AN EVOLUTIONARY APPROACH FOR SOLVING DIFFERENTIAL EQUATIONS

Transcription

1 AN EVOLUTIONARY APPROACH FOR SOLVING DIFFERENTIAL EQUATIONS M. KAMESWAR RAO AND K.P. RAVINDRAN Depamen of Mechanical Engineeing, Calicu Regional Engineeing College, Keala-67 6, INDIA. Absac:- We eploe he possibiliy of solving diffeenial equaions by using evoluionay algoihm. The ial soluion is consuced by incopoaing a Neual Newok componen and a non-neual pa so as o saisfy he iniial/bounday condiions. The effecive minimisaion of he eo funcion gives he closed analyic fom of he soluion fo he given diffeenial equaion wih specified bounday/iniial condiions. We y o minimise he eo funcion by using he Evoluionay algoihm. Compaison of he soluions hus obained and he soluion obained analyically is done. The pesen wok eveals he abiliy of Evoluionay algoihms fo solving Diffeenial equaions. The pesence of he Neual Newok componen makes he soluion hus obained amenable o design on hadwae. Key wods:- Neual Newoks, Diffeenial Equaions, Evoluionay algoihm, iniial/bounday condiions.. Inoducion The mahemaical analysis of any physical sysem is done by fomulaing he Diffeenial equaion govening he sysem wih he appopiae iniial/bounday condiions and solving i. As sysems become comple, i is highly difficul and impossible o obain a closed fom soluion fo he govening equaion [GE]. Thus he compuaional scheme employed becomes he conesone in he analysis of he physical sysem. We can a he mos obain he soluion a discee poins by appoimaing he ems in he GE by Taylo seies, o obain a sysem of equaions. By solving hese equaions, he soluion a discee poins of he domain is obained. Also, by he disceizaion of he domain ino elemens and using appoimaion funcions fo he soluion in each elemen, he soluion a he nodal values can be found, fom which he values a any poin inside he elemens can be found by using inepolaion mehods. All hese mehods have he shofall of he lack of closed fom soluion and limied diffeeniabiliy. In he pevious wok employing Neual Newoks fo solving diffeenial equaions, soluions wee obained by consucing he ial soluion using neual newoks o saisfy he iniial/bounday condiions. The eo funcion ove a numbe of poins in he domain was obained by he collocaion mehod and his eo funcion was minimised employing gadien-based mehods []. Also, Diffeenial equaions wee solved by disceizaion of he poblem domain o obain a sysem of equaions. The soluion of his sysem of equaions is mapped ono a Hopfield neual newok. The effecive minimisaion of he enegy funcion being he opimisaion poblem ha gives he equied soluion [-]. Anohe appoach fo solving diffeenial equaions using Neual Newoks was o employ B -splines as basis funcions and mapping he soluion fom of he coefficiens of hese ono a feed fowad Neual Newok. Hee, each spline is eplaced wih he sum of piecewise linea acivaion funcions ha coespond o hidden unis [4-5]. This mehod consides local basis funcions and in geneal equies many splines in ode o yield accuae soluions. Moeove, i is no easy o eend his echnique o muli-dimensional domains. In he pesen wok he minimisaion of he eo funcion poposed in [] is minimised using Evoluionay Algoihm. The minimisaion of he eo funcion is iself he aining of he Neual Newok. As he gadien descen mehods have he endency o ge apped in he local minima, he evoluionay algoihm is used in he opimisaion of he eo funcion and hence he aining of he Neual Newok. The pobabiliy of he Evoluionay Algoihm o convege o global minima is moe han convenional calculus based echniques. The usage of evoluionay algoihm may enhance he effeciveness of he mehod of solving diffeenial equaions by using Neual Newoks. In addiion o he

