Stochastic approach for elliptic problems in perforated domains
This addresses computational bottlenecks for engineers and scientists working with perforated materials like metals or filters, though it appears incremental as a neural network adaptation of existing stochastic methods.
The authors tackled the computational challenge of solving elliptic PDEs in perforated domains by proposing a neural network-based mesh-free method that incorporates a derivative-free loss approach using stochastic representation. Their method demonstrated efficacy in handling various perforation scales through stringent numerical tests.
A wide range of applications in science and engineering involve a PDE model in a domain with perforations, such as perforated metals or air filters. Solving such perforated domain problems suffers from computational challenges related to resolving the scale imposed by the geometries of perforations. We propose a neural network-based mesh-free approach for perforated domain problems. The method is robust and efficient in capturing various configuration scales, including the averaged macroscopic behavior of the solution that involves a multiscale nature induced by small perforations. The new approach incorporates the derivative-free loss method that uses a stochastic representation or the Feynman-Kac formulation. In particular, we implement the Neumann boundary condition for the derivative-free loss method to handle the interface between the domain and perforations. A suite of stringent numerical tests is provided to support the proposed method's efficacy in handling various perforation scales.