Physics-Informed Neural Networks for Magnetostatic Problems on Axisymmetric Transformer Geometries

Physics-informed neural networks (PINNs) have shown their potential for solving direct problems in many engineering domains including electromagnetics. We employ fully connected and convolutional PINNs in order to predict the magnetic vector potential and resulting inductances and coupling for axisymmetric transformer geometries commonly used in inductive power systems. Both approaches are compared quantitatively and validated against reference solutions obtained from a numerical solver. Fully connected PINNs tend to train faster and more accurately for single geometries, whereas convolutional PINNs show an outperformance in terms of their generalization capability and are able to predict inductances and coupling for a wide range of transformer geometries accurately in a matter of milliseconds. The combination of high accuracy and fast inference paves the way for PINN-based topology optimization in the field of power electronics.


Physics-Informed Neural Networks for Magnetostatic Problems on Axisymmetric Transformer Geometries
Philipp Brendel , Vlad Medvedev , and Andreas Rosskopf Abstract-Physics-informed neural networks (PINNs) have shown their potential for solving direct problems in many engineering domains including electromagnetics.We employ fully connected and convolutional PINNs in order to predict the magnetic vector potential and resulting inductances and coupling for axisymmetric transformer geometries commonly used in inductive power systems.Both approaches are compared quantitatively and validated against reference solutions obtained from a numerical solver.Fully connected PINNs tend to train faster and more accurately for single geometries, whereas convolutional PINNs show an outperformance in terms of their generalization capability and are able to predict inductances and coupling for a wide range of transformer geometries accurately in a matter of milliseconds.The combination of high accuracy and fast inference paves the way for PINN-based topology optimization in the field of power electronics.

I. INTRODUCTION
S IMULATION has become an indispensable tool in modern engineering, revolutionizing the way products are designed, tested, and optimized.Resulting computer-based models replicate real-world systems and allow engineers to analyze their behavior under different conditions.By utilizing mathematical algorithms and computational power, simulations enable engineers to approximate complex phenomena without having to conduct challenging or expensive physical experiments.In the field of power electronics, the finite difference method, finite-element method (FEM), and boundary element method are the most established approaches for simulating and designing electronic components and systems.Even though algorithmic, as well as hardware performance have continually improved over the last decades, there are still several limitations.The optimization of power electronic systems and components can be very time consuming and manufacturing constraints are hard to implement.Moreover, simulation models and measured data are hard to combine in order to describe systems accurately over an entire life cycle-e.g., in the context of a digital twin.The authors are with the Fraunhofer Institute for Integrated Systems and Device Technology IISB, 91058 Erlangen, Germany (e-mail: philipp.brendel@iisb.fraunhofer.de;vlad.medvedev@iisb.fraunhofer.de;andreas.rosskopf@iisb.fraunhofer.de).
Color versions of one or more figures in this article are available at https://doi.org/10.1109/JESTIE.2023.3346798.
Digital Object Identifier 10.1109/JESTIE.2023.3346798 Due to these limitations, data-driven machine learning (ML) methods have already been employed in several engineering domains [1], [2], [3].The approach demonstrated in [4] uses neural networks (NNs) and supervised learning to describe the correlation of coil geometries and the corresponding power distributions and magnetic fields in induction hardening based on simulation data.However, such data-driven NNs can easily reach their limits when large amounts of computationally expensive training data are required to obtain sufficiently accurate predictions.
Although the first attempts of combining supervised NNs and physics date back to the end of the previous century [5], [6], this methodology was only brought to wider attention by Raissi et al. [7] in 2019, who coined the term "physics-informed NNs" (PINNs) and demonstrated their potential for solving partial differential equations (PDEs).PINNs combine the advantages of both ML-based and numerical methods to accomplish fast inference times of physical quantities as well as explicit supervision by the governing physical laws during the training process.
In the field of electromagnetics, PINNs have already been applied to predict the magnetic flux distribution based on a given magnetization [8], and in a hybrid setup combined with data for simple electro-and magnetostatic problems [9].More recent results demonstrate their potential for the design of a single coil or choke [10], or coil systems used for induction heating and hardening [11], [12].The aforementioned examples use fully connected NNs (FCNNs) as basic architecture to approximate the physics and allow only limited degrees of freedom in their geometric setups.Due to their generally higher computational costs [13], convolutional NNs (CNNs) are less frequently used in the context of PINNs but have also shown their generalization potentials, e.g., for microlens design in optics [14], [15].
To bridge the gaps between these two fundamentally different approaches to PINNs, we compare and benchmark the capabilities of fully connected and convolutional PINNs (ConvPINN) for the prediction of magnetic vector potentials in axisymmetric transformer geometries, commonly used in inductive power transfer systems [16].We demonstrate that the less common convolutional PINN approach can push the limits in terms of generalization and speed to predict relevant quantities such as inductances and coupling for a wide range of transformer geometries.
The rest of this article is organized as follows.In Section II, the axisymmetric transformer geometry, its variations, and the underlying physical model are introduced.Section III describes the two different PINN methodologies, which are compared and benchmarked in terms of their prediction accuracy for power electronic metrics in Section IV.Finally, Section V concludes this article.

