Solving Diffusion Equations Using Physics-Informed Neural Networks: A Biological Application

A large percentage of processes in the fields of physics, chemistry, biology, economics, and sociology are expressed mathematically by partial differential equations (PDEs). Solving PDEs is often computationally expensive and challenging, particularly for equations with unknown parameters. Traditional methods, such as finite difference or finite element methods, rely on discretization techniques that can become computationally intensive for high-dimensional problems or require detailed knowledge of the unknown parameters. Recently, there has been an increasing interest in the study of deep learning techniques and has recently attracted more attention for solving PDEs. Physics-informed neural networks (PINNs) have become a strong framework among them. An efficient machine learning framework which includes the physical rules controlling a system into neural network training is known as PINNs. Due to this framework, PINNs can solve PDEs effectively without requiring traditional discretization methods. The Fisher-Kolmogorov-Petrovsky-Piscounov (Fisher-KPP) equation is the main subject of this investigation. It was first put up as a model for the transmission of a beneficial gene in a population in the ۱۹۳۰s. In terms of mathematics, it belongs to the broad category of reaction-diffusion equations. We solve Fisher-KPP using PINNs using deep learning. The neural network design is defined by three hidden layers, each of which has ۲۰ neurons and the ReLU activation function. The model is trained using the Adam optimization technique, and the optimal model is obtained at step ۱۰,۰۰۰: ۱.۱۱e-۰۱ for train loss and ۱.۱۹e-۰۱ for test loss. This study serves as a first step toward a deeper understanding of the topic under study and establishes the foundation for future research in this field.

کلیدواژه ها:

partial differential equations ، biology ، deep learning ، physics-informed neural network ، Fisher-Kolmogorov-Petrovsky-Piscounov