2 feaues menioned in [], he poposed appoach will ensue a bee convegence o global opima. Descipion of he mehod The mehod of consucion of ial soluion is eplained wih he following geneal diffeenial equaion poposed in []. G(, Ψ(), Ψ(), Ψ()) =, D () subjec o he bounday condiions (BC s), eihe Diichle o Neumann, whee n n = (,,,...n ) R,D R denoes he definiion of he domain and Ψ ( ) is he soluion o be compued. Applicaion of he collocaion mehod educes he poblem o G(i, Ψ(i ), Ψ(i ), Ψ(i )) =, i ^ D () subjec o he consains imposed by he BC s. Le Ψ (,p) denoes a ial soluion wih adjusable paamees p, and hen he poblem is ansfomed o Min (G(i, Ψ (i,p), Ψ (i,p), Ψ (i,p))) ^ i D () consained by he BC s. The ial soluion Ψ (,p) employs a feed fowad Neual Newok and he paamees p coespond o he weighs and biases of he neual achiecue [6],[7]. The ial fom of he soluion by consucion saisfies he bounday condiions, which is achieved by wiing i as sum of wo ems Ψ () = A() + F(, N(,p)) (4) whee N(,p ) is single oupu feed fowad neual newok wih paamees p and n inpu unis fed wih he inpu veco. The em A( ) conains no adjusable paamees and saisfies he bounday condiions. The second em is consuced so as no o conibue o he BC s, since Ψ () mus also saisfy hem. The weighs and biases of he Neual Newok in his em ae o be adjused and his consiues he opimisaion poblem. As he ial soluion is consuced so as o saisfy he BC s, he opimisaion poblem becomes an unconsained opimisaion poblem. The employmen of Gadien mehods [] may make he soluion o convege a a local minimum insead of he global minimum. Also he minimisaion using Gadien mehod equies he compuaion of he Gadien of he eo funcion wih espec o he newok paamees and an efficien opimisaion package as well. On he ohe hand, he employmen of he Evoluionay algoihm doesn equie Gadien compuaion and is vey easy o implemen. I has demonsaed good ageemen wih he analyic soluion.. Evoluionay Algoihm The Evoluionay Algoihm elies on he pinciples of naual selecion and suvival of he fies. I sas he seach pocess fom a collecion of poins ahe han a single poin. I doesn equie any auiliay knowledge like Gadien of he eo funcion fo is seach, bu insead, needs he value of eo funcion alone [8]. Evoluionay Algoihms ae usually applied o assign values ha allow a paicula pefomance o finess o he paamees of a sysem. An Evoluionay Algoihm begins wih a populaion of seveal samples of he same sysem wih vaiables assigned andomly o abiaily [9]. The diffeen samples undego a se of ials in ode o compae hei pefomances wih he equied one. In his way, each sample eceives an evaluaion. Samples ha have obained he bes finess ae hen seleced and epoduced wih andom cossove and muaion in ode o compose a new geneaion. The cossove pocess consiss of wo seps. Fis, selecing he newly epoduced sings andomly and hen geneaing new sings by swapping all chaaces fom andomly seleced cossove sies. The muaion opeaion consiss of copying he values of vaiables by adding lile andom changes. The new geneaion undegoes he same se of ials. The pocess is coninued unil desied accuacy of he soluion is obained. In he pesen wok we use Evoluionay Algoihm fo he minimisaion of he unconsained eo funcion. The vaiables fo he unconsained opimisaion poblem ae he weighs and biases of he Neual Newok. In he pesen mehod we used eal numbe coding, as he numbe of vaiables is lage and i is difficul o do binay coding. Fom epeimens i was found ha he values of he weighs ange fom o +, he iniial populaion was geneaed accodingly. An appopiae value fo he eo funcion was chosen as he cieion fo eminaion.

3 The paamees of he Evoluionay Algoihm used ae shown in Table. Ψ a / e = (8) Populaion size Sing lengh Numbe of cossove sies 6 The Neual Newok achiecue fo OD s is shown in Fig. and he paamees used in he Newok ae shown in Table. Type of coding Real numbe coding Objecive funcion Minimising he eo funcion Vaiables Weighs of he inpu o hidden laye, hidden laye o he oupu laye and biases in he hidden laye I/P O/P Table Evoluionay algoihm paamees Illusaion wih eamples The following eamples illusae he mehod and compaison of he soluion wih he analyic soluion.. Odinay Diffeenial Equaions In his secion he Odinay Diffeenial Equaions ae deal... Poblem The Diffeenial Equaion and Bounday condiion ae: dψ d + + = Ψ BC s: Ψ ()=, [, ] (5) The ial soluion can be wien as, Ψ () = A() + F(, N(,p)) (6) Fo his poblem his educes o Ψ () = +.N(,p) (7) The ue soluion (analyic) is, Fig. Neual Newok Achiecue fo ODE s Numbe of layes Numbe of inpu nodes Numbe of hidden nodes Type of squashing funcion Numbe of oupu nodes Sigmoid funcion /(+e - ) Table Paamees used in Neual Newok The numbe of aining poins aken was en, wih unifom spacing in he poblem domain. Fig. shows accuacy of he soluion obained by he Evoluionay Algoihm as compaed wih he analyic soluion, Hidden laye

4 while he plo of analyical soluion is given in Fig.. The soluion accuacy obained hee is compaable wih he esuls in []. soluion, while he plo of analyical soluion is given in Fig.5. soluion accuacy f() Fig. Soluion accuacy fo Poblem Fig.4 Plo of Analyical soluion fo Poblem II f() Fig. Plo of analyical soluion fo Poblem... Poblem-II This is anohe Odinay Diffeenial Equaion and he Bounday condiion used in his poblem ae: dy = d y soluion accuacy Fig.5 Soluion accuacy fo Poblem II. Paial Diffeenial Equaions In his secion he soluion of paial diffeenial equaions (PDE s) ae discussed. Fig.6 shows he Neual Newok achiecue used fo PDE,s..9 B.C. a y () = ; (9) The ue soluion fo his poblem is, ( ) 4e + ya = ( ); () The ial fom of he soluion used in his eample is: y = + ( )N(,p); () Fig.4 shows accuacy of he soluion obained by Evoluionay Algoihm as compaed wih he analyic I/P weighs biases O/P weighs Fig.6 Neual Newok Achiecue fo PDE s

