| Abstract: |
| Deep learning has gained significant development in the field of scientific computing, especially in its application to solve problems related to differential operators using deep neural networks. However, the utilization of neural networks to solve problems involving singularities still faces challenges. In this talk, we will discuss the failure of deep learning methods for the singular variational problems exhibiting the Lavrentiev phenomenon. For such problems, we show the standard deep Ritz method and some variants fail to detect the singular minimizers. We then introduce a guiding term that renders the neural network to explore solutions as desired during training. Numerical experiments demonstrate that the method achieves much better approximations than the previous methods. Furthermore, we apply the same algorithm to solve problems with regular solutions to show the robustness of the proposed method. |
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