A Unified CutFEM Formulation for Finite-Strain Elasticity: Energy Minimisation and Corner Singularities
arXiv:2607.02334
2026
Training
2 ideas extracted · analyzed Aug 30, 2026
What the math gives to ML
The transferable contribution is the construction of one augmented scalar energy whose automatic differentiation produces consistent bulk, boundary, interface, and stabilization terms. This can become a principled objective for neural fields on irregular, moving, or decomposed domains, replacing manually assembled and often dimensionally inconsistent residual penalties. The corner analysis also identifies a concrete approximation bottleneck: mixed-boundary singularities cap convergence independently of whether the discretization is fitted, unfitted, polynomial, or neural. The most promising implementations are an energy-derived Nitsche loss for patchwise neural fields and an explicit singularity-enriched ansatz near boundary junctions.
Ideas from this paper
Unverified
2026
Add an explicit local power-law singular basis to a neural field near mixed Dirichlet-Neumann junctions, allowing the neural network to learn only the smoother remainder. Use the predicted or fitted singular exponent to concentrate collocation points near the junction. This directly targets the regularity bottleneck identified by the paper, where increasing polynomial degree or network capacity cannot overcome a convergence cap under uniform resolution.
Useful6/10
Difficulty5/10
Novelty7/10
Unverified
2026
Represent a solution on an unfitted domain with local neural subnetworks and train them using one augmented energy containing the bulk physical energy, symmetric Nitsche boundary or interface terms, and a derivative-jump ghost penalty. Automatic differentiation of this scalar objective supplies all gradients and avoids independently tuning inconsistent PDE residual, flux, and boundary losses. The method is especially suited to moving geometries, cut-cell domains, and domain-decomposed neural…
Useful6/10
Difficulty5/10
Novelty6/10