Physics Embedded Neural Network: A Novel Data-free Numerical Method for Solving Computational Physics Problems

A novel approach for solving partial differential equations (PDEs) using neural networks for scientific computing is introduced. The proposed approach, referred to as physics-embedded neural network (PENN), features a unique architecture that incorporates the PDE and boundary conditions information directly within the final fully-connected layer of the feed-forward neural network (NN). The key aspect of PENN is the parallel numerical embedding of a differential equation associated with physical problems within the activation function of the network’s final layer. This integration leads to a new class of computational solvers competitive with classical methods like the Finite Element Method (FEM) and capable of addressing traditionally difficult problems. This direct solution method eliminates classical training and testing phases typical in data-driven physics-based NNs. The effectiveness of PENN is demonstrated through various cases involving second-order inhomogeneous PDEs in 2-Dimensional (2D) problems, including the solution of electromagnetic (EM) wave equations to illustrate its engineering applications. Comparisons with FEM show that PENN achieves significantly lower errors with exponential convergence, outperforming other NN-based methods like Physics Informed Neural Networks. Furthermore, the application of PENN in transfer learning is explored, showing up to five times reduction in simulation time compared to direct PENN solutions. To further extend the capabilities of PENN, the architecture is updated to incorporate B-spline interpolation during trial solution generation, leveraging B-splines' smoothness and local support to improve solution representation. This integration leads to more precise and computationally efficient PDE solutions, particularly beneficial for complex geometries or large domain simulation. Numerical experiments validate this interpolation-based PENN, demonstrating superior convergence speed over the original PENN. Overall, the study advances the development of efficient and accurate methods for computationally solving PDEs through neural network optimization and highlights the potential benefits of incorporating transfer learning techniques in numerical modeling of physical phenomenology.Advisor: Shubhendu Bhardwaj