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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.24651 |
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| _version_ | 1866911713516847104 |
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| author | Zhu, Bokai Zhang, Qinghui Rabczuk, Timon |
| author_facet | Zhu, Bokai Zhang, Qinghui Rabczuk, Timon |
| contents | We propose a Weak-form Physics-Informed Neural Operator (WINO), a data-free framework that combines the efficiency of neural operators with the geometric flexibility of the $φ$-finite element method ($φ$-FEM). $φ$-FEM is an unfitted method that accommodates geometric variations without body-fitted meshes, where the domain geometry is represented by the level-set function $φ$. To impose the boundary conditions, Dirichlet problems adopt the $φ$-FEM lifting so only the homogeneous displacement contribution is learned, whereas traction-driven Neumann problems additionally predict the auxiliary fields necessary for the unfitted weak formulation. Parameters are trained by minimizing squared weak-form residuals aligned with $φ$-FEM together with squared penalties on the cut-cell auxiliary equations, which removes the need for large paired datasets of converged reference solutions. After training, WINO outputs can seed the nonlinear $φ$-FEM solvers as neural operator warm starts (NOWS), which reduce iteration counts relative to traditional cold-started solvers. Numerical benchmarks show that WINO achieves high accuracy below 0.04 across all benchmarks, while reducing total computational time by 50--80\% compared with purely data-driven methods. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_24651 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | WINO: A Weak-Form Physics Informed Neural Operator for Hyperelasticity on Variable Domains Zhu, Bokai Zhang, Qinghui Rabczuk, Timon Numerical Analysis Machine Learning We propose a Weak-form Physics-Informed Neural Operator (WINO), a data-free framework that combines the efficiency of neural operators with the geometric flexibility of the $φ$-finite element method ($φ$-FEM). $φ$-FEM is an unfitted method that accommodates geometric variations without body-fitted meshes, where the domain geometry is represented by the level-set function $φ$. To impose the boundary conditions, Dirichlet problems adopt the $φ$-FEM lifting so only the homogeneous displacement contribution is learned, whereas traction-driven Neumann problems additionally predict the auxiliary fields necessary for the unfitted weak formulation. Parameters are trained by minimizing squared weak-form residuals aligned with $φ$-FEM together with squared penalties on the cut-cell auxiliary equations, which removes the need for large paired datasets of converged reference solutions. After training, WINO outputs can seed the nonlinear $φ$-FEM solvers as neural operator warm starts (NOWS), which reduce iteration counts relative to traditional cold-started solvers. Numerical benchmarks show that WINO achieves high accuracy below 0.04 across all benchmarks, while reducing total computational time by 50--80\% compared with purely data-driven methods. |
| title | WINO: A Weak-Form Physics Informed Neural Operator for Hyperelasticity on Variable Domains |
| topic | Numerical Analysis Machine Learning |
| url | https://arxiv.org/abs/2605.24651 |