Neural Subdomain Solver for Magnetostatic Field Computations of a Quadrupole Magnet
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In our work, we construct a neural network (NN)-based ansatz function in the parametric domain of Non-Uniform Radial B-Splines (NURBs), and solve the magnetostatic boundary value problem on a quadrupole magnet. To decompose the magnet, we employ conforming, multi-patch NURBs which decompose the geometry into a set of subdomains. On each subdomain the NURBs' push-forward mappings can be used to define local NN-based ansatz functions in the parametric domain that impose the tangential continuity of the magnetic vector potential at material interfaces. The set of individual, local ansatz functions make up the global ansatz function on the physical domain. Our approach exhibits a number of benefits compared to physics-informed neural networks (PINNs). First and foremost, hard-constrained boundary and interface conditions remove the necessity for penalty factors, making the approach heuristics-free. In addition, geometries of arbitrary size can be simulated as the NNs are optimised in the parametric domain, which is the unit square in the case of NURBs. As a consequence, NN inputs are always normalised. Finally, the error of our approach compared to a finite element reference solution is 0.45% relative mean squared error, which is a significant improvement compared to classical PINNs.
