: January 31, 2007
statistical properties of a field structure. Such reconstruction
results in an accurate short-term prediction as well as a sta-
tistically proper long-term prediction of deterministic chaotic
field evolution [13].
In this article, a statistical method of field generators, which
is based on normalized radial basis function neural network
(RBFNN), is used to model the spatiotemporal dynamics of
laser welding melt pool images. The stochastic field evolution
is modeled from sample state vectors reconstructed from
recorded spatiotemporal data. The field evolution equation
is estimated non-parametrically from the samples, using the
conditional average estimator which determines the governing
equation of RBFNN. The goal of this article is to find an
optimal dimensionality of the neural network, i.e., to determine
its optimal structure and an adequate number of sampling
patterns, which will result in the best quality Q of field
generator prediction.
Accurate modeling of laser welding images, together with a
criterion function specified by the operator of the laser system,
provides the basis for optimal control of the laser welding
process.
II. DESCRIPTION OF RBFNN
A. Non parametric statistical modeling
Experimental analysis of process dynamics is based on
a representative record of the field ϕ = ϕ(s), where the
variable s represents space as well as time components s =
s(r,t). Most commonly, the spatiotemporal field evolution
of ϕ(s) is described analytically by a system of nonlinear
partial differential equations or integrodifferential equations.
An analytical form of the model can be estimated from the
recorded data, based on spatial and temporal derivatives [15],
[16], [17]. In the case of experimentally obtained data, it is
difficult to estimate derivatives. Therefore, for a more general
approach, a model of field evolution should be expressed in
terms of recorded data only.
In our model, field evolution is expressed in terms of data
recorded at equally spaced discrete points in space and time.
We assume that the dynamics of the field can be described in
terms of the generator equation
ϕ(s) = G (ϕ(s′ ∈ S(s)), σ) , (1)
where ϕ(s′ ∈ S) represents the past distribution of the record,
while ϕ(s) represents its future distribution. S represents
http://arxiv.org/abs/0704.0611v1
Fig. 1. Illustration of point s and its surroundings s′ ∈ S . The future
distribution of field ϕ (s) is located in plane t+1, while the surrounding
points s′ ∈ S , which represent the past distribution of field, are located in
planes t, t−1, t−2...
the surroundings of point s. The field generator G provides
for determination of the future field distribution from its
past distribution. σ is a model parameter depending on the
experimental setup and will be specified in greater detail later.
An arbitrary point s and its surroundings s′ ∈ S are illustrated
in Fig. 1. The source of information for modeling the field
generator is a field record containing joint sample pairs ϕ(s)
and ϕ(s′ ∈ S). These joint sample pairs form a sample vector
Vi(s) = (ϕi(s), ϕi(s
′ ∈ S)). To make further derivation
more transparent, the past field distribution ϕi(s
′) and the
future field distribution ϕi(s) will be denoted by xi and yi,
respectively. Hence Vi(s) = (yi,xi).
The samples Vi are interpreted as random variables and can
therefore be used to express the joint probability distribution
function (PDF) by the kernel estimator [14]
fN (V) =
ψ(V −Vi, σ), (2)
in which ψ denotes an acceptable kernel function such as
the Gaussian function ψ(x − xi, σ) = 1/(
2πσ)exp(−(x −
2/2σ) and N is the number of sample pairs.
Once the samples from the field record have been taken, the
question of how to determine the optimal predictor becomes
relevant. We consider as an optimal predictor of the future
field distribution y from a given value x the value ŷ at which
the mean square prediction error is minimal:
E[(y − ŷ)2|x] = min(ŷ). (3)
Here E[ ] denotes averaging over all points in a field record
at a given time t. The solution of Eq. (3) yields together with
PDF from Eq. (2) the conditional average estimator
ŷ(x) =
yiψ(x− xi, σ)
ψ(x− xj , σ)
yiCi(x), (4)
where coefficients of the expansion Ci(x) represent basis
functions that measure the similarity between the temporary
vector x and vector xi from the field record. The conditional
average estimator described by Eq. 4 represents a radial basis
function neural network in which the recorded data xi,yi rep-
resent the memorized contents of neurons, x and ŷ(x) are the
input and the output of the network, while the basis functions
Ci(x) correspond to activation functions of neurons. Since
Ci(x) = 1, the conditional average estimator represents
a normalized RBFNN. In this function, the parameter σ can
be interpreted as the width of receptive fields of neurons.
