Transfer learning based physics–informed neural networks to solve the Allen–Cahn equation on curved surfaces

Abstract Deep learning-based algorithms have recently been utilized to solve several nonlinear partial differential equations (PDEs) on curved surfaces. The mesh-free nature of these algorithms reduces computational complexity, especially for curved surfaces, where generating meshes is significantly challenging. The purpose of this paper is to generate an accurate solution of the highly nonlinear Allen–Cahn equation on various curved surfaces using a deep learning-based algorithm. However, generating a convergent solution of the Allen–Cahn equation for sharp interphase is significantly challenging, especially for curved surfaces, where the curvature of the geometry plays a crucial role in the solution accuracy. Due to the sharp interphase between two layers, the equation becomes very stiff; as a result, instability is a common practice. To mitigate this, we employ a transfer learning-based approach with PINNs to solve the Allen–Cahn equation on curved surfaces. A thorough numerical study is carried out to show the effectiveness of the numerical results presented for various complex closed curved surfaces with several initial conditions. Benchmark examples are provided for some cases to compare against the previous literature, along with a comparison with the analytic solution for the sphere case to test the convergence and accuracy of the present solution.