## Computational Tools for Differentiable Multi-Physics Electric Machine Models

*Andrés Beltrán-Pulido, Sihun Kim, Dionysios Aliprantis (Elmore Family School of Electrical & Computer Engineering, Purdue University)*

*Ilias Bilionis (School of Mechanical Engineering, Purdue University)*

This research aims to develop a computational framework for differentiable multi-physics electric machine (EM) models, which can enable more efficient and accurate modeling, analysis, and design for various applications (e.g., electric vehicles). The funding from the Grainger CEME has supported one graduate and one undergraduate student. We use the physics-informed neural networks (PINN) framework [1] to build an EM model for solving partial differential equation-constrained optimization problems. We have made significant progress on essential tasks, including developing a tool for describing EM geometries using signed distance function (SDF)-based indicator functions, enabling us to model the geometry of electric machines efficiently and accurately. For example, Figure 9 depicts the cross-section of the EM under study represented by SDF-based indicator functions.

Additionally, we have successfully leveraged the PINN framework to solve the quasi-magnetostatic problem for different operating conditions, yielding a physics-informed, differentiable model for rapidly predicting the EM electromagnetic fields. Finally, we have also set up finite element (FE) simulations for comparison with the proposed PINN framework, allowing us to validate further and refine our approach. Figure 10 illustrates the accuracy of our trained PINN-based model, which yielded an average absolute error in the magnetic vector potential (MVP) of 0.46 mWb with respect to FE. This level of accuracy is relatively high overall, with the highest absolute errors near the air gap and shaft. By using deeper networks, we can improve the accuracy at the cost of increased computational time. Our next steps will focus on further refining and expanding the framework for multi-physics optimization, with an emphasis on incorporating thermal physics considerations. We expect this effort to yield openly available codes and methods, extensive simulation data sets, and journal publications.

[1] A. Beltrán-Pulido, I. Bilionis, and D. Aliprantis, “Physics-Informed Neural Networks for Solving Parametric Magnetostatic Problems,” in IEEE Transactions on Energy Conversion, vol. 37, no. 4, pp. 2678-2689, Dec. 2022.