B. Quality of predictor
Working towards optimal modeling of future field distribu-
tions requires a quantitative estimation of modeling quality. We
therefore introduce a testing field y and define the prediction
quality Q, based upon the difference between the predicted
field ŷ and the testing field y as:
Q = 1− E[(ŷ − y)
E[(ŷ − ˆ̄y)2] + E[(y − ȳ)2]
. (5)
Here ˆ̄y and ȳ stand for the average values of predicted field
ŷ and testing field y, i.e., E[ŷ] = ˆ̄y and E[y] = ȳ. A perfect
prediction ŷ = y yields Q = 1, while uncorrelated ŷ and y
result in Q = 0.
C. Prediction of field evolution
The prediction process consists of three steps:
1) Learning, that corresponds to setting up the basis of joint
sample pairs (ϕi(s), ϕi(s
′ ∈ S)) = (xi, yi) from the
field record,
2) predicting the field ŷ by using the conditional average
estimator from Eq. (4),
3) and, if the testing field exists, comparing predicted field
with testing field and calculating prediction quality Q.
In order to achieve the highest quality of prediction for the
process, answers to the following crucial questions are needed:
• How to find the surrounding S of a given point s, which
gives the best prediction of field ϕ̂(s) at this point?
• How to determine an optimal number of joint sample
pairs (ϕi(s), ϕi(s
′ ∈ S)) = (xi, yi)?
These questions will be addressed in the following chapters.
III. TIME EVOLUTION OF MELT POOL
Characteristic dynamic properties of laser welding process
in space and time can be experimentally obtained by recording
the surface optical activity of the melt pool. With respect to
the energy supplied to the material, various dynamic regimes
of the welding process can be distinguished. In Fig. 2 visual
records of two different welding regimes are shown, a deep
welding regime (a) and a heat conduction welding regime (b).
In the following discussion, only the deep welding regime is
considered.
Dynamics of the welding regime are here represented by
a record of 1000 images of size 32 × 32 points in space
with sampling time 1/220 s. This experimental record forms a
three-dimensional field of light intensity ϕ(s = r, t) in two-
dimensional space {(rx,i, ry,i); i = 1, ..32, j = 1..32} and
time {tk; k = 1..1000}. Due to local energy supply, the field
is non-homogeneous in space. Consequently, we model its
evolution locally at each spatial point separately. A model
of field evolution, i.e., the learning sample is formed from
the first 800 images. We then predict the time evolution of the
field and compare it with the next 200 images, which represent
10 20 30
10 20 30
10 20 30
10 20 30
10 20 30
10 20 30
10 20 30
10 20 30
Fig. 2. Time series of laser welding records for two different welding
regimes: (a) a deep welding regime , and (b) a heat conduction welding regime.
The next time-step images denoted by ? are unknown and must be predicted.
the testing sample. Based on the quality Q of these predicted
images, we optimize our prediction procedure, i.e., and define
the structure of the surroundings S, and the optimal number
of joint sample pairs N and parameter σ.
A. Optimal value of parameter σ
Parameter σ in the conditional average estimator ŷ (Eq. (4))
was to this point left undetermined. However, as shown in
Fig. 3, obtained for the deep welding regime, the quality of
prediction depends on the value of σ. The learning sample
consisted of 800 images and the surrounding of predicted
field distribution in point s = (r, t) was taken to be just one
neighboring point with the same space coordinate and the time
coordinate being one step behind s′ = (r, t−1). Based on the
learning sample, ten images from the testing interval have been
predicted and compared with the corresponding images from
the testing field. The value Q shown in Fig. 3 is the average
quality of these ten images.
As Fig. 3 shows, the quality exhibits a strong σ dependence
at the beginning of the interval, and reaches its largest value
for σ approximately equal to 4. For larger values, it becomes
a weakly decreasing function of σ. Since the optimal quality
is reached for σ ≈ 4, this value is used in our further
calculations.
As shown in Eq. (4), prediction of the field distribution
at a given point is determined on the basis of similarity
between the field distribution surrounding this point and the
5 10 15 20 25 30
Fig. 3. Dependence of prediction quality Q on the value of parameter σ
from conditional average estimator ŷ. Surrounding set S is taken to be only
one point with the same spatial position and a neighboring position in time.
Number of learning images in the learning field is set to 800.
field distribution in surrounding points taken from the learning
field. If we keep in mind that the parameter σ defines the width
of the Gaussian kernel function ψ (see comment to Eq. 2), we
can conclude, that for very small σ, only those joint sample
pairs from the learning field which have a field distribution in
the surrounding points (xi) very similar to the field distribution
in the surroundings of the point to be predicted (x) contribute
to the predicted value of the field. On the other hand, for larger
σ those joint sample pairs with larger difference x − xi also
contribute to the prediction of ŷ. In the limit of large σ, almost
all joint sample pairs contribute equally to ŷ.