II. MODEL
The design of inductive components such as coils, chokes, or transformers is commonly based on standard components for windings and ferrites.However, in the domain of automotive and especially aerospace customized components and arrangements with much more degrees of freedom (DoF) are commonly used to optimize the efficiency, construction space, or costs of the entire system.To demonstrate the potential of PINNs in such a process, we introduce a generalized axisymmetric (2.5-D) transformer model and summarize the underlying physical laws in this section.

A. Geometric Model
We consider a rectangular, axisymmetric domain Ω = [0 cm, 4 cm] × [0 cm, 7 cm] where the vertical z-axis represents the rotational axis of symmetry, cf.Fig. 1.The ferrite material within Ω is arranged in two L-shaped substructures F1 and F2 that each contain two rectangles connected at the lower-bottom and upper-right parts of the geometry, respectively.From a design optimization perspective, a common problem is to find the optimal geometry among a set of many feasible setups.Therefore, we introduce the following geometric variations and notations for further reference throughout this work, cf.Fig. 1.
2) Geo2a (6 DoF): Generalized geometry with variation of position, width, and height of two coils (red and green) within the two L-shaped ferrite pieces.Cross-sectional coil areas remain constant.3) Geo2b (18 DoF): Generalized geometry with coil variation as in Geo2a and additional variation of widths and heights on each of the four rectangles F1a, F1b, F2a, and F2b.Furthermore, the positions of substructures F1 and F2 are varied via the coordinates of the bottom-left and top-right corner of the respective L-shaped cross-sections (indicated by asterisks in Fig. 1).Table III describes the varied parameters and their feasible ranges in more detail.

B. Physical Model
In order to extract relevant engineering quantities such as inductances and coupling from a given geometric model, the quasi-static Maxwell equation needs to be solved in an axisymmetric manner.The general quasi-static derivation from Maxwell's equations with the introduction of the magnetic vector potential and electric scalar potential is well established and elaborated, e.g., in [17] and [18].The resulting formulation for the magnetic vector potential A with a given current density J and permeability μ is given as For the materials present in our problem, permeabilities are given as μ coil = μ air = 1.25 × 10 −6 H/m and μ ferrite = 2000 • μ air .The excitation is assumed to be unilateral in the ϕ-direction and due to a current of 45 A on a fixed cross-sectional coil area a coil = 0.36 cm 2 , which yields J ϕ (r, z) = 1.25 × 10 6 A/m 2 within the excited coil.In order to transform (1) to cylindrical coordinates (r, ϕ, z), the curl operator is adapted following the basic principles of calculus [19].Due to the unilateral excitation in the ϕ-direction, the vector potential only has a ϕ-component A ϕ as well, cf.[20], and is described by the axisymmetric formulation Homogeneous Dirichlet conditions on the boundary ensure that there is no magnetic flux across the boundary ∂Ω, cf.[18], [21], so we set ( In domains with large permeability jumps at material interfaces, the strong solution of ( 2) may be of bad regularity.To overcome this potential issue, Beltrán-Pulido et al. [10] propose the minimization of the integral on the coupling field coenergy instead, which is equivalent to solving the weak formulation of ( 2) and (3).We obtain for piecewise constant permeabilities μ and refer to this equivalently as "weak formulation" in the following.The transformation to cylindrical coordinates (r, ϕ, z) with the aforementioned assumptions follows analogously with an additional multiplicative factor r due to the formulation of volume integrals in cylindrical coordinates as: Due to the different orders of magnitudes present in the problem, a nondimensionalization scheme similar to the one used in [10] is applied to rescale variables and constants to numerically similar ranges via Both coordinates are scaled equally by r * = z * = 0.07 m to obtain the nondimensionalized domain Ω = [0, 4  7 ] × [0, 1].The constant on the magnetic vector potential is empirically set to A * = 1.0 mWb/m according to the expected order of magnitude in our experiments and μ * = 1.25 × 10 −4 H/m to scale the nondimensionalized permeabilities of air and ferrite to 0.01 and 20, respectively.By setting 2) and (5) can identically be described in terms of the nondimensionalized variables and constants.In the following, we omit the overlines and treat all quantities as nondimensionalized.
With the magnetic vector potential being described by ( 2) or (5), self-inductances L 11 and L 22 are due to excitations in coils C1 and C2, respectively.They can be calculated by integrating the resulting vector potentials A 1 or A 2 over the corresponding coil cross-section [22], [23].The coupling k between two coils is Fig. 2. FC-PINN for prediction of the magnetic vector potential A ϕ (r, z) based on continuous input coordinates (r, z).The generalized version (bottom left) uses additional geometric input parameters (λ 1 , λ 2 , . .., λ n ) to predict A ϕ (r, z, λ) for parametrized geometries.
given via the symmetric mutual inductance . Consequently, the quasi-static Maxwell equation needs to be solved twice for each geometric setup with different current densities J ϕ in order to compute all inductances and coupling between the coils.

