Deep neural network method for solving the fractional Burgers-type equations with conformable derivative

Abstract

In this article, we introduce the modified physics-informed neural network (PINN) method for finding data-driven solutions of three classes of time-fractional Burgers-type equations under the conformable sense. Since conformable derivative satisfies the chain rule, automatic differentiation can be applied to compute it directly to avoid truncation and other numerical discretization. In addition, the locally adaptive activation function and two effective weighting strategies are introduced to improve solution accuracy. As a result, three numerical examples indicate that the modified PINN method gives an efficient and reliable solution.

A novel discretized physics-informed neural network model applied to the Navier–Stokes equations

“…The solution within each subdomain is approximated by the neural network model specific to that subdomain, denoted as uθ i . Therefore, the global solution u θ , which encompasses the entire domain Ω, is obtained by aggregating the solutions from individual subdomains, as shown in equation (15). To enforce interface constraints effectively, the loss function for each subdomain includes terms that penalize deviations from these constraints.…”

“…A notable development is the introduction of automatic differentiation [1], which makes it possible to bypass numerical discretization and mesh generation, leading to a new machine learning-based technique known as PINN [2,3]. Since the introduction of the PINN, it has been widely used to approximate solution to PDE [4][5][6][7][8][9][10][11][12][13][14][15][16], particularly highly nonlinear ones with non-convex and oscillating behavior, such as Navier-Stokes equations, that pose challenges for traditional numerical discretization techniques. The problematic part of Navier-Stokes equations is because of the convective term in the equations introduces a nonlinearity due to the product of velocity components.…”