Right Feature, Right Place: Residual-Driven Construction of Cauchy-Adaptive Physics-Informed Neural Networks

Physics-informed neural networks (PINNs) solve partial differential equations by fitting a neural representation to the equation residual. Many solutions of scientific interest, from rogue waves and solitons to anomalous diffusion, carry isolated complex poles and algebraic tails, and at a finite size a PINN's accuracy depends on whether its neurons can represent this structure. Progress on PINN accuracy has centred on optimization, and adaptive methods tune features within one representation family chosen in advance. We introduce the Cauchy-Adaptive PINN (CAPINN), which lets the PDE residual choose each neuron's class, centre, and scale from a Cauchy-enriched dictionary of Cauchy, tanh-derived, and Gaussian neurons, building a compact basis over successive time intervals without solution labels. Standard PINN benchmarks favour tanh networks through their locally tanh-like fronts, so we evaluate CAPINN on a problem roster balanced for analytic structure. CAPINN is more accurate than vanilla PINNs and physics-informed Gaussians on every problem with poles or algebraic tails, and its basis adapts to each solution: Cauchy neurons place their poles along the withheld singularity trajectories, while smooth solutions are fitted with Gaussians alone. Front-dominated benchmarks remain the territory of tanh-based solvers. At a fixed neuron budget, how the representation is allocated is a lever complementary to optimization, and residual-driven allocation yields compact, interpretable solvers.