Physics-Informed Neural Networks for 1-D Steady-State Diffusion-Advection-Reaction Equations

This work aims to solve six problems with four different physics-informed machine learning frameworks and compare the results in terms of accuracy and computational cost. First, we considered the diffusion-advection-reaction equations, which are second-order linear differential equations with two boundary conditions. The first algorithm is the classic Physics-Informed Neural Networks. The second one is Physics-Informed Extreme Learning Machine. The third framework is Deep Theory of Functional Connections, a multilayer neural network based on the solution approximation via a constrained expression that always analytically satisfies the boundary conditions. The last algorithm is the Extreme Theory of Functional Connections (X-TFC), which combines Theory of Functional Connections and shallow neural network with random features [e.g., Extreme Learning Machine (ELM)]. The results show that for these kinds of problems, ELM-based frameworks, especially X-TFC, overcome those using deep neural networks both in terms of accuracy and computational time.