Designing complex engineering systems often relies on simulation and optimization. Examples include electric motors or high-frequency devices. These processes are based on the Finite Element Method (FEM) or Boundary Element Methods (BEM). These methods are highly accurate but can become computationally expensive, especially when exploring many scenarios, such as in optimization tasks.
In our research, we aim to combine the best of two worlds. We integrate the mathematical rigor of FEM or BEM with the speed of machine learning. The neural networks are trained on a set of high-fidelity simulations to “learn” the behavior of complex systems. Our approach differs from physics-informed neural networks (PINNs). While PINNs directly approximate the solution of a physical problem, our method learns the solution via basis coefficients. These coefficients relate to a predefined set of basis functions, such as those used in FEM or BEM. We recommend using high-order (spline) basis functions for both geometry and solution representation. The increased density of the linear systems does not harm the machine learning. On the other hand, the use of basis functions ensures that the network’s predictions respect physical properties like continuity and boundary conditions while being efficient to compute. Once trained, the network provides near-instant predictions for new scenarios, e.g., when varying the geometry, e.g. control points of splines, making them highly useful for optimization tasks in industrial applications, where computational efficiency is essential.
The work groups CEM (TU Darmstadt), SAM (ETH Zürich) and DCSE (TU Delft) collaborated to apply the method to an electromagnetic scattering problem. This work illustrates its potential in practice.