Estimation of effective models by machine learning

Transcription

1 International workshop on numerical methods and simulations for materials design and strongly correlated quantum RIKEN Estimation of effective models by machine learning NIMS Ryo Tamura Collaborator: U. Tokyo NIMS Koji Hukushima 25/Mar/27

2 Motivation Experimental results H. Kitazawa et al. / Physica BH Kitazawa (999) et al. / Physica B (999) Candidate models 89 bij (si sj )2 z 2 Jij si sj dij [si sj ] Di (si ) si sj (si rij )(sj rij ) rij rij Fig. 2. High-field magnetization curves for GdPd Al single " crystals at 4.2 K. Fig. 2. High-field magnetization curves for GdPd Al single " Fig.. Temperature dependences of (A) thecrystals specific heat at K. zero at 4.2 field and (B) the susceptibility at. T with field cooled process pendencesfrom of (A) specific at zero 4 the K for GdPd heat Al. The inset shows low-field magnetiz " at 5 K. tibility at ation. Tcurves with field cooled process l. The inset shows low-field magnetiz" input Model selection moment develops above very small fields along the cuniaxial-type magnetic properties of GdPd Al, we have " axis. M /H(¹) along the a-axis abruptly increases below introduced the 2D triangular lattice Heisenberg Hamil ¹ and shows a peak at 4.5 K. Below ¹, M (H) increases tonian with weak Ising-type anisotropy discussed by small fields along the c bove very uniaxial-type magneticmiyashita properties[6]. of GdPd Al, we have with increasing field and bends with a weak ferromagbecause observed features in several " the a-axis abruptly increases below introduced the 2D triangular lattice Heisenberg Hamil- with this model: netic moment at higher field than that along the c-axis. GdPd Al are qualitatively consistent "anisotropy discussed by at 4.5 K.When Below ¹, M (H) increases Ising-type ferromagnetic components M()tonian for eachwith axes weak are () the overall isothermal magnetization curve along the and bends with by a weak ferromag- from higher Miyashita observed features estimated the extrapolation fields to[6]. zero, Because c-axisseveral at 4.2 K; (2) the H ¹ phase in diagram along the () and Malong () at zero temperature yield ferromagnetic c-axis determined bywith tracing anomalies as a function of her field M than that the c-axis. GdPd Al are qualitatively consistent this model: " " moments.242 /Gd, respectively. temperatures and fields [7]; along (3) existence of two successcomponents M()offor each#/gd axesand are.95() the overall isothermal magnetization curve the # ¹ from and higher ¹ might be attributed to the ordering of spin iveh ¹ phase phase transitions and (4)along non-collinear spin structure apolation fields to zero, c-axis at 4.2 K; (2) the diagram the $ component along the c-axis and normal to the c-axis, with at least two different sites of Gd ions at zero field ero temperature yield ferromagnetic c-axis determined by tracing anomalies as a function of respectively. and low temperatures derived from the $''Gd Mo ss/gd and.95 /Gd, respectively. temperatures [7]; measurement (3) existence[3]. of two success- we propose that The isothermal high-field magnetization curves and at fields bauer To conclude, # attributed to the ordering of spin ive phase transitions and (4) non-collinear spin structure 4.2 K are displayed in Fig. 2. M increases linearly with GdPd Al is a new candidate of 2D triangular lattice " " e c-axis an and normal to up thetoc-axis, with least two different sites of GdThe ions-ray at zero field scattering or neuincrease of field H "6.2 T. A clearat/3 plateau antiferromagnets. magnetic "$ of the full moment 7 for Gd"# ions appears in the field tron experiments are desired to determine a real spin and low temperatures derived from the $''Gd Mo ss# range of H "6.2 T to H ".8 T along only the structure in GdPd Al. igh-field magnetization curves " at bauer measurement [3]. To conclude, we" propose that "$ M increases againwith above H GdPd and isal saturated n Fig. 2.c-axis. M increases linearly is a new candidate of 2D triangular lattice " " " the full moment amplitude of Gd"# ions" above with p to H "6.2 T. A clear /3 plateau antiferromagnets. The -ray magnetic scattering or neu"$ H"#"22.7 T. On the other hand, M along the a-axis References " ions appears in the field for Gd"# tron experiments are desired to determine a real spin and M* along the a*-axis [2 ] increase linearly with # T along only the T to H increasing ".8 in GdPd. A. Do nni, A. Furrer, H. Kitazawa, M. Zolliker, J. Phys.: fields and are saturatedstructure above H"# and Al"[] " again above H and is saturated Condens. Matter. 9 (997) 592. H *"#"25.5 T with the same amplitude as that along " [2] H. Kitazawa, A. Mori, S. Takano, T. Yamadaya, A. Matnt amplitude of Gd"# ionsthat above the c-axis. It is clear the c-axis is magnetically fasushita, T. Matsumoto, Physica B (993) 66. exchange e other vored hand,bymanisotropic along the a-axis or dipole dipole Referencesinterac[3] E. Colineau, J.P. Sanchez, J. Rebizant, J.M. Winand, Solid ordered state. tions in the *-axis [2 ] increase linearly with State Commun. 92 (994) 95. To our knowledge, only and a few compounds, e.g. hexag[] A. Do nni, A. Furrer,[4]H.S.Kitazawa, M.Phys. Zolliker, J. Phys.: d are saturated above H"# Miyashita, J. Soc. Jpn. 55 (986) 365. onal layered compounds C Eu [4], RbFe(Mo O ) and Condens. [5] T Sakakibara, M. Date, J. Phys. Soc. Jpn. 53 (984) the same amplitude as that along % & Matter. 9 (997) CsFe(SO ) [5] whose magnetic ions[2] form triangulara. Mori, S H.aKitazawa, Takano, T. Yamadaya, A. Mat&is magnetically fathat thelattice, c-axis show a ferrimagnetic plateau with sushita, /3 of the full [6] Physica T. Inami, Y. Ajiro, T. Goto, J. Phys. T. Matsumoto, B (993) 66. Soc. Jpn. 65 (996) exchange or dipole dipole interac- process. To account for moment in the magnetization [7] H. Kitazawa et al., in preparation. [3] E. Colineau, J.P. Sanchez, J. Rebizant, J.M. Winand, Solid state. State Commun. 92 (994) 95., only a few compounds, e.g. hexag[4] S. Miyashita, J. Phys. Soc. Jpn. 55 (986) 365. unds C Eu [4], RbFe(Mo O ) and [5] T. Sakakibara, M. Date, J. Phys. Soc. Jpn. 53 (984) % & se magnetic ions form a triangular magnetic plateau with /3 of the full [6] T. Inami, Y. Ajiro, T. Goto, J. Phys. Soc. Jpn. 65 (996) Machine learning output input L regularization L2 regularization Full search + Cross validation Plausible effective model for experimental results (selection of model parameters in candidate model) 2

