A barrier to data-driven discovery is that existing AI methods often do not meet the needs of scientific models, i.e. they must respect or incorporate physical laws, process and operational constraints. We propose to analyse the flotation behaviour inside a columnar cell a Genetic Algorithm-based method with directly measured data and other data calculated through flotation hydrodynamics. Typical variables measured in a columnar cell are air flow (Jg), slurry flow (Jl), froth height (hf), % solids (Cp), particle size p80 and p95, reagent type and dosage. Adding hydrodynamic variables such as: bubble size db, gas holdup Eg and bubble surface area flux Sb which are difficult to measure in real time. A genetic model is built and solutions are obtained to explain the recovery, the "winning" model uses the chromosome with the smallest error and shortest length, but the "shape" of the result equation does not comply with the laws of physics.

1. Application of Genetic Algorithms to
describe flotation behavior in a
columnar cell
Leon, Mario1

2. Introduction - Hydrodynamics of Flotation
Machine
Learning
• Problems too large and too
complex.
• Unable to capture all effects for
realistic simulation.
• Constantly evolving data.
• Enough accuracy and precision
Data from the
instruments installed
in the cell column
https://benmoseley.blog/my-research/so-what-is-a-physics-informed-neural-network/
Where are:
• Hydrodynamics of
Flotation?
• Virtual sensors?

3. Our Approach: Physics-Informed Machine Learning

4. Fluid Dynamic Model - Columnar Cell Flotation
• Two-phase CFD models of a Columnar Cell
(Water + air bubbles) for the study of Gas
Holdup.
• Fluid Dynamic Models, Stokes, Sedimentation.
• Bubble density.
• Drift-Flux model for bubble diameter.
• Models based on empirical equations to validate
the models.
• Kinetic models for recovery and residence time.
• Calibration of models direct measurements (Gas
holdup).
• ...
Ikukumbuta Mwandawande, 2016

5. New Approach: Genetic Algorithm (GA) Application
Classical Regression
Symbolic Regression
Classical numerical regression requires the structure of the
expression to be fixed a priori
Symbolic regression allows us to obtain both the structure
of the expression and the values of the coefficients in an
automatic way
yi = β0 + β1sin(x1+x2)+ β2exp(x3) + ε

6. Column Cell Model
Air Flow
Slurry and wash
water Flow Rate
Froth Height
Gas Holdup
Copper Feed
grade
Collector
Dosage
Frother Dosage
Mineral and
Slurry Density
Bubble Surface
Area Flux
Recovery
Copper
Concentrate
grade

7. Data - Column Cells
The data was obtained from 3 different copper mines using
Columnar Cells in their flotation circuits.
For this analysis we have data from instruments installed in the
cells, the total is equivalent to a period of one year and 6
months with a frequency of 6 minutes.
The data was split into groups for training, testing and validation
in continuous data proportions of 60%, 30% and 10%.
The data was analyzed before processing in search of outliers. It
was found that 5% of the data corresponded to shutdowns,
maintenance, start-up processes, defective instruments, etc.

8. MEP Model - Multi-Expression Programming
• The MEP algorithm starts by creating a random
population of numerical expressions.
• Two parent individuals are selected by a selection
procedure.
• The parents recombine to obtain two offspring.
• These new offspring individuals are considered for the
mutation process.
• The best offspring will replace the worst individual in the
current population, provided it has a better fit.
• Very destructive crossover, which generates a lot of
diversity.

9. MEP Model Configuration
Parameter Set Value
Functions
1
Addition, subtraction, multiplication, division,
power, square root, square root, exponential,
inverse, negative, square
2
Multiplication, division, power, power, square
root, exponential, inverse, negative, square
Measurement of error 1, 2
MAE: Mean absolute error
MAD: Median absolute error
n° of sub-populations 1, 2 50
Sub-population size 1, 2 300
Tree length 1, 2 80, 40, 20
Crossover probability 1, 2 90%
Mutation probability 1, 2 1%, 5%
n° of generations 1, 2 60
n° of runs 1, 2 3000
The MEP model was processed with a
computer equipped with Xeon Processor,
4 cores, 3 GHz, 24 GB ram.
OS: Windows 10 Pro 64-bit,
Processing time one model: ~ 1 day,
n° of processed models: 10

10. Model
Error MAE %
Cu grade
Expression Constant
1 0.407
a0 = 0.0720
a1 = 0.3167
2 0.408 a0 = 0.0111
3 0.410
a0 = 0.3566
a2 = 0.3115
4 0.411
a0 = 0.3521
a2 = 0.0748
1
3
0
0
1
0
1
x
x
a
x
a
a
y 
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
3
1
0
0
1
x
x
x
a
y 
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
0
1
0
2
0
0
1
0
0
0
0
exp
1
a
x
a
a
x
x
x
a
a
a
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y 
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Results
The variables are as
follows:
X0 Feed Cu grade,
X1 % solids,
X2 Wash water,
X3 Froth height,
X4 Airflow,
Y Concentrated Cu grade

11. Results
The Models f2, f3 y f4 have the same fit and shape as the "winning" model f1.
This expression perfectly follows the fluctuations of the measured grade as a function of the
measured variables: X0 Feed Cu grade, X1 % solids, X3 Froth height and X4 Airflow.

12. The models obtained with symbolic regression are numerically accurate over the range of the data
(Domain).
The best models do not contain the variable X2 Washing water, this is because this variable has a MAD /
median ratio of 5%, therefore it is close to constant.
The models do not have a "classical" structure or form of a physical model.
Analysis of the models with data outside the training range generates unexpected results, therefore,
further analysis is required to apply them to behavioral predictions.
One approach that can be applied to avoid the effect of obtaining models with "unphysical shapes" is to
use only dimensionless variables, e.g. Reynold's number, concentrate output / feed flow ratio and etc.
Conclusions

13. (1) The author would like to acknowledge the support of IntelliSense.io/BASF in the development of this publication.
References:
• Mihai Oltean and Crina Groşan, “Evolving digital circuits using multi expression programming”
• Crina Groşan, “Evolving mathematical expressions using Genetic Algorithms”
• Mihai Oltean and Crina Groşan, “Evolving evolutionary algorithms using multi expression programming”, 2021
• Sourabh Katoch & Sumit Singh Chauhan & Vijay Kumar, “A review on genetic algorithm: past, present, and future”,
2020
• F. Nakhaei, M. Irannajad, M. Yousefikhoshbakht, “Simultaneous optimization of flotation column performance
using genetic evolutionary algorithm”, 2016
• J.H. Holland, “Adaptation in Natural and Artificial Systems”, 1975
• Cramer, N.L., “A representation for the adaptive generation of simple sequential programs”, 1985
• Koza, J., “Genetic programming: On the programming of computers by means of natural selection”, 1992
• Mohammadzadeh S., Bolouri Bazaz , Amir H. Alavi b “An evolutionary computational approach for formulation of
compression index of fine-grained soils”, 2013
Acknowledgements / References