WINO: A Weak-Form Physics Informed Neural Operator for Hyperelasticity on Variable Domains

Abstract

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. When labeled reference data are available, an optional data-augmented variant (WINO+data) can further combine this physics-informed loss with a supervised term. 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 substantial accuracy of WINO together with total training times of about 15%-70% of those of supervised φ-FEM-FNO across all cases, without requiring reference-solution generation.

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BibTeX RIS

BibTeX

@misc{zhu2026wino,
  title = {{WINO: A Weak-Form Physics Informed Neural Operator for Hyperelasticity on Variable Domains}},
  author = {Bokai Zhu and Yizheng Wang and Qinghui Zhang and Timon Rabczuk},
  howpublished = {arXiv},
  year = {2026},
  month = aug,
  doi = {10.48550/arxiv.2605.24651},
  eprint = {2605.24651},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2605.24651},
}

RIS

TY  - UNPB
TI  - WINO: A Weak-Form Physics Informed Neural Operator for Hyperelasticity on Variable Domains
AU  - Bokai Zhu
AU  - Yizheng Wang
AU  - Qinghui Zhang
AU  - Timon Rabczuk
PY  - 2026
DA  - 2026/08/27
PB  - arXiv
DO  - 10.48550/arxiv.2605.24651
UR  - https://arxiv.org/abs/2605.24651
AB  - 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. When labeled reference data are available, an optional data-augmented variant (WINO+data) can further combine this physics-informed loss with a supervised term. 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 substantial accuracy of WINO together with total training times of about 15%-70% of those of supervised φ-FEM-FNO across all cases, without requiring reference-solution generation.
ER  -

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