Distance-gated adaptive neural Ritz learning for nonconvex variational problems with localized phase transitions

Neural variational solvers provide mesh-free approximations of energy-minimization problems, but localized, geometry-dependent interfaces in nonconvex energies remain difficult. To address this, we propose Bio-PINN, a distance-gated adaptive neural Ritz method that minimizes an empirical variational energy with a neural trial map. A smooth boundary-distance gate modulates the deformation-variability sampling score and expands the proposal region from mechanically active boundaries toward the far field, while a retain-resample-release update preserves low-discrepancy coverage. We further establish deterministic release, retention, resampling, and near-to-far propagation properties of the sampler. Our method is evaluated on cell-induced remodeling of perforated fibrous extracellular-matrix domains, a challenging vector-valued benchmark involving pericellular densification and intercellular tether formation. Numerical experiments show that our method more faithfully reproduces the characteristic uniform, needle-like low-J_θ densification patterns near cell boundaries and within intercellular gaps. This physically consistent behavior is accompanied by a 29.7%-39.7% reduction in the boundary-adjacent near-phase Jacobian deviation.