Relative cooling power modeling of lanthanum manganites using Gaussian process regression

Efficient solid-state refrigeration techniques at room temperature have drawn increasing attention due to their potential for improving energy efficiency of refrigeration, air-conditioning, and temperature-control systems without using harmful gas in conventional gas compression techniques. Recent developments of increased magnetocaloric effects and relative cooling power (RCP) in ferromagnetic lanthanum manganites show promising results of further developments in magnetic refrigeration devices. By incorporating chemical substitutions, oxygen content modifications, and various synthesis methods, these manganites experience lattice distortions from perovskite cubic structures to orthorhombic structures. Lattice distortions, revealed by changes in lattice parameters, have significant influences on adiabatic temperature changes and isothermal magnetic entropy changes, and thus RCP. Empirical results and previous models through thermodynamics and first-principles have shown that changes in lattice parameters correlate with those in RCP, but correlations are merely general tendencies and obviously not universal. In this work, the Gaussian process regression model is developed to find statistical correlations and predict RCP based on lattice parameters among lanthanum manganites. This modeling approach demonstrates a high degree of accuracy and stability, contributing to efficient and low-cost estimations of RCP and understandings of magnetic phase transformations and magnetocaloric effects in lanthanum manganites.


Introduction
Energy efficiency and sustainability are priority topics in modern society. Refrigeration and air conditioning account for a signicant amount of power consumption among various end uses of energy in both commercial and residential areas. 10 Most refrigeration technology relies on the conventional gas compression (CGC) technique, which has drawn increasing criticisms due to its lack of efficiency and use of air-pollutant gas. Recent developments of magnetic refrigeration (MR) technology, based on the magnetocaloric effect in magnetic materials particularly near room temperature, have offered an exciting alternative to vapor compression refrigeration. 20 Advantages of MR technology over CGC include, but not limited to, almost ten-fold higher cooling efficiency in magnetic refrigerators, much smaller footprints, complete solid-state operation, and being environmentally friendly. 21 Furthermore, recent developments in high-temperature superconductors with enhanced critical temperature and magnetic elds that can be generated have prompted developments of high-efficiency MR devices with superconducting magnetic eld sources. 8,22,[28][29][30] An early development of a gadolinium (Gd) rare earth metal with a large magnetocaloric effect (MCE) marked a signicant starting point in developing room-temperature MR, but its application in large-scale commercial usage was greatly limited due to the very high price of Gd. 9 Therefore, numerous research has been conducted to search for new materials with large MCEs, large relative cooling power (RCP), and cheap prices.
Among these materials, ferromagnetic lanthanum manganites, with the general formula, La 1ÀxÀy RE x A y Mn 1Àz TM z O 3 where RE is a rare earth element that partially or totally substitutes lanthanum, A is an element of the IA or IIA group, and TM is a transition element that partially substitutes Mn, are of practical importance. These materials have unique properties such as small magnetic and thermal hysteresis, a large MCE around Curie temperature T C , and a broad working temperature range. Furthermore, manganites are inexpensive to prepare, chemically stable, and highly electrically resistive. 6 The parent LaMnO 3 compound is semiconducting and orders antiferromagnetically at 150 K, but a formation of mixed valence in Mn ions via a double exchange mechanism between Mn 4+ and Mn 3+ can induce ferromagnetism. A wide range of T C from $150 K to 375 K can be obtained by, for example, substitution of a divalent ion (Ca 2+ , Ba 2+ , Sr 2+ , etc.) or a monovalent ion (Na 1+ , K 1+ , etc.) for La 3+ , and an excess of oxygen. Furthermore, the ground state of manganites can be tuned by partial substitution of La 3+ by a trivalent rare earth, or in a La-free Pr or Nd manganites. These perovskite-based structures show lattice distortions as a result of modications from the cubic structure by the deformation of the MnO 6 octahedron arising from the Jahn-Teller effect and/or changes in the connective pattern of the MnO 6 octahedra in the perovskite structure. 26,27 Values of T C , magnetic entropy changes DS m , adiabatic temperature changes DT ad , and the resultant RCP are strongly dependent on the doping mechanisms and thus lattice distortions. Qualitative analysis on the effect of dopant types and levels on RCP of lanthanum manganites has been conducted through experiments, mainly by varying synthesis methods (solid-state reaction, wet chemistry, sol-gel, etc.), morphologies (particle size, shape, etc.), crystalline states, and nal forms (powder, pellet, lm, etc.). 1,3,4,[11][12][13][14][16][17][18][23][24][25] Quantitative analysis through thermodynamics models and rst-principle models has been utilized to aid the understanding of magnetothermal responses of these materials and facilitate the searching of new candidates for MR devices. 2,5,7,15 However, these models require a signicant amount of data inputs, such as variables for equations of state, exchange coupling energies, and magnetic moments of magnetocaloric materials, which can only be obtained by extensive measurements.
In this work, the Gaussian process regression (GPR) model is developed to elucidate the statistical relationship between RCP and lattice parameters of orthorhombic lanthanum manganites. The model generalizes well in the presence of only a few descriptive features, where intelligent algorithms are able to learn and recognize the patterns. This modeling approach demonstrates a high degree of accuracy and stability, contributing to efficient and low-cost estimations of RCP and understandings of which based on lattice parameters. As one of the computational intelligence techniques, the GPR model has already been utilized in other materials systems to predict signicant physical parameters in different elds of applications. [31][32][33][34][35][36][37][38][39][40][41][42][43][44] On one hand, the model can serve as a guideline for searching for doped-manganites with a large RCP value by screening the lattice parameters. On the other hand, the model can be used as part of machine learning to aid the understanding of the magnetic phase transformation in various types of doped-manganites.
The remaining of this work is organized as follows. Section 2 proposes the GPR model. Section 3 describes the data and computational methodology. Section 4 presents and discusses results, and Section 5 concludes. Recall a linear regression model, y ¼ x T b + 3, where 3 $ N(0,s 2 ). A GPR aims at explaining y by introducing latent variables, l(x i ) where i ¼ 1, 2, ., n, from a Gaussian process such that the joint distribution of l(x i )'s is Gaussian, and explicit basis functions, b. The covariance function of l(x i )'s captures the
A GP is dened by the mean and covariance. Let m(x) ¼ E(l(x)) be the mean function and k(x,x 0 ) ¼ Cov[l(x),l(x 0 )] the covariance function, and consider now the GPR model, y ¼ b(x) T b + l(x), where l(x) $ GP(0,k(x,x 0 )) and . k(x,x 0 ) is oen parameterized by the hyperparameter, q, and thus might be written as k(x,x 0 |q). In general, different algorithms estimate b, s 2 , and q for model training and would allow specications of b and k, as well as initial values for parameters.  The current study explores four kernel functions, namely exponential, squared exponential, Matern 5/2, and rational quadratic, whose specications are listed in eqn (1)-(4), respectively, where s l is the characteristic length scale dening how far apart x's can be for y's to become uncorrelated, s f is the signal standard deviation, r ¼ , and a is a positive-valued scale-mixture parameter. Note that s l and s f should be positive. This could be enforced through q such that q 1 ¼ log s l and q 2 ¼ log s f .
Similarly, three basis functions are investigated here, namely constant, linear, and pure quadratic, whose specications are listed in eqn (5)-(7), respectively, where To estimate the GPR model, a Bayesian optimization algorithm is utilized. With a Gaussian process model of f(x), the algorithm evaluates y i ¼ f(x i ) for N s points x i taken at random within the variable bounds, where N s points stand for the number of initial evaluation points and 4 is used. If there are evaluation errors, it takes more random points until N s successful evaluations are arrived-at. The algorithm then repeats the following two steps: (1) updating the Gaussian process model of f(x) to obtain a posterior distribution over functions Q(f|x i ,y i for i ¼ 1, . ,n); (2) nding the new point x that maximizes the acquisition function a(x). It stops aer reaching 30 iterations. The acquisition function, a(x), evaluates the goodness of a point, x, based on the posterior distribution function, Q. This work employs the lower-condence-bound (LCB) acquisition function, which looks at the curve G two standard deviations, s Q , below the posterior mean, m Q , at each point: G(x) ¼ m Q (x) À 2s Q (x). Therefore, G(x) is the 2s Q lower condence envelope of the objective function model. The algorithm then maximizes the negative of G: The optimization is carried out on s, the noise standard deviation. q and b are estimated by maximizing the log likelihood function.

