Physics-informed Neural Networks for Solving Second-order Boundary Value Problems Comparison with FEM, FD Methods

Recently, physics-informed neural networks (PINNs) have become an encouraging computational approach to solving differential equations through the use of an explicit encoding of physical laws into the learning step of neural networks. The paper carries out a detailed comparison and contrast of PINNs with two well-established numerical approaches, i.e., Finite Element Method (FEM) and Finite Difference (FD) method in terms of solving second-order, boundary value problems. It is assumed to be a representative benchmark problem defined over a bounded domain with given boundary conditions, and to which an analytical solution exists to evaluate the accuracy of the numerical methods. The suggested PINN framework is built as a feedforward neural network framework that has a trial solution strategy that provides a natural way to meet the boundary conditions. Automatic differentiation is used to calculate necessary derivatives effectively and precisely so as not to require numerical approximation schemes. In the training, the L-BFGS optimization algorithm is used along with the Sobol quasi-random collocation points to guarantee efficient sampling of the computational domain and enhance the convergence characteristics. Moreover, mathematical underpinnings of the PINN formulation, as well as loss function construction and training mechanisms are addressed and compared with the respective formulations in FEM and FD methods. A large number of numerical experiments are performed to test the performance of the three methods in terms of accuracy, convergence properties, and computational efficiency. The findings reveal that PINNs are as accurate as classical numerical methods with a number of benefits, including mesh-free nature, ability to work with complex domains, and the natural implementation of physical constraints. Whereas FEM and FD approaches are more efficient when it comes to solving low-dimensional problems, PINNs offer a more general framework that can be applied to more complicated cases and the higher dimensionality. In general, in this paper, the promise of physics-informed learning as a well-built and versatile substitute to the conventional numerical approaches is emphasized. The results can be added to the existing literature on the relevance of PINNs to computational physics and engineering, especially those problems where traditional methods are limited by geometric complexity or data integration needs.