B. Optimal number of joint sample pairs
If the prediction of welding pool images is to be part of a
laser welding control system, the prediction operation has to
be performed in the shortest time interval possible. Since the
number of operations needed to predict a field distribution in a
given point increases linearly with the number of joint sample
pairs (see Eq. 4), it is necessary to find the smallest number of
joint sample pairs which is still able to give predicted images
of good quality.
In Fig. 4 we show the dependence of the quality of predic-
tion on the number of images N defining the learning field.
As in the case of Fig. 3, the surrounding of the predicted
field distribution in point s = (r, t) was taken to be just
one neighboring point with the same space coordinate and
the time coordinate being one step behind s′ = (r, t−1).
Parameter σ is set to 4. Again Q is taken to be the average
quality of ten predicted images, which were compared with
the corresponding images from the testing field.
As one can see from Fig. 4, Q increases rather strongly with
small values of N (N < 400), while for N > 600 an increase
in N does not result in a significant improvement of prediction
quality. Therefore, in further calculations we apply N = 600.
C. Choosing the surrounding S
Our next goal is to find an optimal structure of RBFNN
which yields the best quality of prediction in the shortest time
200 400 600 800
Fig. 4. Dependence of prediction quality Q on number of images defining
the learning field. The surrounding set S is taken to be only one point with
the same spatial position and a neighboring position in time. Parameter σ is
set to 4.
interval. The structure of surrounding set S plays an important
role in this optimization process since each additional point in
the surrounding increases the dimensionality of vectors xi and
therefore the time needed to predict the field distribution in a
given point. Our task is to find the smallest surrounding of
point s, which results in high prediction quality.
In Fig. 5 the prediction quality is presented for various
selections of surrounding set S. Parameters N and σ are
600 and 4, respectively. As before, Q represents the average
quality of ten predicted images which were compared with the
corresponding images from the testing field. All the member
points of the first six surrounding sets in the diagram lie in
the plane t−1. Member points of other surrounding sets lie in
several planes. For each of these, only those planes containing
the member points are plotted..
As can be seen in Fig. 5, the smallest surrounding sets give
the best quality of prediction - see sets Nr. 1-3 and 7-10. If
more points belonging to the same time-plane are added to
S, prediction quality is decreased- compare, for example sets
Nr. 1 and Nr. 6 or Nr. 14 and Nr. 15. In contrast, surrounding
sets containing points from two planes, t−1 and t−2, give a
slightly better Q than sets containing only points from t−1 -
compare for example sets Nr. 1 and Nr. 7. However, an addition
of multiple time-planes reduces the quality (see set Nr. 12).
As the best quality is obtained for set Nr. 7, this surrounding
set is considered optimal in further calculations. We would
like to stress, that in Fig. 5 only those surrounding sets which
seemed to have the potential to give the best quality were taken
into account. The optimal structure of S was chosen on the
basis of selected sets. To be sure that the chosen structure was
really optimal, it is necessary to calculate the prediction quality
of all the subsets containing all combinations of neighboring
points. Since the number of points in our learning set is
32×32×600, a calculation of Q for all sets would become
a computationally prohibitive task.
D. Optimal prediction of melt pool evolution
After determining the optimal parameters of our RBFNN
model, we next show the discrepancy between the predicted
images of the laser welding melt pool and the corresponding
0 5 10 15
8 97 10
14 15
Fig. 5. Dependence of prediction quality Q (*) on the structure of the
surrounding set S . Parameters are N = 600 and σ = 4. The netlike patterns
(1-16) describe the position of surrounding points, while the netlike pattern
denoted by P describes the position of prediction point s. In the netlike
patterns, only those time planes which contain points from S are plotted.
images from the testing field. In Fig. 6 we therefore present
predicted laser welding images and corresponding images
from the testing field for the optimal structure of RBFNN.
Parameters are N = 600, and σ = 4, while the surrounding set
S has only two member points, both having the same spatial
position as the predicted point, but neighboring positions in
time. Since the quality of prediction is 0.93 (see Fig. 5), a
very good similarity between the predicted and corresponding
image from the testing field is expected. Comparison of
predicted images and images from the testing field in Fig. 6
indeed exhibits a good resemblance. However, we would like
to draw attention to surface smoothness. As can be seen, the
predicted surface is smoother than the original surface. This
can be easily understood if the origin of prediction of images in
the conditional average estimator (Eq. 4) is taken into account.