Performance evaluation
Performance of the proposed GPR models is evaluated using the root mean square error (RMSE), mean absolute error (MAE), and correlation coefficient (CC) in eqn (8), (9), and (10) respectively, where n is the number of data points, T exp i and T est i are the i-th (i ¼ 1, 2, ., n) experimental and estimated magnetic cooling power, and T exp and T est are their averages.
3 Empirical study

Description of dataset
The experimental data used, shown in  Fig. 1 reveals nonlinear relationships, which are modeled through the GPR.

Computational methodology
MATLAB is utilized for computations and simulations in this work. All observations are used to train the nal GPR model given the relative small sample size. The stability of the GPR approach is conrmed by bootstrap analysis.

Comparison with previous study
The nal GPR model is detailed in Fig. 2, whose performance is compared with that based on the SVM regression in ref. 19 Fig. 2 shows good alignment between GPR predicted and experimental data.

Prediction stability
Given the small sample size (see Table 1) used, the prediction stability of the GPR is assessed through bootstrap analysis in Fig. 3, which shows that the modeling approach maintains high CCs, low RMSEs, and low MAEs over the bootstrap samples. This result suggests that the GPR might be generalized for magnetic cooling power modeling of manganite materials based on larger samples. Table 2 shows that GPR predictions are not so sensitive to choices of kernels or basis functions. Because predictions based on different kernel-basis function pairs are so close and nearly visually indistinguishable, they are not plotted for comparisons. However, it is worth noting that estimated model parameters are different across these kernel-basis function pairs.

Conclusions
The Gaussian process regression (GPR) model is developed to predict relative cooling power of manganite materials based on lattice parameters. The high correlation coefficient between the predicted and experimental magnetic cooling power, the low prediction root mean square error and mean absolute error, and stable model performance suggest the usefulness of the GPR for modeling and understanding the relationship between lattice parameters and relative cooling power. This modeling approach is straightforward and simple and requires less parameters as compared to thermodynamics models and rst-principle models. It can be used as part of computational intelligence approaches for new magnetocaloric materials searches.

Conflicts of interest
There are no conicts to declare.