A Learning-Based Ansatz Satisfying Boundary Conditions in Variational Problems

Abstract: Recently, innovative adaptations of the Ritz Method incorporating deep learning have been developed, known as the Deep Ritz Method. This approach employs a neural network as the test function for variational problems. However, the neural network does not inherently satisfy the boundary conditions of the variational problem. To resolve this issue, the Deep Ritz Method introduces a penalty term into the functional of the variational problem, which can lead to misleading results during the optimization process. In this work, an ansatz is proposed that inherently satisfies the boundary conditions of the variational problem. The results demonstrate that the proposed ansatz not only eliminates misleading outcomes but also reduces complexity while maintaining accuracy, showcasing its practical effectiveness in addressing variational problems.