III. PHYSICS-INFORMED NNS
In this section, we describe the fully connected and convolutional approaches to PINNs and demonstrate their application for the transformer model introduced in the previous section.

A. Fully Connected PINN (FC-PINN)
Following the original approach by Raissi et al. [7], we apply an FCNN to approximate the solution A ϕ (r, z) for the quasistatic Maxwell equation-in either weak or strong form-based on inputs (r, z) ∈ Ω.In the following, we refer to this as fully connected PINN (FC-PINN), cf.Fig. 2.
The FC-PINN is trained by sampling N Ω points (r i , z i ) ∈ Ω according to a quasi-random distribution based on Hammersley sets [25], which have empirically been shown to be beneficial for nonadaptive sampling in PINNs [26].The training points are used as input to an FCNN whose output is trained via a physicsdriven loss function L FC pde .The choice of L FC pde determines whether FC-PINN is trained to predict a strong solution for (2) ("s") or to predict a minimum for the weak formulation in (5) ("w").The strong solution is obtained by minimizing the mean-squared error (MSE) of the residual of (2) for each with The minimization of ( 5) can be posed as a stochastic optimization problem within FC-PINN in the sense that the integrand of ( 5) is minimized on the set of training points (r i , z i ), cf.[10].Consequently, we obtain the following loss function for the weak Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply. solution: As the coupling field coenergy in (8) may yield negative values, we compute the mean error (ME) instead of the MSE as the loss metric in (9).The spatial gradients in ( 6) and ( 8) are calculated by means of automatic differentiation [27], a fast and commonly used differentiation technique in ML that approximates the gradients of the output A ϕ ) with respect to the inputs (r, z).
The homogeneous Dirichlet conditions in (3) can be enforced by hard constraints as introduced in [24].The main idea is to apply a transformation on the output A NN in the final layer, such that all (r, z) ∈ ∂Ω fulfill the boundary condition by design.For the nondimensionalized domain Ω = [0, 4  7 ] × [0, 1] this is achieved via To deal with the singularities at r i = 0 in ( 6) and ( 8), we enforce the loss to be zero for r = 0 while A ϕ (0, z) = 0 is guaranteed by the hard constraint in (10).
In order to solve the generalized geometric setups Geo2a and Geo2b with FC-PINN, an additional input vector λ = (λ 1 , λ 2 , . .., λ n ) is introduced, which contains a set of continuous design parameters such as width, height, and location of ferrite and coil material, cf.bottom left in Fig. 2. In this setup, permeability μ(r, z, λ) and current density J(r, z, λ) also depend on the parametrization vector λ, so the FC-PINN is trained to capture the physical meaning of a set of continuous input variables while minimizing the PDE residual for each coordinate pair (r, z) with μ and J depending on λ.When training the generalized FC-PINN, N Ω tuples (r, z, λ) are sampled from the aforementioned Hammersley distribution within a higherdimensional hypercube, whose dimensionality depends on the considered degrees of freedom, cf.Section IV.More details on the parameters described by λ as well as their varied parameter ranges are found in the Appendix.
Both ungeneralized and generalized FC-PINNs are implemented in Python using the framework DeepXDE by Lu et al. [28].

