Reproducing kernel neural networks for nonlinear integro-differential equations

Inspired by physics-informed neural networks (PINNs), this paper proposes a novel framework termed the reproducing kernel neural networks (RKNNs) for solving nonlinear integro-differential equations (NIDEs). This methodology utilizes reproducing kernel theory to overcome the limitations of automatic differentiation when dealing with integral operators. Specifically, the RKNNs framework establishes a reproducing kernel integral solver that enables simultaneous implementation of automatic differentiation for integer-order differential operators and numerical discretization for integral operators. RKNNs exhibit effectiveness and generalization ability. The reproducing kernel integral solver of RKNNs is not only data-driven but also yields approximation that converges to the integral term in L²-norm. Extensive numerical experiments demonstrate that the proposed method exhibits the abilities to effectively solve time-dependent and time-independent NIDEs and stabilizes the training process.