Multi-Fidelity Surrogate Based on Single Linear Regression

Transcription

1 echncal Note Mult-Fdelty Surrogate Based on Sngle near Regresson Ymng Zhang, Nam-o Km, Chanyoung Park, Raphael. atka Department o Mechancal and Aerospace Engneerng Unversty o Florda Ganesvlle, Florda, 326 {ymngzhang52, nkm, cy.park, hatka}@ul.edu Introducton Varous rameworks have been proposed to predct mechancal system responses by combnng data rom derent deltes or desgn optmzaton and uncertanty quantcaton as revewed by Fernández- Godno et al.[] and Peherstorer et al. [2]. Among all rameworks, the Bayesan ramework based on Gaussan processes [3] has the potental o hghest accuracy. owever, the Bayesan ramework requres optmzaton or estmatng hyper-parameters, and there s a rsk o estmatng napproprate hyperparameters as Krgng surrogate oten does, especally n the presence o nosy data. We propose an easy and yet powerul ramework or practcal desgn and applcatons. In ths techncal note, we revsed a heurstc ramework [4] whch mnmzes the predcton errors at hgh-delty samples usng optmzaton. he system behavor (hgh-delty behavor) s approxmated by a lnear combnaton o the low-delty predctons and a polynomal-based dscrepancy uncton. he key dea s to consder the low-delty model as a bass uncton n the mult-delty model wth the scale actor as a regresson coecent. he desgn matrx or least-square estmaton conssts o both the low-delty model and dscrepancy uncton. hen the scale actor and coecents o the bass unctons are obtaned smultaneously usng lnear regresson, whch guarantees the unqueness o ttng process. Besdes enablng ecent estmaton o the parameters, the proposed least-squares mult-delty surrogate (S-MFS) can be applcable to other regresson models by smply replacng the desgn matrx. hereore, the S-MFS s expected to be easly appled to varous applcatons such as predcton varance, D-optmal desgns, uncertanty propagaton [6, 7] and desgn optmzaton. east-squares Mult-delty Surrogate A heurstc mult-delty surrogate (MFS) usng polynomal response surace (PRS) has been proposed[4] and demonstrated reasonable eectveness and robustness. he MFS provdes a t combnng a small number o expensve hgh-delty data and less expensve low-delty data as a uncton o desgn parameters. he MFS s bult wth two surrogates, ˆ ( x) and ˆ( x ), whch are the PRS tted to the low delty data and the dscrepancy data as ˆ ( x) ˆ ( x) ˆ ( x ) () where the scale actor ρ and dscrepancy uncton ˆ( x ) are obtaned through optmzaton as, ˆ ( x ) ˆ x d ˆ x d mn : ( ) ( ) (2) d ˆ ( x ) y (3) where the hgh-delty data set s denoted as ( x, y ) ncludng n samples. In ths techncal note, we propose the least-squares estmaton o the scale actor ρ and the coecents o the dscrepancy uncton. he MFS n Eq. () can be expanded wth the dscrepancy uncton beng represented by p monomal bases, as

2 where X x denotes the ˆ ( x) ˆ ( x) ˆ ( x) ˆ ( x )+ = X p x th monomal/bass, and b s the coecent o b (4) X x. In the proposed S- MFS, the tradtonal desgn matrx s augmented by ncludng the low-delty model wth unknown scale parameter ρ. In least-squares estmaton, the relatonshp between hgh-delty samples and model predctons can be gven as where Y XB e (5) X () ˆ () () () () y ( x ) X x p x b e Yn, n ( p ), X B p, en ( n) ˆ ( n) ( n) ( n) ( n) ( ) X X y p x x x b e p (6) In the above equaton, Y s the vector o hgh-delty samples, X s the augmented desgn matrx, B s the parameter vector and e s the vector or resdual errors. By augmentng the desgn matrx, the scale parameter and unknown coecents o dscrepancy unctons are estmated smultaneously. In addton, the low-delty model s consdered as addtonal bass uncton. An addtonal advantage o the proposed S- MFS s that t s unnecessary to buld the surrogate model o the low-delty. It only requres to evaluate the low-delty model at the same locatons wth the hgh-delty samples. he unknown parameters n S-MFS are obtaned by usng the standard regresson technque as B X X X Y (7) he scale actor mples the level o trend smlarty between ˆ ( x ) and ˆ ( x ), and plays a crtcal role to approxmate mult-delty data. Negatve values or extremely large values o ndcates a rsky predcton, whch are lkely to be assocated wth undesrable low-delty models, napproprate surrogate orms, or nadequate samples. he dscrepancy uncton ˆ( x) s lkely to be a low-order PRS when had a smlar trend as x. here are nce propertes o S-MFS. Frstly, the S-MFS could be easly appled to more than twodelty models by augmentng the desgn matrx wth multple low-delty models wth multple scale actors, although we showed the S-MFS or two-delty models n ths techncal note. Secondly, comparng wth the heurstc approach, the S-MFS s more convenent or analytcal study o varous applcatons such as the predcton varance, D-optmal desgn o experments, uncertanty propagaton, and desgn optmzaton, whle ncorporatng the eect o the mult-delty models. Numercal perormance o the least-squares mult-delty surrogate We selected an exponental uncton [, 9] wth two varables to test the S-MFS. he hgh-delty model 2 x s gven n Eq. (). A synthetc nose ollowng a normal dstrbuton,.2 x N has been added to the hgh-delty samples. We also moded the orgnal low-delty model [] wth a larger scale actor and added a quadratc uncton as n Eq. (9). hese varatons make the mult-delty modelng more challengng. Major settngs o the test uncton are summarzed n able. he responses o x (no nose) and x have reasonable spatal complexty as shown n Fg..

