Enhanced Solution for the Advection–Diffusion–Reaction Equation Using the Physics-Informed Neural Network Technique

This study focuses on the use of Physics-Informed Neural Networks (PINNs) to solve the 1D Advection–Diffusion–Reaction (ADR) equation. The performance of the PINN model is evaluated in comparison with the classical Crank–Nicolson Finite Difference Method (CNFDM) and validated against analytical solutions to assess improvements in accuracy, robustness, and flexibility. Quantitative analysis reveals that the PINN achieved a high level of accuracy with absolute errors ranging from approximately 2.13×10−4 to 1.17×10−3 across the spatial domain. The study utilizes a neural network architecture with two hidden layers of 80 neurons each, optimized through a two-stage training process involving Adam and L-BFGS optimizers. This work contributes to the growing field of physics-informed machine learning by demonstrating the strengths and quantitative reliability of the PINN technique for solving complex partial differential equations in transport phenomena.