Mechanics-trained neural coordinate mapping for B-spline analysis of crack-tip and corner singularities

What the math gives to ML

The paper provides a constructive monotone coordinate transformation whose admissibility is guaranteed by parameterization rather than by a penalty or post hoc repair. A normalized positive neural density produces a map with fixed endpoint anchors and strictly positive derivative, while a prescribed near-tip power converts a singular physical behavior r^lambda into the smoother computational behavior s^(q lambda). This is directly transferable to neural fields and PINNs as a learnable input warp: optimize the coordinate map through the solved PDE energy or residual so that network capacity is concentrated near singularities. The strongest initial test is a Deep Ritz or PINN model on an L-shaped domain, comparing the learned warp against identity and fixed power coordinates at equal parameter count and training FLOPs.