Physics

Computational physics regularly encounters mathematical singularities: point sources in wave equations, field concentrations at material boundaries, stress behavior near crack tips. Physics-informed neural networks and reduced-order models can struggle when learning functions with this singular structure—standard architectures with smooth activations systematically underestimate gradients near poles, creating "dead zones" in the learned representation. ZeroProofML addresses this by operating in an extended arithmetic system where division-by-zero produces tagged values (±∞, Φ) rather than errors or NaN propagation. When learning rational approximations to singular solutions, the framework maintains mathematical rigor while providing stable gradients and deterministic behavior.

Potential applications:

• Surrogate modeling: Learn fast approximations of expensive simulations involving singular solutions (wave scattering, electromagnetics), particularly for parametric studies where the singularity location varies with design parameters
• Physics-informed neural networks: 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, 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, maintaining accuracy across wider parameter ranges