A Probabilistic Computation of Artiﬁcial Neural Network and Genetic Algorithm in Finding the Minimum-Norm-Residual Solution to Linear Systems of Equations Rodrigo S. Jamisola, Jr. Dept. of Electronics and Communications Engineering De La Salle University - Manila jamisolar@dlsu.edu.ph Elmer P. Dadios Dept. of Manufacturing Engineering and Management De La Salle University - Manila dadiose@dlsu.edu.ph Marcelo H. Ang, Jr. Dept. of Mechanical Engineering National University of Singapore mpeangh@nus.edu.sg Abstract— Artiﬁcial neural network and genetic algorithm have been extensively used in solving many real-world en- gineering problems. In this work these computational meth- ods are used to solve linear systems of equations in ﬁnding the minimum-norm-residual solution, using a probabilistic approach. This work will show the efﬁcacy of probabilistic artiﬁcial neural network and probabilistic genetic algorithm in ﬁnding solutions to determined, overdetermined, and un- derdetermined systems. This work does not claim superiority over other neural network or genetic algorithm computational implementations, nor superiority over other linear solvers, but is presented as an alternative approach in solving root-ﬁnding or optimization problems. Experimental results for randomly generated matrices with increasing matrix sizes will be pre- sented and analyzed. This work will be the basis in modeling and identifying the dynamics parameters of a humanoid robot through response optimization at excitatory motions. I. INTRODUCTION Artiﬁcial intelligence methods have received wide accep- tance to the research community in the past decade. The major reason for this is their ability to ﬁnd the best possible solution to real-world problems, which are difﬁcult to deal with using traditional mathematical techniques. This work is an extension of a previous work that deals with determined linear systems of equations computed through probabilistic artiﬁcial neural network and probabilistic genetic algorithm. This further deals with determined, overdetermined, and underdetermined linear systems of equations by ﬁnding the minimum-norm-residual solution. The results are compared to QR factorization. The increased popularity of artiﬁcial neural network ad- dressed problems including autoregressive moving-average model model parameter estimation [1], identifying linear discrete time systems [2], augmentation of controller for underwater vehicles [3], and 2D pattern recognition problems [4]. For systems of linear equations, artiﬁcial neural network has been used to solve linear system of equations [5] including time-varying systems [6] and linear and quadratic programming [7]. Genetic algorithm’s wide applications include trafﬁc engi- neering optimization [8], learning and structuring of artiﬁcial neural network [9], linear array synthesis problem [10], op- timizing reactive power planning [11], optimizing piecewise linear function [12], solving linear bilevel programming [13], and automated linear modeling[14]. The methods presented in this work will be the basis in the modeling and identiﬁcation of the humanoid robot dynamics. In addition, the difﬁculty in using traditional method of computation in identifying the failure-tolerant workspace for redundant manipulators [15], [16] can hopefully be addressed using the computational methods discussed in this work. A. Overview on Probabilistic Artiﬁcial Neural Network Artiﬁcial neural network (ANN), being a function mapping computational approach, does not give too much emphasis on how the network is implemented but rather on getting the desired output from the given set of inputs. In general, the neural network implementation can be treated as a black box. This work will try to move away from the traditional computational approach of the ANN by attempting to ﬁnd the appropriate values of the weights, which is the unknown vector x, in order to arrive at a desired solution of Ax = b. The desired output is the input vector b, and matrix A will remain constant all throughout the computation. This method will use backpropagation technique in ANN to adjust the weights values x at every computational cycle. The strategy therefore is to ﬁnd the appropriate stepping technique such that when the values of the weights are adjusted at every iteration, the resulting weights move closer to the correct values. The correct values of the weights are declared to be found when the computed value of Ax is close enough to the desired input value b within the required tolerance ε . A predeﬁned maximum number of iterations is set such that if the desired ε is not achieved up to the maximum iterations, a random walk from the ﬁnal output is performed to start a new iterative computation. The random walk is the method of escaping from the local minimum in the artiﬁcial neural network computation presented in this work. Probabilistic neural networks applications include pattern [17] and power quality classiﬁcation [18], volume segmen-