LPINNs: First-Layer Gated Localization for Physics-Informed Neural Networks

Physics-informed neural networks (PINNs) use one shared representation over the computational domain, which can become difficult to optimize on long domains and for high-order operators. We study a minimal alternative: multiply the first hidden activation of an otherwise unchanged dense PINN by input-dependent localization functions, giving first-layer units receptive fields without partitioning the domain or adding interface losses. We screen 13 families of localization functions, in up to three parameterizations each, on a nonlinear harmonic oscillator (HO), a heat equation on a long spatial interval, and a manufactured four-dimensional (4D) fourth-order problem, with ten paired seeds throughout. Three configurations give large reductions in solution error at matched budgets: (i) Fixed Gaussian localization functions on the 2π HO domain cut mean solution RMSE from 4.8369×10-1 to 8.83×10-3 at 3k epochs. (ii) The inverse-quadratic family with learnable centers and widths cuts it from 2.896×10-1 to 3.06×10-2 on the 8π heat domain at 10k epochs. (iii) Fixed bump localization functions cut it from 1.75947×101 to 2.260×10-1 on the 4π 4D domain at 10k epochs. Every paired seed improves in these three comparisons. The screen also shows that the mechanism is not a free win: on HO only 2 of 13 families beat the baseline, and 10 of the remaining 11 are 9 to 23 times worse; on 4D four families are non-finite and five are more than three orders of magnitude worse than the baseline. The inverse-quadratic family is the only one that beats the baseline on all three equations. Overall, these results show that first-layer localization can provide measurable improvements to baseline PINNs on long-domain and high-order problems.