A Learning-Based Ansatz Satisfying Boundary Conditions in Variational Problems
This addresses a specific bottleneck in physics-informed machine learning for researchers, though it is incremental.
The paper tackled the issue of neural networks not inherently satisfying boundary conditions in the Deep Ritz Method for variational problems, proposing an ansatz that eliminates misleading results and reduces complexity while maintaining accuracy.
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.