3 3 2 23x 9x 292x 6 ( x ) exp () 3 2 2x2 x 5x 4x 2 ( x) [ ( x.5, x2.5) ( x.5, max, x2.5 )] [ ( x.5,.5) (.5,max x2 x, x.5 2 )] ( x ) x2 (9) able. Major settngs or the exponental test uncton Input varables Range o x Range o x Nose or x x, x2 [,] [.4, ] [-7, 4.949] 2 N,.2 5 gh-delty model ow-delty model y x 2.2 x. Fgure.Responses o the test uncton n 2D space We generated two sets o samples accordng to 2 2 and 33 ull actoral desgns as shown n Fg. 2(a) and Fg. 2(b). y and y were computed rom x and x, respectvely, at the ull actoral desgns. Instead o the low-delty surrogate ˆ ( ) x, the exact low-delty model x s used to avod approxmaton error n buldng the low-delty surrogate. It s also possble to buld ˆ ( x ) based on only low-delty data or repeated calls o S-MFS n practcal applcatons. he S-MFS was evaluated usng relatve root-mean-square error (RMSE) or the overall accuracy and maxmum error or the worst ndvdual predcton. he relatve RMSE and maxmum error were obtaned based on test grds as shown n Fg. 2(c). he key parameters and perormance o S-MFS were summarzed n able 2. For the 2 2 ull actoral desgn, S-MFS had the relatvely small RMSE o 9.% when approxmatng the complex test uncton wth nose usng a lnear dscrepancy uncton. he large maxmum relatve error, 3.37%, was manly due to a small uncton value. Whle ntroducng more samples usng the 33 desgn, all the evaluaton metrcs (e.g. the relatve RMSE, the maxmum relatve/absolute errors) decreased notceably as n able 2. PRS ttng or only hgh-delty samples were gven n able 3 as a comparson. For the 2 2

4 desgn, relatve RMSE o PRS ttng was 5 tmes larger than that o S-MFS, and the absolute maxmum error o PRS ttng was tmes larger than that o S-MFS. For the 33 desgn, PRS ttng mproved whle ntroducng more samples, but was stll sgncantly neror to the S-MFS regardng the speced evaluaton metrcs (a) 2 2 ull actoral desgn or both x x and.2. x (b) 33 ull actoral desgn or both x x and.2. x Fgure 2. Desgn o experments to buld and test the least squares mult-delty surrogate (S-MFS) (c) test grds to evaluate predctons o x able 2. S-MFS based on 2 2 and 33 ull actoral desgns Model parameters or S-MFS 2 2 ull actoral desgn =3.4, near dscrepancy 33 ull actoral desgn =3.6, Quadratc dscrepancy Relatve RMSE (%) relatve error (%) absolute error 9.% 3.37%.2.52%.3%.26 able 3. PRS ttng or only hgh-delty samples based on 2 2 and 33 ull actoral desgns Order o PRS Relatve RMSE (%) relatve error (%) absolute error 2 2 ull actoral desgn near 43.% 92.23% ull actoral desgn Quadratc 26.% 56.2% 6.99 Acknowledgments hs work was supported by the U.S. Department o Energy, Natonal Nuclear Securty Admnstraton, Advanced Smulaton and Computng Program, as a Cooperatve Agreement under the Predctve Scence Academc Allance Program, under Contract No. DE-NA237 Reerences. Fernández-Godno, M. G., Park, C., Km, N.-., and atka, R.. "Revew o mult-delty models," arxv preprnt arxv:69.796, Peherstorer, B., Wllcox, K., and Gunzburger, M. "Survey o multdelty methods n uncertanty propagaton, nerence." and optmzaton. echncal Report R-6-. Aerospace Computatonal Desgn aboratory, Department o Aeronautcs and Astronautcs, Massachusetts Insttute o echnology, Cambrdge, MA, USA, Kennedy, M. C., and O'agan, A. "Bayesan calbraton o computer models," Journal o the Royal Statstcal Socety: Seres B (Statstcal Methodology) Vol. 63, No. 3, 2, pp

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