Adaptive Physics-Informed Neural Networks for One-Dimensional Reaction-Diffusion Models with Non-Lipschitz Nonlinearity and Free Boundaries

Abstract

We develop a novel framework based on Physics-Informed Neural Networks (PINNs) combined with Adaptive Dynamic Sampling (ADS) to solve one-dimensional reaction-diffusion equations with non-Lipschitz nonlinearities (singular kinetics, sublinear growth). The proposed two-network architecture simultaneously approximates both the solution profile and the free boundary location, thereby addressing the inherent coupling between the Partial Differential Equation (PDE) solution and the unknown interface. By dynamically concentrating training points within the support of the active solution (effective region (support)), ADS-PINNs overcome the limitations of uniform sampling and significantly improve accuracy. We validate the framework on one-dimensional nonlinear eigenvalue and boundary value problems with known analytical solutions. Our results demonstrate that ADS-PINNs accurately capture the free boundary and extend classical monotonicity results by confirming that the interface location increases monotonically with respect to the nonlinearity exponent p and depends sensitively on the boundary parameter d. Numerical experiments show close agreement with analytical benchmarks, achieving relative errors on the order of 10⁻² for the free boundary and 10⁻⁸ for the solution norm.