Abstract

A deep neural network comprises multiple interconnected layers of neurons, where information flows sequentially from the input layer through various hidden layers. Deep neural networks learn transformations that define relationships between input and output data by minimizing a cost functional, iteratively improving predictions to meet specified constraints. To identify dynamic systems from input-output data, deep neural networks have been employed to approximate integro-differential operators, predict responses to new inputs, or infer equation parameters based on output data (inverse problems); these architectures, known as physics-informed neural networks. Physics-informed neural networks are widely applied to solve integro-differential equations due to several advantages: (i) they eliminate the need for fine mesh discretization, enabling efficient computation in complex geometries; (ii) they accommodate unconventional boundary and initial conditions, reflecting real-world scenarios more accurately; (iii) they derive solutions from limited or noisy data, making them suitable for problems with scarce experimental information; (iv) they handle multiple coupled differential equations; (v) they enable control over differential equations by specifying expected or target solutions; and (vi) they can identify governing equations by learning physical parameters directly from data. This work reviews various deep neural network architectures that have been implemented and validated for solving parabolic-elliptic differential equation systems. © Published under licence by IOP Publishing Ltd.