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This article introduces WINO, a novel neural operator framework that integrates physics-informed learning with the $\varphi$-finite element method to solve hyperelasticity problems on variable domains without requiring large labeled datasets. By leveraging weak-form residuals and auxiliary field predictions for boundary conditions, the approach significantly reduces training time compared to supervised baselines while maintaining high accuracy in geometrically complex scenarios. The work represents a significant advancement in machine learning applications for numerical analysis and scientific computing within computational mechanics.
WINO: A Weak-Form Physics Informed Neural Operator for Hyperelasticity on Variable Domains
arXiv:2605.24651v2 Announce Type: replace-cross 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 $\varphi$-finite element method ($\varphi$-FEM). $\varphi$-FEM is an unfitted method that accommodates geometric variations without body-fitted meshes, where the domain geometry is represented by the level-set function $\varphi$. To impose the boundary conditions, Dirichlet problems adopt the $\varphi$-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 $\varphi$-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 $\varphi$-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 $\varphi$-FEM-FNO across all cases, without requiring reference-solution generation.
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 $\varphi$-finite element method ($\varphi$-FEM). $\varphi$-FEM is an unfitted method that accommodates geometric variations without body-fitted meshes, where the domain geometry is represented by the level-set function $\varphi$. To impose the boundary conditions, Dirichlet problems adopt the $\varphi$-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 $\varphi$-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 $\varphi$-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 $\varphi$-FEM-FNO across all cases, without requiring reference-solution generation.
Numerical Analysis (math.NA); Machine Learning (cs.LG)
arXiv:2605.24651 [math.NA]
(or arXiv:2605.24651v2 [math.NA] for this version)
https://doi.org/10.48550/arXiv.2605.24651 Focus to learn more arXiv-issued DOI via DataCite
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