Gradient-based adaptive neural network technique for two-dimensional local fractional elliptic PDEs

Abstract

This paper introduces gradient-based adaptive neural networks to solve local fractional elliptic partial differential equations. The impact of physics-informed neural networks helps to approximate elliptic partial differential equations governed by the physical process. The proposed technique employs learning the behaviour of complex systems based on input-output data, and automatic differentiation ensures accurate computation of gradient. The method computes the singularity-embedded local fractional partial derivative model on a Hausdorff metric, which otherwise halts the computation by available approximating numerical methods. This is possible because the new network is capable of updating the weight associated with loss terms depending on the solution domain and requirement of solution behaviour. The semi-positive definite character of the neural tangent kernel achieves the convergence of gradient-based adaptive neural networks. The importance of hyperparameters, namely the number of neurons and the learning rate, is shown by considering a stationary anomalous diffusion-convection model on a rectangular domain. The proposed method showcases the network’s ability to approximate solutions of various local fractional elliptic partial differential equations with varying fractal parameters.