Shearlet-Based Energies for Anisotropic Phase-Field Models
Final Report Abstract
In this project, we established innovative methods to solve free-discontinuity problems. In the first part of this project, we introduced a novel approach based on the shearlet transform that can be used to replace classical, non-linear energies by more accessible shearlet-based energies. This approach is inspired by similar results for wavelet-based energies. During this first part of the project, we identified several significant errors in the established arguments for the wavelet case. As a consequence, we could not obtain the initially anticipated results, which led us to consider an alternative approach to solve free discontinuity problems. Deep neural networks have been shown to a) approximate the solutions of free-discontinuity problems optimally b) can be trained efficiently to minimise general energies. From this perspective, systems of deep neural networks appear to be very well suited to solve general free discontinuity problems. Because of these considerations, we established a comprehensive theoretical foundation for the application of deep neural networks for free discontinuity problems. Among other results, we identified the approximation behaviour of deep neural networks in Sobolev norms and found a connection to various finite element spaces. Moreover, we demonstrated that deep neural networks can solve linear parametric PDEs as efficient as the so-called reduced basis method. Furthermore, we studied topological properties of the space of deep neural network, which provides a novel point of view to understand the underlying optimisation problem of deep learning.
Publications
- Gamma-convergence of a shearlet-based Ginzburg–Landau energy
Philipp Petersen, Endre Süli
- Approximation properties of hybrid shearlet-wavelet frames for Sobolev spaces, Advances in Computational Mathematics, 45(3), 1581–1606, 2019
Philipp Petersen, Mones Raslan
(See online at https://doi.org/10.1007/s10444-019-09679-9)