B. Convolutional PINN
As a second approach to PINNs, we adapt the framework by Lim and Psaltis [14] to solve the quasi-static Maxwell equation with a CNN architecture originally intended for image-to-image regression tasks [29].However, in [14], it is employed to predict the E-field as a physical quantity of interest based on a microlens shape that is given as binary input image.A discretization scheme based on the commonly used Yee [30] grid is applied to approximate the spatial derivatives that are part of the PDE formulation.The PDE residual is used as a physics-driven loss function and applied pixelwise on the output of the CNN.
Fig. 3 shows our adaption, which we refer to as ConvPINN in the following.ConvPINN uses two binary input images representing the geometry of ferrite material and the excited coil, respectively.The pixel resolution N r × N z is defined uniformly in both directions with pixel sizes Δ r = Δ z .Pixels are indexed starting from the bottom left of the domain, so ., N r and j ∈ 1, . .., N z .The finite difference scheme described in [14] is applied on the output image to approximate the spatial derivatives and we refer to Section B in the Appendix for a more detailed description on that.In combination with the binary input images defining μ(r i , z j ) and J ϕ (r i , z j ), the pixel-based PDE residual is composed as given by ( 6) and (8), respectively.In total, we obtain the following loss formulations for prediction of strong and weak solutions to the quasi-static Maxwell equation with ConvPINN: The homogeneous Dirichlet boundary conditions are enforced by explicitly setting the boundary pixels to zero after the final convolution layer.
As we assume equal permeabilities for copper and the surrounding air, the position of the nonexcited coil does not affect the magnetic vector potential resulting from the excited coil.Therefore, only the excited coil is used as explicit secondchannel input in ConvPINN.Without this assumption, the nonexcited coil could additionally be passed as a third-input channel to distinguish it from air, but this approach is beyond the scope of this work.
By design, ConvPINN complements the solution of generalized geometric setups, such as Geo2a and Geo2b from Section II as different ferrite and coil geometries can be passed as inputs during training and no additional input (like λ in FC-PINN) is required.Consequently, the ConvPINN is trained to interpolate Altogether, the main differences in the two presented PINN approaches can be summarized as follows: FC-PINN receives a continuous coordinate tuple (r, z) as input and predicts the corresponding vector potential A ϕ (r, z) in a pointwise manner, whereas ConvPINN receives pixel-discrete input images representing the material and current density distribution and predicts the magnetic vector potential A ϕ (r i , z j ) in a pixel-discrete manner for i ∈ 1, .., N r and j ∈ 1, . .., N z .While both architectures minimize the same physics-driven loss functions, FC-PINN approximates the spatial derivatives via automatic differentiation, whereas ConvPINN relies on a pixel-based discretization scheme to compute the derivatives on the output image.
See Table IV for the hyperparameters used for training both FC-PINN and ConvPINN.

IV. RESULTS
In this section, we evaluate both FC-PINN and ConvPINN on the different geometric setups introduced in Section II.Predictions of PINNs are quantitatively compared with a numerical reference solution obtained via the open-source FEM software FEMM [31].The mean absolute error (MAE) on the predicted magnetic vector potentials A PINN as well as percentage errors (PE) on resulting self-inductances L PINN  11 , L PINN  22 , and couplings k PINN are computed to obtain both global MAE and applicationspecific metrics for the prediction accuracy of FC-PINN and ConvPINN Training and evaluation of FC-PINN and ConvPINN is executed on an NVIDIA A100 GPU with 80 GB memory and CUDA 12.0.Reference solutions from FEMM are obtained with the corresponding Python interface pyFEMM on an Intel Xeon E5-2667 CPU with 128 GB RAM.

