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.
@article{arxiv.2605.24651,
title = {WINO: A Weak-Form Physics Informed Neural Operator for Hyperelasticity on Variable Domains},
author = {Bokai Zhu and Qinghui Zhang and Timon Rabczuk},
journal= {arXiv preprint arXiv:2605.24651},
year = {2026}
}