Fraunhofer Institute for Integrated Systems and Device Technology IISB
Fraunhofer Institute for Integrated Systems and Device Technology IISB

Physics-informed Neural Networks for Engineering Applications

© Fraunhofer IISB
Fig. 1: a) Inductive systems as used in practical applications transferred to a parametric simulation model, trained by pure physics using a PINN. b) Approximating the magnetic vector potential enables a fast and accurate determination of the relevant engineering quantities.
© Fraunhofer IISB
Fig. 2: Evaluation of the PINN prediction accuracy by comparison of the simulated near E-field from an EUV mask and corresponding lithographic metrics: (a) 3D EUV mask. Use case: horizontal line on the top of MoSi multilayer, feature size: 20 nm with a pitch of 40 nm (wafer scale), 80 nm with no biasing (mask scale), absorber: 60 nm thick, TaBN material. Illumination: a plane wave under ϕ = 6°; θ = 0°. (b) Amplitude computed numerically with the waveguide method. (c) Amplitude predicted by PINN. (d) Image difference with MAPEXY = 0.18 %; MAPE = 0.91 %; RMSE = 4.1E-3 a.u. (e) Cross sections of simulated aerial images. (f) Simulated lithographic process windows. (g) relevant lithography metrics.

Simulation has become an indispensable tool in modern engineering to solve complex problems without or at least with a reduced number of experiments. However, both established numerical and data-driven approaches, face inherent limitations that hinder their effectiveness in capturing complex physical phenomena. Numerical methods, though widely used, often struggle with high computational costs and the curse of dimensionality, particularly in scenarios characterized by intricate geometries or multiscale interactions. Additionally, these methods heavily rely on mesh generation, posing challenges when dealing with evolving or irregular domains. On the other hand, data-driven approaches can suffer from the need for extensive labeled data and may struggle to generalize accurately in regions with sparse or no observations. Recognizing these challenges, physics-informed neural networks (PINNs) have emerged as a promising paradigm that integrates the strengths of both numerical and data-driven methods, offering a novel approach to address the shortcomings of current simulation techniques:

  • Physics-informed: PINNs seamlessly integrate physical principles into the learning process, enabling accurate modeling of complex systems as well as inter- and extrapolation based on fundamental laws.
  • Data efficiency: PINNs require less or even no labeled training data compared to purely data-driven approaches, making them effective in scenarios with limited or expensive data collection.
  • Separation: The stages of model development and training (with high computational effort) and the subsequent application or inference phase are split. The resulting PINN-simulator commonly requires only a fraction of a second to approximate complex physics.
  • Adaptability to irregular geometries: PINNs excel in handling problems with irregular geometries, as they do not rely on structured meshes, overcoming a common limitation in traditional numerical methods.
  • Multiscale capabilities: PINNs can effectively capture and model multiscale phenomena, enabling the study of intricate interactions between
    different scales in a system.
  • Flexibility: Directly coupled equations including established or artificial boundary conditions can be approximated just by evaluating (instead of solving) the equations. This strategy reduces the need for explicit enforcement and provides a more flexible and robust approach.

Our research group AI-augmented Simulation works on the application of PINNs in several domains of power electronics and semiconductor technology. Two examples are shown below.

 

Magnetostatic Problems on Transformer Geometries

Figure 1, see also paper in IEEE Journal of Emerging and Selected Topics in Industrial Electronics

Coils and transformers are essential components for the transmission and storage of energy. Training PINNs with the rotational symmetric magnetostatic formulation of Maxwells equations results in an ultra-fast approximator (1-10 ms) for all relevant engineering quantities of a transformer. Based on a U-Net PINN consisting of an encoder-decoder architecture, the spatial magnetic vector potential distribution is calculated for any pixel-based input geometry of two coils and corresponding ferrite structures. With a deviation of 1 % (compared to reference FEMM simulations) of the inductances and coupling this PINN solution opens the door to live optimization of inductive components and systems.

 

Modeling of Diffraction from 3D Mask and Imaging in EUV Lithography

Figure 2, see also paper at 2023 Photonics & Electromagnetics Research Symposium (PIERS) and paper at 2024 SPIE Advanced Lithography + Patterning

The increasing demands on computational lithography and imaging in the design and optimization of lithography processes necessitate rigorous modeling of EUV light diffracted from the mask. The implemented PINN model solves the scattered field computation problem by seamlessly embedding the residual of Maxwell’s equation into the loss function and minimizing it through gradient-based optimizers. The developed PINN-based EMF solver enables a more efficient computation of light diffraction from reflective 3D EUV masks and lithographic imaging. The PINN demonstrates significant speed-up compared to numerical solvers and high accuracy in modeling the near field and the far field, process windows, as well as 3D mask effects.