Approximating the Algebraic Solution of Systems of Interval Linear Equations with Use of Neural Networks Nguyen Hoang Viet Michal Kleiber Institute of

Approximating the Algebraic Solution of Systems of Interval Linear Equations with Use of Neural Networks

Nguyen Hoang VietMichal Kleiber

Institute of Fundamental Technological Research. Polish Academy of Sciences.

Email: {viet,mkleiber}@ippt.gov.pl

N.H. Viet & M Kleiber. IFTR, Polish Academy of Sciences.

Outline

Introduction to systems of linear interval equations (SILE),

Solving SILE as an optimization task, Neural networks in solving SILE, Result and conclusions.


Introduction to SILE

Interval real number:

Operations on interval real numbers:

Endpoint notations:



Distance between IRVs:



System of Interval Linear Equations:

SILEs can be found in many real world applications (for example in FEM methods), where all uncertainties can be describe in term of impreciseness.

The two most popular types of solution: the united solution and the algebraic solution.



Algebraic solution: an interval vector so that the product:

gives an interval vector equal to , where the product of interval real numbers:



Algebraic solution (...)And in effect:

It is well known that solving the algebraic SILE is an NP-hard problem.


Solving SILE as an optimization task

The cost function:

where:

The cost function is not differentiable, event not continuous.



The product of IRNs is approximated by a differentiable function:

The modified cost function:



The modified cost function is now differentiable:


Neural network for IRN multiplication

Network architecture:

All neurons are sigmoidal. Such networks represent a differentiable function wrt. to each input variable.


Neural network for IRN multiplication

Hybrid network for computing the gradient wrt. input signals:


Results and Conclusions

A set of 50 networks with 5 hidden neurons were trained with use of 500 training samples and 200 validating samples (randomly generated).

These NNs were tested against a test set composed of 2000 samples.

The best one was chosen to be used in SILE solving.

NN training and cost function minimization were done with use of the Scaled Conjugate Gradient Algorithm.



The left side interval matrices and right side interval vectors were generated in order to test the performance of the proposed approach

Results for various sizes of the problem:



Example 1 (f=1.38e-30, 312 iterations)



Example 2: (f=7.8e-5, 42 iterations)



A new approach for solving the NP-hard problem SILE was proposed.

Small neural network, no need to update. The results can be obtained in real time for

large systems. An alternative for other approach, for

example the GA. Similar techniques are being developed for

solving system of fuzzy linear equations.

Thank you for your attentionThank you for your attention

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Approximating the Algebraic Solution of Systems of Interval Linear Equations with Use of Neural Networks Nguyen Hoang Viet Michal Kleiber Institute of