3 As the first stage

4 As the first stage m ex (H) Machine learning H H = J ij s i s j b ij (s i s j ) 2 d ij [s i s j ] s i s j r 3 ij D i (s z i ) 2 3 (s i r ij )(s j r ij ) r 5 ij

5 Forward modeling and Bayes modeling P (m(h, x) x) P (m ex (H) m(h, x)) x = H = i,j J ij s i s j + m(h, x) m ex (H) P (x m ex (H)) = P (mex (H) x)p (x) P (m ex (H)) P (B A) BA

6 Thermal average - forward modeling Forward modeling P (m(h, x) x) x = {model parameters} H= i,j Jij si sj + P (mex (H) m(h, x)) magnetization m(h, x) observed magnetization mex (H) Definition of magnetization as thermal average of spins N H Trsi e hsi ih,x = m(h, x) = hsi ih,x H Tre N s i= Conditional probability of m(h, x) given x N P (m(h, x) x) = m(h, x) hsi ih,x N s i= Magnetization is uniquely obtained when the model parameters are given. 5

7 Observation noise - forward modeling Forward modeling P (mex (H) m(h, x)) P (m(h, x) x) x = {model parameters} H= i,j magnetization Jij si sj + m(h, x) observed magnetization mex (H) Existence of observation noise in mex (H) ex m (H) = m(h, x) + " Assumption P (") / exp observation noise ex " Conditional probability of m (H) given m(h, x) ex ex 2 P (m (H) m(h, x)) / exp (m (H) m(h, x)) 2 2 6

