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

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.

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

Introduction to SILE

Interval real number:

Operations on interval real numbers:

Endpoint notations:

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

Introduction to SILE

Distance between IRVs:

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

Introduction to SILE

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.

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

Introduction to SILE

Algebraic solution: an interval vector so that the product:

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

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

Introduction to SILE

Algebraic solution (...)And in effect:

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

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

Solving SILE as an optimization task

The cost function:

where:

The cost function is not differentiable, event not continuous.

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

Solving SILE as an optimization task

The product of IRNs is approximated by a differentiable function:

The modified cost function:

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

Solving SILE as an optimization task

The modified cost function is now differentiable:

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

Neural network for IRN multiplication

Network architecture:

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

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

Neural network for IRN multiplication

Hybrid network for computing the gradient wrt. input signals:

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

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.

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

Results and Conclusions

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:

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

Results and Conclusions

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

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

Results and Conclusions

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

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

Results and Conclusions

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