Energy-based PINNs for solving coupled field problems: concepts and application to the multi-objective optimal design of an induction heater

TL;DR

The paper addresses solving coupled magnetic-thermal field problems arising in induction heating using energy-based PINNs (ePINNs). It shows that minimizing the variational energy functional $F$ offers faster convergence, reduces the required derivative order, and minimizes hyperparameter dependence compared to residual-based PINNs. The authors build a multi-component framework: a magnetics network $mNN$ trained on FEA to obtain Joule losses, an energy-based tePINN to predict temperature via a learned $F$-based loss, and a hypernetwork $tHNN$ that outputs tePINN weights as a function of geometry for rapid parametric and inverse design. The approach achieves high-accuracy temperature fields in a graphite plate under induction heating, with substantial speedups over FEA in single- and multi-objective optimization, and demonstrates the potential of unsupervised, differentiable design workflows with exact derivatives for engineering systems.

Abstract

Physics-informed neural networks (PINNs) are neural networks (NNs) that directly encode model equations, like Partial Differential Equations (PDEs), in the network itself. While most of the PINN algorithms in the literature minimize the local residual of the governing equations, there are energy-based approaches that take a different path by minimizing the variational energy of the model. We show that in the case of the steady thermal equation weakly coupled to magnetic equation, the energy-based approach displays multiple advantages compared to the standard residual-based PINN: it is more computationally efficient, it requires a lower order of derivatives to compute, and it involves less hyperparameters. The analyzed benchmark problems are the single- and multi-objective optimal design of an inductor for the controlled heating of a graphite plate. The optimized device is designed involving a multi-physics problem: a time-harmonic magnetic problem and a steady thermal problem. For the former, a deep neural network solving the direct problem is supervisedly trained on Finite Element Analysis (FEA) data. In turn, the solution of the latter relies on a hypernetwork that takes as input the inductor geometry parameters and outputs the model weights of an energy-based PINN (or ePINN). Eventually, the ePINN predicts the temperature field within the graphite plate.

Figures (9)

• Figure 1: Benchmark setup (half of it due to symmetry).
• Figure 2: Particular of the mesh (graphite plate and induction heater) used in the FEA.
• Figure 3: Estimating the accuracy of the mHH for a random test inductor geometry ($\xi_1$ = 13.63 mm, $\xi_2$ = 13.59 mm, $\xi_3$ = 9.59 mm, $\xi_4$ = 12.67 mm). A) Joule losses within the graphite plate. B) Absolute error taking the FEA solution as ground truth.
• Figure 4: Max absolute error in the temperature field solution (ground truth is the FEA solution). The residual-based PINN takes $\eta_{2}=10^3$ in the loss function
• Figure 5: Estimating accuracy of the tePINN prediction in case of a given inductor geometry ($\xi_1$ = 14.8 mm, $\xi_2$ = $\xi_3$ = $\xi_4$ = 15 mm). The heat source distribution comes from FEA solution. A) Temperature field within the graphite plate. B) Absolute error taking the FEA solution as ground truth.