Solving partial differential equations with hybridized physic-informed neural network and optimization approach: Incorporating genetic algorithms and L-BFGS for improved accuracy

Partial differential equations (PDEs) are essential mathematical models for describing a wide range of physical phenomena. Numerically, Physic-Informed Neural Networks (PINNs), a variant of artificial neural networks, present a promising method for solving PDEs. However, due to limitation in accuracy and stability, various adaptive PINN variants have been proposed. We have designed a novel approach that adopted self-adaptive PINN (SA-PINN) with two optimization techniques: the genetic algorithm (GA) and the limited-memory Broyden-Fletcher-Goldfarb-Shanno (L-BFGS) algorithm. Self-adaptive PINN modifies the weights in the loss function to be fully trainable, enabling the ANN to learn and stabilize the PINN in approximating the difficult regions of the solution. GA initializes the population of ANN trainable parameters to optimize the training process with less number of iterations, while L-BFGS is used to find the best solution accurately. Our proposed approach, named SA-PINN-GA-LBFGS, is tested on solving several benchmark PDE problems including elliptic, parabolic, and hyperbolic types. We compare our results with state-of-the-art methods, demonstrating that SA-PINN-GA-LBFGS provides higher accuracy and greater efficiency.