With the rise of deep learning, physics informed neural networks [1] became an alternative to finite element simulation in many fields of materials and device design. Physics informed neural networks bring several advantages as compared to traditional numerical methods for solving partial differential equations like finite differences or finite elements: 1. There is no need for mesh generation. 2. Inverse problems may be solved effectively. 3. A whole family of problems may be solved with a single neural network. We demonstrate how physics informed neural networks can be applied in magnetostatics and micromagnetics. The approach is like the Ritz method for computing the magnetic field or the magnetization. However, instead of finite element basis functions [2], we use dense neural networks to approximate the unknowns. The underlying physics is incorporated through the loss function which is minimized during training of the neural network. We first applied physics informed neural network to solve a classical inverse problem in magnetism. We searched for the magnetization in a hollow cylinder that produces a uniform field in the hole and has zero field outside. The mean relative error between the neural network estimate and the analytic solution for the magnetization was 0.27 percent. We computed the demagnetization curve of a Nd2Fe14B particle. The difference between the finite element solution and the neural network estimate was 0.9 percent. We computed the lowest eigenvalue of Brown's linearized micromagnetic euqation to get the nucleation field for a soft magnetic defect in a hard magnetic matrix. The mean realtive error for the nucleation field of an Fe inclusion in Sm2Fe17N3 as function of defect size was 0.4 percent as compared to the analytic solution [3]. Because of the ability to solve multiple problems simulateneously, physics informed neural networks have a great potential for optimization and tailoring the magnet's internal structure to achieve a specific demagnetiation curve. The financial support by the Austrian Federal Ministry for Digital and Economic Affairs, the National Foundation for Research, Technology and Development and the Christian Doppler Research Association is gratefully acknowledged.

Physics informed neural networks for computational magnetism

First Order Reversal Curve (FORC) diagrams have over 30 years of history as a tool to characterise ensembles of magnetic particles reaching from applications in geophysics to detect the prevalence of carriers of paleomagnetic signals [1] to the measurement of magnetic recording media properties [2]. Using FORC diagrams as a purely quantitative tool remains an open topic of active research (see e.g. [3]), but they have proven to be useful as semi-quantitative instrument to determine traces of magnetic phases in a material mixture [4]. Based on this present understanding we introduce the idea of using FORC diagrams to detect the crystalline structure of the cobalt binder phase in tungsten carbide (WC-Co), a hard metal used in a wide range of high-tech applications (e.g. drills or cutting inserts) [5]. The goal is to interpret FORC diagrams from experimental measurements by micromagnetic simulations as a source of “big data” for a fast characterisation of tungsten carbide binders. Our proposed method goes significantly beyond state-of-the art techniques used for characterisation of WC-Co in particular, but also makes a contribution to the interpretability of FORC diagrams in general by applying machine learning to the large amount of data points available through FORCs. Here, we present our first results towards such an automated interpretation of FORC diagrams, which requires a substantial amount of training data from micromagnetic simulations. Such simulations provide the opportunity to assemble mixtures of particles with a known distribution of various properties, which should serve as labelled training data. We use energy minimization [6] to compute the magnetization curves of Co particles with sizes of 100 nm and uniaxial anisotropy. A reversal curve is computed starting at each point Hr on the major loop where the magnetization M has changed by a defined amount. Depending on the alignment of the anisotropy axes with respect to the field H different numbers and configurations of domain wall nucleation-propagation-annihilation sequences occur during a full reversal loop, leading to a large variation in the number of generated FORCs. To mitigate the necessary computational efforts to acquire the training data of large ensembles of more than 100 particles we developed a Python-based FORC analysis framework and used scikit-learn’s Random Forest Regressor [7] to predict the surface M(H,Hr) from a subset of simulations. Our first results for an ensemble of non-interacting cubes with varying anisotropy axes show that the FORC diagrams calculated from the predicted M(H,Hr) surface give a valid representation of the FORC diagram calculated directly from the fully simulated ensemble (Fig.1). This approach will enable us to speed up the generation of training data considerably, which provides the basis to understand the microstructure of WC-Co samples containing Co particles with dimensions beyond the 100 nm scale by using FORCs as a fast and non-destructive characterisation tool. Fig. 2 shows two examples of simulated FORCs and their respective FORC diagrams. In Fig. 2a the external field is aligned more parallel to the particle’s easy axis than in Fig. 2b. As FORC diagrams only show pronounced peaks for irreversible processes, they can be directly associated with the jumps present in the FORCs shown in the upper part. Analytical models to explain the peaks’ positions and heights can only be found for simplified models. Following [8], the two peaks in Fig. 2a can be modelled analytically using a curvilinear hysteresis loop, but this model does not reveal the exact reversal process in the material. Combining micromagnetic simulations with machine learning will enable us to train models using realistic assumptions about the material of interest. As a first step the presented approach will make it possible to get data for the interpretation of FORC diagrams within minutes instead of weeks by replacing time-consuming micromagnetic simulations with sufficiently trained machine learning models. The financial support by the Austrian Federal Ministry of Climate Action, Environment, Energy, Mobility, Innovation and Technology (BMK) in the KI-Carbide project (#877141) is gratefully acknowledged.

**References:**

[1] A. R. Muxworthy and D. J. Dunlop: Earth and Planet. Sci. Lett. 203, pp 369-382 (2002) [2] M. Winklhofer and G. T. Zimanyi, J. Appl. Phys. 99, 08E710 (2006) [3] D. A. Gilbert, G.T. Zimanyi, R.K. Dumas et al., Sci. Rep. 4, 4204 (2014) [4] C. Carvallo et al.: Phys. Earth Planet. Inter. 154, pp 308–322 (2006) [5] J. García et al., Int. J. Refr. Met. 80, pp 40-68 (2019) [6] L. Exl, J. Fischbacher, A. Kovacs et al., Comp. Phys. Comm. 235, pp 179-186 (2019) [7] F. Pedregosa et al, JMLR 12, pp 2825-2830 (2011) [8] C. Pike and A. Fernandez, J. Appl. Phys. 85, pp 6668-6676 (1999)

**Images:**

![](https://assets.underline.io/uploads/markdown_image/1/image/9de8549ddbc61514a201eecc9b7f0b4d.jpg)
Fig. 1 FORC diagrams computed (a) directly from simulating a full ensemble of 125 non-interacting hcp-Co particles with varying alignment of the anisotropy axis with respect to the external field and (b) by using only a subset of 20 simulations as input to a Random Forest Regressor to predict the M(H, Hr) surface

![](https://assets.underline.io/uploads/markdown_image/1/image/2c361c16a150eeb7f7d6915c3daf9b09.png)
Fig. 2 Examples of simulated FORCs and calculated FORC diagrams for a 100 nm cube of hcp Co (uniaxial anisotropy). Colorbars indicate the peak height of the FORC distribution (see also Figure 1). The external field in (a) is aligned more parallel to the easy axis than in (b). The visible peaks in the FORC diagrams (bottom) correspond to irreversible jumps of the magnetization (top).

Thomas Schrefl

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