DeepRitzSplit Neural Operator for Phase-Field Models via Energy Splitting
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arXiv
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| Autores principales: | , , , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866908980190642176 |
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| author | Huang, Chih-Kang Gagnon, Ludovick Založnik, Miha Appolaire, Benoît |
| author_facet | Huang, Chih-Kang Gagnon, Ludovick Založnik, Miha Appolaire, Benoît |
| contents | The multi-scale and non-linear nature of phase-field models of solidification requires fine spatial and temporal discretization, leading to long computation times. This could be overcome with artificial-intelligence approaches. Surrogate models based on neural operators could have a lower computational cost than conventional numerical discretization methods.
We propose a new neural operator approach that bridges classical convex-concave splitting schemes with physics-informed learning to accelerate the simulation of phase-field models. It consists of a Deep Ritz method, where a neural operator is trained to approximate a variational formulation of the phase-field model. By training the neural operator with an energy-splitting variational formulation, we enforce the energy dissipation property of the underlying models.
We further introduce a custom Reaction-Diffusion Neural Operator (RDNO) architecture, adapted to the operators of the model equations. We successfully apply the deep learning approach to the isotropic Allen-Cahn equation and to anisotropic dendritic growth simulation. We demonstrate that our physically-informed training provides better generalization in out-of-distribution evaluations than data-driven training, while achieving faster inference than traditional Fourier spectral methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_18261 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | DeepRitzSplit Neural Operator for Phase-Field Models via Energy Splitting Huang, Chih-Kang Gagnon, Ludovick Založnik, Miha Appolaire, Benoît Analysis of PDEs Machine Learning Numerical Analysis The multi-scale and non-linear nature of phase-field models of solidification requires fine spatial and temporal discretization, leading to long computation times. This could be overcome with artificial-intelligence approaches. Surrogate models based on neural operators could have a lower computational cost than conventional numerical discretization methods. We propose a new neural operator approach that bridges classical convex-concave splitting schemes with physics-informed learning to accelerate the simulation of phase-field models. It consists of a Deep Ritz method, where a neural operator is trained to approximate a variational formulation of the phase-field model. By training the neural operator with an energy-splitting variational formulation, we enforce the energy dissipation property of the underlying models. We further introduce a custom Reaction-Diffusion Neural Operator (RDNO) architecture, adapted to the operators of the model equations. We successfully apply the deep learning approach to the isotropic Allen-Cahn equation and to anisotropic dendritic growth simulation. We demonstrate that our physically-informed training provides better generalization in out-of-distribution evaluations than data-driven training, while achieving faster inference than traditional Fourier spectral methods. |
| title | DeepRitzSplit Neural Operator for Phase-Field Models via Energy Splitting |
| topic | Analysis of PDEs Machine Learning Numerical Analysis |
| url | https://arxiv.org/abs/2604.18261 |