A. Geo1 (Fixed)
In order to investigate the performance of both FC-PINN and ConvPINN on the weak and strong formulations introduced in Section II, Table I  To analyze the convergence behavior of each PINN approach, Fig. 4 shows the evolution of ΔL PE  11 during training on Geo1 with μ rel = 2000 for different amounts of domain points and pixels, respectively.The FC-PINN trained with N Ω = 25 000 points reaches an accuracy of ΔL PE 11 < 0.2% very fast after around 20 s but diverges (also known as overfitting in ML) shortly after to an invalid albeit loss-minimizing solution.The corresponding trial with N Ω = 100 000 converges slower but reaches a more accurate solution at ΔL PE 11 ≈ 0.01%, eventually.The higher amount of domain points seems to stabilize the training on the PDE residuals although some form of divergence from the ground truth can also be observed in this case after around 120 s.The convergence of ConvPINN seems more stable in general and Authorized licensed use limited to the terms of the applicable license agreement with IEEE.Restrictions apply.shows a clear relation between pixel resolution and achievable accuracy on this geometry.This makes sense from a physical point of view as the coarser pixel grid may prevent a sufficient resolution of the spatial derivatives at material interfaces.
We conclude that the weak formulation is fundamental for solving the quasi-static Maxwell equation with either FC-PINN or ConvPINN and use the corresponding loss function exclusively in the following.Furthermore, convergence speed and accuracy of both approaches depend heavily on a sufficient amount of domain points and pixels, respectively, so we consider N Ω = 100 000 and N r × N z = 512 × 896 as default settings going forward.

B. Geo2a (6 DoF)
In order to evaluate FC-PINN and ConvPINN with respect to their generalization capabilities, we compare predictions on inductances and coupling for randomly generated samples according to geometry variation Geo2a, Section II and Table III.To this end, several PINN trainings were conducted with varying network sizes, learning rates, and activation functions as main hyperparameters of interest.Both FC-PINN and ConvPINN are given an equal amount of training time along which the prediction accuracy is repeatedly evaluated based on a small validation dataset of 10 randomized samples.The best validation result in terms of the MAE is selected and evaluated on a larger benchmark dataset of 100 samples.Fig. 5 shows the corresponding predictions for FC-PINN and ConvPINN with reference inductances and couplings (via FEMM) ranging from 0.3 to 0.4 μH and 60% to 95%, respectively.FC-PINN-indicated by orange circles-yields insufficient average errors on this rather basic generalization task with avg(ΔL PE 11 ) = 7.71%, avg(ΔL PE 22 ) = 9.04%, and avg(Δk PE ) = 10.75%.On the other hand, ConvPINN-indicated by blue stars-shows significantly higher accuracy with average errors of avg(ΔL PE 11 ) = 0.78%, avg(ΔL PE  22 ) = 0.77%, and avg(Δk PE ) = 0.56%.We conclude that ConvPINN is the superior approach in terms of predicting the magnetic vector potential on a range of different transformer geometries and test its limits on the more complex generalization task described by Geo2b next.

C. Geo2b (18 DoF)
We present the results on use case Geo2b solely for ConvPINN using the weak loss formulation in (12) and refer to Section II and Table III for more details on the underlying geometric variations.As described before, we consider the best training result among several trials based on the MAE over 10 validation samples and evaluate it on 100 randomized benchmark samples.In this setup, a wider range of values is covered for both inductances (0.05 − 0.3 μH) and couplings (25% − 90%) and we plot the predictions of ConvPINN against the reference solutions from FEMM in Fig. 6.
Despite the increased amount of DoFs with respect to Geo2a, ConvPINN reaches a comparable degree of accuracy with average errors of avg(ΔL PE 11 ) = 0.84%, avg(ΔL PE 22 ) = 1.02%, and avg(Δk PE ) = 0.81%.Fig. 7 shows the geometry with the highest observable PE (ΔL PE 11 = 3.95%), as well as the corresponding predictions of the magnetic vector potential in both excitation cases.None of the 100 benchmark samples results in an error above the 4% threshold in any of the metrics ΔL PE 11 , ΔL PE 22 , or Δk PE , cf.Fig. 8.
It is notable throughout our results that thin coil cross-sections as shown in Fig. 7 tend to cause higher errors.This tendency might be caused by the need for a larger receptive field, i.e., a deeper CNN, or the fact that such geometries may be stochastically underrepresented in the randomly generated input shapes used for training.Both of these explanations should be investigated further in future work.
Fig. 9 shows the evolution of the average errors in terms of ΔL PE  11 , ΔL PE 22 , and Δk PE across training time for our benchmark result on Geo2b.Although we presented the final prediction accuracy after about 35 h of training above, this evolution shows that significantly less time would have to be spent on training if a slightly worse prediction accuracy, e.g., 1% − 3%, would be sufficient from a design perspective.

