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Article: Solving nonconvex energy minimization problems in martensitic phase transitions with a mesh-free deep learning approach
| Title | Solving nonconvex energy minimization problems in martensitic phase transitions with a mesh-free deep learning approach |
|---|---|
| Authors | |
| Issue Date | 15-Aug-2023 |
| Publisher | Elsevier |
| Citation | Computer Methods in Applied Mechanics and Engineering, 2023, v. 416 How to Cite? |
| Abstract | In this paper, we propose a novel mesh-free method for solving nonconvex energy minimization problems for martensitic phase transitions and twinning in crystals using the deep learning approach. These problems pose significant challenges to analysis and computation because they involve multiwell gradient energies with large numbers of local minima, each involving a topologically complex microstructure of free boundaries with gradient jumps. We use the Deep Ritz method, which represents candidates for minimizers using parameter-dependent deep neural networks, and minimize the energy with respect to network parameters. The key feature of our method is a newly proposed activation function called SmReLU, which captures the structure of minimizers where traditional activation functions fail. Our mesh-free approach allows for the approximation of free boundaries, essential to this problem, without any special treatment, making it extremely simple to implement. We demonstrate the success of our method through numerous numerical computations. |
| Persistent Identifier | http://hdl.handle.net/10722/331691 |
| ISSN | 2023 Impact Factor: 6.9 2023 SCImago Journal Rankings: 2.397 |
| DC Field | Value | Language |
|---|---|---|
| dc.contributor.author | Chen, Xiaoli | - |
| dc.contributor.author | Rosakis, Phoebus | - |
| dc.contributor.author | Wu, Zhizhang | - |
| dc.contributor.author | Zhang, Zhiwen | - |
| dc.date.accessioned | 2023-09-21T06:58:03Z | - |
| dc.date.available | 2023-09-21T06:58:03Z | - |
| dc.date.issued | 2023-08-15 | - |
| dc.identifier.citation | Computer Methods in Applied Mechanics and Engineering, 2023, v. 416 | - |
| dc.identifier.issn | 0045-7825 | - |
| dc.identifier.uri | http://hdl.handle.net/10722/331691 | - |
| dc.description.abstract | <p>In this paper, we propose a novel mesh-free method for solving nonconvex energy <a href="https://www.sciencedirect.com/topics/engineering/minimization-problem" title="Learn more about minimization problems from ScienceDirect's AI-generated Topic Pages">minimization problems</a> for martensitic phase transitions and twinning in crystals using the <a href="https://www.sciencedirect.com/topics/engineering/deep-learning" title="Learn more about deep learning from ScienceDirect's AI-generated Topic Pages">deep learning</a> approach. These problems pose significant challenges to analysis and computation because they involve multiwell <a href="https://www.sciencedirect.com/topics/mathematics/energy-gradient" title="Learn more about gradient energies from ScienceDirect's AI-generated Topic Pages">gradient energies</a> with large numbers of local minima, each involving a topologically complex microstructure of free boundaries with gradient jumps. We use the Deep <a href="https://www.sciencedirect.com/topics/mathematics/ritz-method" title="Learn more about Ritz method from ScienceDirect's AI-generated Topic Pages">Ritz method</a>, which represents candidates for minimizers using parameter-dependent <a href="https://www.sciencedirect.com/topics/engineering/deep-neural-network" title="Learn more about deep neural networks from ScienceDirect's AI-generated Topic Pages">deep neural networks</a>, and minimize the energy with respect to network parameters. The key feature of our method is a newly proposed <a href="https://www.sciencedirect.com/topics/engineering/activation-function" title="Learn more about activation function from ScienceDirect's AI-generated Topic Pages">activation function</a> called SmReLU, which captures the structure of minimizers where traditional activation functions fail. Our mesh-free approach allows for the <a href="https://www.sciencedirect.com/topics/computer-science/approximation-algorithm" title="Learn more about approximation from ScienceDirect's AI-generated Topic Pages">approximation</a> of free boundaries, essential to this problem, without any special treatment, making it extremely simple to implement. We demonstrate the success of our method through numerous <a href="https://www.sciencedirect.com/topics/mathematics/numerical-computation" title="Learn more about numerical computations from ScienceDirect's AI-generated Topic Pages">numerical computations</a>.<br></p> | - |
| dc.language | eng | - |
| dc.publisher | Elsevier | - |
| dc.relation.ispartof | Computer Methods in Applied Mechanics and Engineering | - |
| dc.title | Solving nonconvex energy minimization problems in martensitic phase transitions with a mesh-free deep learning approach | - |
| dc.type | Article | - |
| dc.identifier.doi | 10.1016/j.cma.2023.116384 | - |
| dc.identifier.volume | 416 | - |
| dc.identifier.eissn | 1879-2138 | - |
| dc.identifier.issnl | 0045-7825 | - |