Physics

Computational physics frequently deals with mathematical singularities. Point sources in wave equations, field concentrations at material boundaries, and stress behavior near crack tips are common examples. Physics-Informed Neural Networks (PINNs) and reduced-order models struggle here. Standard architectures with smooth activations systematically underestimate gradients near poles. They create "Soft Walls" or "Dead Zones" in the learned representation where the physics demands an infinite asymptote.

ZeroProofML addresses this by introducing a Rational Inductive Bias via the Signed Common Meadows (SCM) operator. Instead of treating the singularity as a numerical error to be smoothed over, SCM allows the network to represent the exact rational structure of the solution ($P/Q$). The architecture uses Scale-Detached Gradients to navigate the optimization landscape safely. It captures vertical asymptotes and infinite gradients without numerical collapse.

Potential applications:

• Surrogate Modeling: Learn fast approximations of expensive simulations involving singular solutions (wave scattering, electromagnetics). This is particularly useful for parametric studies where the singularity location varies with design parameters.
• Physics-Informed Neural Networks (PINNs): Represent solutions with known singular behavior (point sources, dipoles, crack fields) without the smoothing artifacts that plague standard architectures.
• Reduced-Order Models: Capture localized features and singularities in projection-based ROMs. This is relevant for parametric PDEs with moving singularities or concentrated loads.
• Solution Operators: Train neural operators (DeepONets, FNO variants) that can handle problems with singular inputs or outputs. This helps maintain accuracy across wider parameter ranges where standard operators fail to generalize.