5 .. Poblem III u u + = y [,] y [,] B.c s ae u (,) = ; u (,y) = ; u (,y) = ; u(,) = sin( π) () Analyic soluion fo his poblem is u a = sin( π).sin( πy) / sinh( π) () The ial soluion o his poblem can be wien as u = y sin( π) + ( )( y)yn (4) This is he govening equaion fo a hea ansfe poblem fo a fla plae wih fou Diichle BC s. The soluion accuacy of he poblem is ploed fo poins in he poblem domain. The accuacy of he soluion is checked wih he analyic soluion of he poblem. Fig.7 shows he plo of soluion accuacy a vaious poins in he domain. The plo is made by aking poins in he domain. The X ais in he plo epesens he poins on X dimension of he poblem domain and Y ais epesens he poins on Y dimension of he poblem domain. The Z ais epesens he soluion accuacy, which is he diffeence beween he analyic soluion and he soluion obained by he Neual Newok mehod.. Poblem IV ϕ(,y) ϕ(,y) + ϕ(,y) = y [,] sin( π)( π y [,] y + y sin( π); B.C. s ae ϕ (,y) =, ϕ (,y) =, ϕ (,) =, ϕ(,)/ y = sin( π) (5) The ial soluion is, ϕ = B(, y) + ( )y[(n(, y,p) The analyical soluion o his poblem is, N(,, p) N(,, p) (6) ] y ϕ = sin( π) (7) a y This is a non-linea paial diffeenial equaion wih boh Diichle and Neumann bounday condiions. The plo of soluion accuacy vesus vaious poins in he domain is shown in Fig. 8. The plo of he soluion accuacy a vaious poins in he domain is shown simila o ha of he pevious poblems. The Z ais epesen he soluion accuacy, which is he diffeence beween he ue soluion and he soluion obained by he Neual Newok mehod. Fom he plo i is eviden ha he soluion accuacy a he boundaies is zeo because of he choice of he ial soluion. The accuacy a ohe poins is posiive as well as negaive. Fig.7 Soluion accuacy fo Poblem III Fig.8 Soluion accuacy fo Poblem IV

6 4 CONCLUSIONS AND SCOPE The applicabiliy of Evoluionay Algoihm fo solving Diffeenial Equaions wih ial soluions having a Neual Newok is eploed. The compaison of he soluions hus obained, wih he analyic soluion showed good ageemen. I should be noed ha ou epeimens wee caied on an odinay PC wih Penium III pocesso wih 56MB RAM. The mehod is o be fuhe modified o apply o Paial Diffeenial Equaions wih iegula domain []. Fuue wok involves compaison of soluions hus obained, wih his mehod and he one s employing Gadien Algoihm as well as Finie Elemen Mehod. Refeences: [] I.E.Lagais, A.Likas and D.I.Foiadis, Aificial Neual Newoks fo Solving Odinay and Paial Diffeenial Equaions, IEEE Tansacions on Neual Newoks, Vol.9, No.5, 998, pp [] H.Lee and I.Kang, Neual Algoihms fo solving Diffeenial Equaions, J.Compu.Phys., Vol.9, 99, pp. -7. [] L.Wang and J.M. Mendel, Sucued ainable newoks fo mai algeba, IEEE In. Join Conf. Neual Newoks, Vol., 99, pp.5-8. [4] A.J.Meade, J., and A.A.Fenandez, The numeical soluion of linea odinay diffeenial equaions by feedfowad neual newoks, Mah.Compu. Modelling, vol.9, no., 994, pp -5. [5] A.J.Meade, J., and A.A.Fenandez, Soluion of nonlinea odinay diffeenial equaions by feedfowad neual newoks, Mah.Compu.Modelling, vol., no.9, 994, pp [6] David.E.Goldbeg, Geneic Algoihms in Seach, Opimizaion & Machine Leaning,, Addison Wesley, 999. [7] William. H.Pess e al, Numeical Recipes in C, nd ediion, Cambidge Univesiy Pess, 99. [8] Simon Haykin, Neual Newoks A compehensive foundaion, nd ed., Penice Hall Inenaional, Inc.,999. [9] N.K.Bose, P.Liang, Neual Newok Fundamenals wih Gaphs, Algoihms and Applicaions, Taa McGaw Hill, 998 [] I.E.Lagais, A.Likas and D.I.Foiadis, Neual Newok Mehods fo Bounday Value Poblems wih Iegula Boundaies, IEEE Tansacions on Neual Newoks, Vol., No.5,, pp