8 Conditional probability - forward modeling Forward modeling P (m(h, x) x) x = {model parameters} H= i,j Jij si sj + P (mex (H) m(h, x)) magnetization m(h, x) observed magnetization mex (H) Conditional probability of mex (H) given x Z P (mex (H) x) / dm(h, x)p (mex (H) m(h, x))p (m(h, x) x) N ex 4 5 / exp m (H) hs i i H,x 2 2 N s i= m (H) where P (m (H) x) is maximize. ex ex observed magnetization 7

9 Bayes modeling P (m(h, x) x) P (m ex (H) m(h, x)) x = H = i,j J ij s i s j + m(h, x) m ex (H) P (x m ex (H)) = P (mex (H) x)p (x) P (m ex (H)) x P (x m ex (H))

10 Prior distribution - Bayes modeling ex P (m (H) x)p (x) ex P (x m (H)) = P (mex (H)) Bayes modeling x = {model parameters} H= i,j P (x) magnetization Jij si sj + observed magnetization m(h, x) mex (H) : Prior distribution (prior knowledge about model parameters) (事前分布) If prior knowledge does not exist, P (x) / const. If x is sparse (number of model parameters is small), P (x) / exp K k= xk K amplitude of regularization (hyperparameter) number of model parameters 9

11 Posterior distribution - Bayes modeling Bayes modeling x = {model parameters} H= i,j ex P (m (H) x)p (x) ex P (x m (H)) = P (mex (H)) observed magnetization magnetization Jij si sj + mex (H) m(h, x) We assume that each magnetization is independently obtained in magnetization curve. L Y P (x mex (Hl )) Assumption P (x {mex (Hl )}l=,,l ) = l= Posterior distribution 2 P (x {mex (Hl )}l=,,l ) / exp 4 observed magnetization curve 2 L 2 l= mex (Hl ) N hsi ihl,x N s i= 2 K k= 3 xk 5

12 How to determine hyperparameter P (x) / exp K x k k= t M = x

13 How to determine hyperparameter P (x) / exp K x k k= t M =9 x Overfitting will be observed

14 How to determine hyperparameter P (x) / exp K x k k= t M =3 x

15 How to determine hyperparameter P (x) / exp K x k k= t M =3 x

16 Cross validation x ( ):= 4 L L/4 l = m ex l N s 2 N hs i i Hl,x i= ( )

17 Validation by theoretical model H = hi,ji J ij si s j b ij (s i s j ) 2 H i s z i b ij = bj ij s i x = {J,J 2,J 3,J 4,J 5,J 6,J 7,b} r = J : n =2 J 2 : n 2 =2 J 3 : n 3 = J 4 : n 4 = J 5 : n 5 =2 J 6 : n 6 = J 7 : n 7 = r 2 = r 3 = r 4 = r 5 = p 3 r 6 =2 r 7 =2

18 Inputted observed magnetization J =,J 2 =4,J 3 =5,J 4 =6,b=. J 5 = J 6 = J 7 = m ex (H) randomly divided into 4 groups H m ex (H) m ex (H) H H m ex (H) m ex (H) H H

19 Simulation methods E(x,,K)= 2 2 L l= m ex (H l ) N s 2 N hs i i Hl,x + i= K k= x k P (x {m ex (H l )} l=,,l ) / exp [ E(x,,K)] Boltzmann distribution (x x ) min {, exp[ (E(x,,K) E(x,,K))]}

20 Simulation methods P (x {m ex (H l )} l=,,l ) / exp apple T E(x,,K) min, exp apple (E(x i,,k) E(x j,,k)) T i T j

21 Estimated model parameters Success

22 Prediction errors av( ) =.4 Type I Type II Type III J J 2 J 3 J 4 J 5 J 6 J 7 b

23 Prediction errors

24 Effective model estimation method P (x {m ex (H l )} l=,,l ) / exp L l= m ex (H l ) N s N hs i i Hl,x i= 2 K k= 3 x k 5

25 Thank you R. Tamura and K. Hukushima, Phys. Rev. B 95, 6447 (27). 2