Optimization of direct extrusion process for Nd-Fe-B magnets using active learning assisted by machine learning and Bayesian optimization

Process optimization of permanent magnet is time-consuming as the microstructure that depends on alloy compositions and process parameters must be optimized to achieve high coercivity. Given a raw material of ﬁxed composition, the optimization of process involves the reﬁnement of grains size, the alignment of crystallographic orientation and the formation of intergranular phase. In this paper, we implemented an active learning pipeline assisted by machine learning and Bayesian optimization (ALMLBO) for predicting magnetic properties from process parameters and propose optimum conditions leading to high coercivity and remanence in Nd-Fe-B anisotropic magnets fabricated by direct hot extrusion. ALMLBO allowed us to optimize the process to exhibit high coercivity, μ 0 H c –1.7 T, and remanence, μ 0 B r –1.4 T, simultaneously, resulting in an excellent maximum energy product, (BH) max –380 kJ/m 3 . We show that an ALMLBO pipeline is an effective tool for optimizing process for Nd-Fe-B anisotropic magnets. © 2021 The Authors. Published by Elsevier Ltd on behalf of Acta Materialia Inc

Nd-Fe-B permanent magnets for electric vehicle traction motors and wind turbines gain satisfactory coercive force at their operating temperatures of ∼ 160 °C by partially substituting Nd with heavy rare earth elements (HRE) such as Dy and Tb in the Nd 2 Fe 14 B phase. Recently, there is a strong demand to achieve high coercivity ( μ 0 H c ) without alloying HREs because of their scarcity and high cost [ 1 , 2 ]. Several approaches have been proposed to increase μ 0 H c , e.g. the formation of (non-ferromagnetic) Nd-rich intergranular phases [3][4][5] , HRE grain boundary diffusion process [ 6 , 7 ] and grain refinement [8][9][10] have been demonstrated in sintered magnets. The grain refinement is effective to increase the coercivity as it leads to the reduction of the stray field from neighboring grains, and the temperature coefficient of coercivity is also improved [11] . Hot deformation of melt-spun ribbons induces a large anisotropic microstructure with platelet-shaped ultrafinegrained magnets [ 12 , 13 ]. Because the average grain size in the hotdeformed magnets is about several hundreds of nanometer, which is more than an order of magnitude smaller than in sintered magnets ( ∼ 3 μm), they are expected to exhibit superior coercivity and its thermal stability compared to the sintered magnets [14] . The * Corresponding author.
E-mail address: LAMBARD.Guillaume@nims.go.jp (G. Lambard). hot deformation process consists of a hot pressing for obtaining a consolidated bulk and following hot deformation process, such as die-upsetting [13] , backward extrusion [15] , and direct extrusion [14] , to develop a strongly textured grain structure. To date, most of the studies on hot-deformed magnets have been performed for die-upset process to realize excellent permanent magnet properties by optimizing the composition [16][17][18][19][20] and the process parameters [20][21][22][23][24] . Recently, large plate-shaped hot-deformed magnets fabricated by the direct extrusion process are adopted in traction motors for hybrid vehicles [25] . However, there are few reports on the evolution of microstructure and resulting magnetic properties on directly extruded magnets. As the size of the extruded magnets are relatively large compared to the die-upset magnets, various process parameters must be optimized to achieve good permanent magnet properties throughout the relatively large plates. In this paper, we adopted an active learning pipeline assisted by machine learning and Bayesian optimization (ALMLBO) to this end.
Active learning is a general framework which includes the "learning" step formed from past experiments and a set of actions resulting from the experimental feedbacks, the "active" step. Many learning-acting cycles form an active learning pipeline. One may note that a trial-and-error method follows an active learning scheme if an update in the theoretical/empirical knowledge and a forward action, modifying an experimental design to fulfill an  List of tunable process parameters θ i ∈ θ i , issued from θ = {HP T , HP L , HE T , HE RS , HE LL , Die ID }, with their accessible domain d i and resolution δd i chosen for the hot extrusion of Nd-Fe-B permanent magnets. HP T and HP L are the hot-press temperature and load, respectively. HE T , HE RS and HE LL are the hot extrusion temperature, ram speed, and load limit, respectively. Die ID is the identification number of a die of a given shape used at the extrusion exit. First, we defined a fixed protocol for building our dataset composed of the magnetic properties = { μ 0 H c , μ 0 B r ,Sq. } with their (BH) max , and tunable process parameters θ = {HP T , HP L , HE T , HE RS , HE LL , Die ID } of hot-extruded Nd-Fe-B magnets, where: HP T and HP L are the temperature and the load for hot pressing, respectively; HE T , HE RS and HE LL are the hot extrusion temperature, ram speed, and load limit, respectively; Die ID is the identification number of a die of a given shape used at the extrusion exit. We used commercial MQU-F TM melt-spun flakes with the nominal composition of Nd 14 Fe 76 Co 3.4 B 6 Ga 0.6 (at%). These flakes were consolidated into a bulk under uniaxial hot-press followed by a direct extrusion process to obtain an extruded magnet with a size of ∼30 mm in width, 5-7 mm in thickness, and ∼70 mm in length, at various hot pressing and extrusion conditions θ . Extrusions were then cut into cuboidal shapes with 6 mm in width, 6 mm in thickness, and 5-8 mm in height for measuring the magnetic properties , and (BH) max , using a B-H tracer (Tamakawa Co., Ltd, TM-BH25-C1) at room temperature. A microstructure analysis was performed by scanning electron microscope (SEM). The SEM analysis was performed using a Carl Zeiss Crossbeam 1540 EsB FIB/SEM. Specimens for the SEM analysis were prepared by a mechanical polishing followed by a surface cleaning by focused ion beam (FIB) to remove surface contamination.