D. Inference Times for PINNs
In order to be competitive with numerical approaches such as FEMM, PINNs can amortize their additional training time by inferring solutions for the given problem a lot faster after     computation.This additional speed-up potential is also reflected in the inference time for FC-PINN in Table II.Nevertheless, considering the training times in the range of 20-200 s for Geo1, cf.Fig. 4, it is clear that neither FC-PINN nor ConvPINN can be competitive to FEMM for solving a single geometric setup like Geo1.
However, if large amounts of geometries need to be evaluated for optimization purposes, generalized PINNs become a viable option as the training time amortizes with single PINN predictions being several orders of magnitudes faster.This speedup can be improved even further by parallelization and batching of samples, e.g., predicting the magnetic vector potential for 64 geometries in parallel with ConvPINN takes around 220 ms on average, i.e., around 3.5 ms per sample and, therefore, a total speedup of almost 300 compared to FEMM.
For optimization purposes, this means, that-once trained-ConvPINN is able to predict inductances and couplings for a wide range of axisymmetric transformer geometries accurately in a matter of several milliseconds.For Geo2b, cf.Section IV-C, Fig. 9, we obtain average errors around 1% after 35 h of training, i.e., ConvPINN with 3.5 ms inference per sample would become more efficient than FEMM as soon as more than ≈ 125 000 samples are to be evaluated during optimization.This threshold is already not far from realistic optimization scenarios, cf.[32], and is likely to be decreased further in the future due to hardware improvements as well as more efficient architectures and training schemes to reduce the required training time.

V. CONCLUSION
We implemented and compared FC-PINN and ConvPINN for the prediction of inductances and coupling on a wide range of axisymmetric transformer geometries.To the best of our knowledge, this work is the first to demonstrate the application of the convolutional approach in ConvPINN for axisymmetric magnetostatic problems.While FC-PINN is trained to predict a continuous solution that tends to be more accurate for single geometric setups, the pixel-discrete ConvPINN performs far better on tasks that involve the solution of a PDE on parametrized geometries.We have shown the necessity of solving for a weak solution with less regularity requirements and studied basic hyperparameters, such as domain points and pixel resolution in FC-PINN and ConvPINN quantitatively.The generalized ConvPINN trained on random geometries predicts inductances and coupling of random geometries with PEs below 1% in a matter of several milliseconds.Considering current trends in the ML-related hardware and software landscape, these figures are likely to be improved even further in the future, e.g., as the usage of deeper CNNs with higher resolutions becomes more viable in this context.Consequently, our result paves the way for topology optimization of transformers via PINNs and proves the relevance and applicability of AI-based methods for engineering tasks in general.The proposed method can be transferred to related design tasks on electromagnetic devices that involve solving PDEs for electric or magnetic fields for a range of different geometries and can also be adapted for nonlinear materials.It should be noted that the fixed pixel-resolution within ConvPINN poses a practical limitation and a disadvantage with respect to the mesh-free FC-PINN approach.More research is required to quantify this limitation and to come up with more advanced architectures and training schemes that enable varying resolutions within ConvPINN.Apart from pushing the limits in terms of accuracy, convergence speed, and geometric complexity with ConvPINN, future research should also raise the generalization potential of concepts, such as latent space representations or hypernetworks for both FC-PINNs and ConvPINNs.

A. Parameter Ranges
Table III lists all varied geometry parameters for Geo2a and Geo2b.For all use cases, we fix the area of coils by setting the height with respect to the width via c i h = 0.36 cm 2 c i w .The ranges for widths and heights are chosen to allow for a deviation of 20% with respect to the baseline geometry Geo1 in both directions.During training and evaluation, samples need to fulfill the following constraints: ferrite and coil structures must not touch, i.e., have at least one pixel-size of spare room (air) between them.All edges of coils C1 and C2 need to stay within the inscribed rectangle of the two L-shaped ferrite structures.Any samples that violate these constraints are discarded automatically.Furthermore, in order to mitigate discretization errors within ConvPINNs, all randomized variables are rounded to the nearest multiple of the corresponding pixel size (depending on resolution N r × N z ) in order to prevent pixels belonging to more than one distinct material.