Predicted ŷ is therefore a weighted average of all those yi, for
which xi is similar to x. Consequently, the surface roughness
is diminished due to conditional averaging.
IV. CONCLUSION
Time evolution of multi-dimensional fields is usually ob-
tained by solving a system of partial differential equations.
However, if the only source of information is a record of the
field, a neural network can successfully replace differential
equations by extracting field evolution properties from the
recorded data. Neural-network-like structures are also expected
to be the working algorithm of living organisms’ intelligence.
In the same way as neural networks, living organisms predict
the evolution of events in their surroundings solely on the basis
a) predicted field b) testing field
51015202530
51015202530
51015202530
51015202530
Fig. 6. Comparison of predicted melt pool images (a) with corresponding
images from the testing field (b) for two randomly chosen testing records. ϕ
stands for field, rx and ry denote spatial coordinates of the record. Parameters
are N = 600 and σ = 4. The surrounding set S has only two member points,
both having the same spatial position (r) as the predicted point s=(r,t), but
different neighboring positions in time, i.e., t−1 and t−2.
of recorded data. It could be conjectured that this operation is
probably performed by extracting simple evolution laws from
recorded data.
In this paper we show how to optimize a statistical modeling
of a field generator performed by the normalized RBFNN, to
efficiently learn spatiotemporal dynamics of multi-dimensional
fields. In our experimental approach, all information about
process dynamics is contained in a measured space-time record
of the characteristic variable. To extract the model of field
evolution from the corresponding discrete sample data, we
employ a non-parametric approach, following a state-space re-
construction technique. The basis of state-space reconstruction
is the formation of sample vectors which are composed of
past and future field distributions. We assume that the field
distribution in a given spatiotemporal point s is correlated
with the field distribution in the spatiotemporal surroundings
of this point, S. The prediction of field distribution in s is
then accomplished as a mapping relation between the field
distribution in the surroundings S and field distribution in s.
Since the optimization of the state-space reconstruction
technique also requires a quantitative measure of the pre-
diction quality, we introduce the quality estimator Q, which
incorporates the difference between the predicted field and
the corresponding testing field. We consider as a proper set
of model parameters those values at which the prediction
quality achieves a maximum. This strategy is used here to
find a proper value of parameter σ and the structure of the
surrounding S utilized in the prediction process. Generally,
an estimation of the proper number of sample points must
also consider the complexity of the experiments, which is
numerically demanding in a multidimensional case [18]. Con-
sequently, we also specify here the proper number N based
upon the analysis of prediction quality.
We demonstrate the proposed method of modeling of the
properties of the laser-heated melt pool. For this purpose, we
employ non-parametric statistical modeling of field evolution
on a spatiotemporal record of the melt pool of the laser
welding process. The major part of the field record is used
for learning, while the minor part of the record serves for
testing. We show how to construct the set of joint sample pairs
containing past and future values of field distributions and pay
special attention to the structure of these sample pairs. We also
present the optimal structure of sample vectors, which gives
the highest resemblance between predicted images and images
from the testing field and has a small number of member points
in order to make the prediction algorithm work quickly.
ACKNOWLEDGMENT
This work was supported by the Ministry of Higher Educa-
tion, Science and Technology of the Republic of Slovenia and
EU-COST.
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	Introduction
	Description of RBFNN
	Non parametric statistical modeling
	Quality of predictor
	Prediction of field evolution
	Time evolution of melt pool
	Optimal value of parameter 
	Optimal number of joint sample pairs
	Choosing the surrounding S
	Optimal prediction of melt pool evolution
	Conclusion
	References

  Efficient control of a laser welding process requires the reliable prediction
of process behavior. A statistical method of field modeling, based on
normalized RBFNN, can be successfully used to predict the spatiotemporal
dynamics of surface optical activity in the laser welding process. In this
article we demonstrate how to optimize RBFNN to maximize prediction quality.
Special attention is paid to the structure of sample vectors, which represent
the bridge between the field distributions in the past and future.

Introduction
	Description of RBFNN
	Non parametric statistical modeling
	Quality of predictor
	Prediction of field evolution
	Time evolution of melt pool
	Optimal value of parameter 
	Optimal number of joint sample pairs
	Choosing the surrounding S
	Optimal prediction of melt pool evolution
	Conclusion
	References
