Page 90 - Proceedings of the 2017 ITU Kaleidoscope

2017 ITU Kaleidoscope Academic Conference

Fig. 1. Workflow to enrich sparse data to dense data
performance of machines or sensors. The computer aided enriched dense enough for data-driven decision making
calculation, also known as numerical simulation, is a extraction within a short phase.
powerful tool to generate sampling datasets. However, the The model reduction procedure adopts a top-to-down
complex numerical simulation could be time-consuming approach and starts from the original partial differential
because of the high resolution and complex process physics. equations and derives approximated analytical solutions or a
set of ordinary equations using many mathematical
3. ENRICHING SPARSE DATA BY REDUCED approaches, physical and phenomenological approaches,
MODELS numerical approaches and data driven model reduction
methods. Especially, there are several model reduction
Generally speaking, the reduced models can enrich the techniques which are convenient enough to use and worth
sparse data from two aspects. On the one hand, the reduced reviewing.
models decrease the required data volume dramatically. The 3.1 Perturbation analysis
“sparse data” become “dense data” because of the
compendious dimensionality. On the other hand, the reduced The objective of perturbation theory is to determine the
models are also characterized by their convenient solvability, behavior of the solution when one variable tends to be very
millions of datasets can be generated from reduced models small, which can lead to the split of two part of solutions for
in acceptable duration. complex system. One part is the temporal solution and
The workflow to enrich sparse data into dense data are shown another part is the long term asymptotic solution. This
in Figure 1. Firstly, the data is extracted from divergent separation of system solutions result in the reduction for the
sources in real manufacturing processes and experimental models. The typical application of this perturbation theory
measurements. Different types of sensors, diagnosis lies in the fields involving differential equations as well as a
techniques and design of experiments are intensively series of engineering problems [2] [3]. Vossen et al. used the
operated to accumulate the original sparse data. At this stage, perturbation and asymptotic analysis to describe the
the data are featured by its high volume, insufficient dynamical behaviors of the free boundaries of the melt
dimensionality, heterogeneous distribution and irregular data during the laser cutting process considering the spatially
format. Thereafter, the massive mathematical and physical distributed laser radiation. A reduced model which can
modelling work for the complex manufacturing process is generate results at real time scale was derived by
performed and validated by the sparse data in the first stage. perturbation analysis for the purpose of predicting the
From the well-built complicated models, the full product roughness [4].
dimensional parameters will be involved and the data
generated from multifarious models is equipped with 3.2 Inertia and central Manifold analysis
standard data format. Since the sampling process by complex
models is time consuming, model reduction techniques are The inertial manifolds are connected with the long term
applied to generate the fast and frugal reduced models and behavior of the solutions of dissipative dynamical systems.
avoid the unnecessary complexity. The reduced models are The reduced phase space of ordinary differential equations
derived to avoid any unnecessary complexity and to reduce and partial differential equations in the long time limit is
the computation time of large-scale dynamical systems by named as central or inertial manifold. Schulz et al. derived a
inducing approximations of much lower dimensions which reduced model by applying inertial manifold method. This
can produce nearly the same input-output response reduced model can calculate the thermal behavior in laser
characteristics. Meanwhile, to ensure the accuracy and the manufacturing processes very fast [5, 6].
applicability on the specific context, the reduced models are
also calibrated and validated by the measured sparse data. 3.3 Buckingham Pi theory
These reduced models can simulate the complex system by
preventing redundant calculation, so the sparse data can be

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