A Neural Element Method

Abstract

Methods of artificial neural networks (ANNs) have been applied to solve various science and engineering problems. TrumpetNets and TubeNets were recently proposed by the author for creating two-way deepnets using the standard finite element method (FEM) and smoothed FEM (S-FEM) as trainers. The significance of these specially configured ANNs is that the solutions to inverse problems have been, for the first time, analytically derived in explicit formulae. This paper presents a novel neural element method (NEM) with a focus on mechanics problems. The key idea is to use artificial neurons to form elemental units called neural-pulse-units (NPUs), using which the shape functions can then be constructed, and used in the standard weak and weakened-weak (W2) formulations to establish discrete stiffness matrices, similar to the standard FEM and S-FEM. Theory, formulation and codes in Python are presented in detail. Numerical examples are then used to demonstrate this novel NEM. For the first time, we have made a clear connection in theory, formulations and coding, between ANN methods and physical-law-based computational methods. We believe that this novel NEM fundamentally changes the way of approaching mechanics problems, and opens a window of opportunity for a range of applications. It offers a new direction of research on unconventional computational methods. It may also have an impact on how the well-established weak and W2 formulations can be introduced to machine learning processes, for example, creating well-behaved loss functions with preferable convexity.