Embedding Monotonicity and Concavity in the training of Neural Networks
by means of Genetic Algorithms.
Application to multiphase flow.


Computers & Chemical Engineering,  accepted (2004)


Laurentiu A. Tarca, Bernard P.A. Grandjean  and Faїçal Larachi
Department of Chemical Engineering & CERPIC
Laval University - Québec, Canada G1K 7P4


It is well established that including monotonic constraints in neural network (NN) development, when such a priori information is available, increases the confidence in predictions and prevents over fitting. An elegant procedure to guarantee mathematically monotonic NNs with respect to some of their inputs is to force the signs of some of the network’s weights in such a way that the network remains monotonic in the entirety of the input domain. In this work, besides monotonicity, a second-order information, namely, the concavity, is used to guide a genetic algorithm - genetic hill climber optimizer to identify the weights of the neural network. Monotonicity and concavity are key conditions in establishing phenomenological correlations when the available data for training are insufficient to cover in depth the n-dimensional input space. In such instances, classical training procedures fail to unveil the main tendencies in data and may suffer local over fitting problems. In this work, the proof-of-concept of embedding monotonicity and concavity information in the training of NNs by means of genetic algorithms will be illustrated in correlating total liquid holdup in randomly packed bed containing counter-current gas-liquid towers.


Neural networks; monotonicity; concavity; genetic algorithm; liquid holdup; multiphase flow reactor

You can get the LH_CCPBed.zip file that contains an Excel worksheet simulator to compute 
the liquid holdup in randomly packed beds with counter current flow.

You may also download our  Excel worksheets simulators for  Trickle-bed or Flooded Bed reactors.