The ALMLBO pipeline was kickstarted with only n = 18 samples  Table 1 . At this stage, more than 66 million combinations of θ i were experimentally possible. Then, a cycle of the ALMLBO pipeline, summarized in Fig. 1 (a), is conducted as follows: (i) a set ε RF of 3 ensembles of n RF Random Forests (RF) [26] regressors, with n RF = n available experi-mental samples and default hyper-parameters of the RF models assigned as in [27] , are trained to uniquely predict each property in i following a leave-one-out [28] cross-validation scheme, and using a mean-squared-error as the loss function to minimize during training. For conciseness, a comparison between predicted and observed values for the average (BH) max (kJ/m 3 ) only is shown here in Fig. 1 (b) before using ALMLBO (gray squares), and after 3 cycles of ALMLBO (blue, red, and green squares, respectively). For a detailed visualization of the individual prediction performance on the properties , Fig. S1 in the supplementary materials can be examined. Overall, a reasonable average root-mean-square-error (RMSE) ∼ 17.2 ± 13.8 kJ/m 3 on the (BH) max is achieved. Furthermore, salient parameters θ s = {HE T , HE LL , Die ID } for predicting properties were identified according to their variable importance as shown in Fig. 2 (a) [26] . Later on, parameters in θ s are kept for optimization, and the remaining parameters θ ࣬θ s are tuned to their most efficient values during experimental processing. This sole reduction of tunable process parameters from θ to θ s for optimization purpose induced a sharp decrease in experimentally possible θ i conditions to only 770, i.e. roughly 5 orders of magnitude lower than before the filtering of θ over saliency. Also, it is worth noting that θ s did not change with further data integration after each ALMLBO cycle; (ii) A set { θ s , }| r of 100 samples, with θ s,i ∈ d i (see Table 1 ) for the i th sample, is randomly evaluated through ε RF to initialize a following Gaussian process regressor at step (iii), and relax from any experimental biases and/or local optima that would have occurred during the preparation of the dataset issued from a previous cycle; (iii) A Gaussian process is trained on { θ s , }| r that serves as a surrogate function S to exploit and explore θ s ; (iv) A set θ s * of 10 samples that maximizes the expected improvement [29] , built over the product μ 0 H c × μ 0 B r × Sq. and chosen as acquisition function, is proposed for experimental feedback. Steps (ii), (iii) represents the Bayesian optimization part in the ALMLBO pipeline here; (iv) θ s * is experimentally evaluated and corresponding * properties are reported. If * are judged high enough, the ALMLBO pipeline is stopped, otherwise another cycle (i)-(iv) is started by adding newly acquired experimental data to the whole dataset. Fig. 2 (b) shows the evolution of (BH) max (kJ/m 3 ) successfully acquired with a B-H tracer before (gray squares) and during 3 cycles of ALMLBO (blue, red, and green squares, respectively). It is worth noting that, before ALMLBO pipeline was initiated, the improvement in (BH) max had reached a plateau which was hard to overcome without ALMLBO. This is essentially because of the number of involved process parameters θ , as well as their non-linear relationship to properties, that rendered difficult the possibility to intuit a θ -to-relationship without assistance from the ALMLBO pipeline.   [26] ) and observed average maximum energy product (BH) max (kJ/m 3 ) from a pre-, 1st, 2nd, and 3rd ALMLBO cycle (gray, blue, red, and green squares, respectively) with vertical and horizontal error bars representing a standard deviation related to a leave-one-out [28] cross-validation and computed from m experimental measurements, respectively (the dashed line shows a perfect prediction for the purpose of visual guidance only) (For interpretation of the references to color in this figure, the reader is referred to the web version of this article).   [ 19 , 20 , 23 , 30-39 ]. gray and colored squares (blue, red, and green) indicate data points acquired without and with ALMLBO, respectively; (b) Demagnetization curves for samples A (black) and B (red) indicated with arrows in Fig. 3 (a) Fig. 2 (a). Furthermore, the distributions of μ 0 H c , μ 0 B r , Sq ., and (BH) max as a function of the process parameters θ , illus- trated in Figs. S2-S5 of the supplementary materials, respectively, reveal the magnetic properties improvement as a non-linear control inference problem where the de-multiplication of data points at a fixed process parameter shows that magnetic properties are highly multi-parametric, and where the Die ID in sub-figures (f) and, more notably, the HE LL in sub-figures (e) exhibits multiple tendencies. Fig. 4 (a) shows a low-magnification backscattered electron (BSE) SEM image showing a typical microstructure of the center part of extruded magnet. The brightly imaging contrast located near the former flake boundaries are mainly neodymium oxides which were present on the surfaces of MQU-F powder. The flakes were elongated along the extrusion direction. Even though there is no significant difference in the low-magnification microstructure between samples A and B, higher magnification in-lens secondary electron SEM images show different features that might cause the difference in their magnetic properties in Fig. 4 .(b)-(e). Inside a flake, platelet-shaped ultra-fine grains with a low volume fraction of Nd-rich phases are present, which are darkly imaged for the Nd 2 Fe 14 B main phase, as seen in Fig. 4 (b), (c). First of all, there is a significant difference in the shape of the Nd 2 Fe 14 B grains. The grains in sample B are more elongated and finer than in sample A. The minor and major axes of ellipses delimiting segmented grains are about 90 ± 59 nm (72 nm) and 242 ± 155 nm (195 nm), respectively, for sample A, and 69 ± 172 nm (43 nm) and 554 ± 445 nm (423 nm), respectively, for sample B (characteristics of segmented grains are given as a mean value associated to a standard deviation, with a median value given between parenthesis, here and below). The aspect-ratio is about 2.9 ± 1.3 (2.6) for sample A, and 10.1 ± 5.4 (9.4) for sample B. Furthermore, the longitudinal axes of the Nd 2 Fe 14 B grains, which correspond to the normal axes to the easy axes of the Nd 2 Fe 14 B grains, are strongly aligned for sample B compared to sample A. The orientation with the hot extrusion axis is about 14.4 ± 46.1 °(-1.6 °) for sample A, and 3.9 ± 9.6 °(3.1 °) for sample B. Such strong alignment of the Nd 2 Fe 14 B grains led to the high μ 0 B r in sample B. are observed around the flake boundaries in sample A, and the ori-entation of ultra-fine Nd 2 Fe 14 B grains around those coarse grains is rather random ( Fig. 4 (d)). Unlike sample A, the microstructure of sample B consists of strongly aligned ultra-fine Nd 2 Fe 14 B grains around the flake boundaries while there is a small volume fraction of coarse grains ( Fig. 4 (e)). These Nd 2 Fe 14 B coarse grains might have caused the low coercivity as these grains are known to be preferential nucleation sites for the reverse domains in the demagnetization process [ 19 , 21 ]. Therefore, the optimization of salient process parameters θ s led to the development of preferential microstructures which consist of strongly aligned and ultra-fine Nd 2 Fe 14 B grains throughout the bulk for simultaneously achieving high remanence and coercivity, respectively. In this study, the relationship between a microstructure and its permanent magnetic quality in terms of coercivity and remanence is discussed rather qualitatively. However, if the quantitative microstructure-property relationship is later established by thoroughly analyzing a massive dataset of microstructures obtained in a series of extrusion experiments, as in Fig. 3 (a), through an extensive image analysis, the design of microstructures for extruded magnets with simultaneously improved coercivity and remanence could be further achieved.
In summary, this study has demonstrated the effectiveness of the ALMLBO in suggesting process parameters for fabricating hotextruded Nd-Fe-B magnets with excellent permanent magnetic properties from a dataset (18 hot-extruded initial samples) more than six orders of magnitude smaller than the total number of experimentally possible hot processing ( ∼66 million), and within a fairly short amount of time ( ∼9 months). The best hot-extruded magnet fabricated using MQU-F powder exhibited high coercivity, μ 0 H c ∼ 1.7 T, remanence, μ 0 B r ∼ 1.4 T, and squareness, Sq. ∼ 99%, resulting in a high maximum energy product (BH) max ∼ 380 kJ/m 3 due to the formation of strongly aligned and ultrafine grains structure throughout the bulk.

Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.