B. Finite Differences (ConvPINN)
As described in Section III, the derivatives for the physicsdriven loss in ConvPINN are calculated from a finite difference scheme based on the Yee-Grid, which is constructed from staggered grids for electric and magnetic field components.We adapt the scheme from the work in [14], which is based on the frequency-domain Maxwell equation without electric current densities and uses ∇ h and ∇ e to denote derivatives approximated on the electric or magnetic grid, respectively.For the magnetostatic problem defined in (1), this yields Under the aforementioned assumptions from Section II and B = ∇ × A, we get the second-order spatial derivatives on the electric grid via the forward differences B r (iΔr, jΔz) = A ϕ (iΔr, (j + 1)Δz) − A ϕ (iΔr, jΔz) Δz B z (iΔr, jΔz) = A ϕ ((i + 1)Δr, jΔz) − A ϕ (iΔr, jΔz) Δr .

TABLE IV ML HYPERPARAMETERS FOR FC-PINN AND CONVPINN FOR DIFFERENT USE CASES
On the other hand, for derivatives defined on the magnetic grid, a backward difference is used, e.g.

C. Hyperparameters
Table IV lists the corresponding hyperparameters for the results on both FC-PINN and ConvPINN if not stated otherwise in Section IV.Note that for ConvPINN the amount of layers refers only to the downsampling / convolving part of the U-Net architecture, the transposed convolution path has the same depth for upsampling.More details on the ConvPINN design can be found in the supplementary material in [14].In both FC-PINN and Conv-PINN, we use exponential decay with a decay rate of 0.5.

Fig. 1 .
Fig.1.Geo1: Baseline geometry with notation for geometry and material.Geo2a: Variation of positions, widths, and heights for coils C1 and C2.Geo2b: Variation of positions, widths, and heights for L-shaped ferrite pieces F1 and F2, and coils C1 and C2.

Fig. 3 .
Fig. 3. ConvPINN for prediction of the magnetic vector potential A ϕ based on two binary input images for the geometry of ferrite and excited coil.

Fig. 4 .
Fig. 4. Evolution of ΔL PE 11 across training time on Geo1 with µ rel = 2000 for FC-PINN and ConvPINN with varying amounts of domain points and pixels.

Fig. 5 .
Fig. 5. Geo2a: Predicted values by FC-PINN and ConvPINN for inductances L 11 , L 22 , and coupling k on 100 randomized samples versus reference values from FEMM.The dotted and dashed lines indicate a deviation from the reference by 5% and 10%, respectively.

Fig. 6 .
Fig. 6.Geo2b: Predicted values by ConvPINN for inductances L 11 , L 22 , and coupling k on 100 randomized samples versus reference values from FEMM.The dotted and dashed lines indicate a deviation from the reference by 5% and 10%, respectively.

TABLE I
GEO1: COMPARISON OF FC-PINN AND CONVPINN FOR STRONG AND WEAK FORMULATION AND DIFFERENT RELATIVE PERMEABILITIES between different input geometries in order to minimize the PDE residual for each of them within a single CNN.
lists prediction accuracies achieved for the fixed geometry Geo1.The discontinuity at material interfaces between ferrite and air is varied with μ rel = μ ferrite μ air ∈ {2, 20, 200, 2000} and we consider different pixel resolutions for ConvPINN.Both approaches are given an equal amount of time for training and the best predictions in terms of MAE and ΔL PE 11 are reported.The usage of the strong formulation yields dysfunctional training for almost all setups and even the least permeability ferrite on FC-PINN results in a high error on ΔL PE 11 above 27%.The MAE increases with permeability in all setups, indicating the increasing physical complexity and bad regularity of solutions at material interfaces.In the case of ConvPINN, the accuracy in terms of both MAE and ΔLPE11 can be improved significantly by doubling the resolution in each dimension toward 512 × 896 pixels.

TABLE II INFERENCE
TIMES PER SAMPLE FOR FEMM, FC-PINN, AND CONVPINN-AVERAGED OVER 100 SAMPLES

TABLE III VARIED
GEOMETRIC PARAMETERS IN USE CASES GEO2A AND GEO2B AND THEIR PERMISSIBLE RANGES