Abstract:
Machine learning methods [1], such as the Artificial Neural Networks (ANNs), have been applied to solve various science and engineering problems. TrumpetNets and TubeNets were recently proposed by the author [2][3] for creating two-way deepnets using the standard finite element method (FEM) [4] and S-FEM [5][6]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. Such advancements have shown that fundamental understanding on physics law-based and data-based methods has found critical for development of novel methods for various types of engineering problems [9][10][12].
This paper discusses general issues related to law-based [4]-[8] and data-based [1][3] methods, including the principle, procedure, predictability, and property of these two types of methods in dealing with different types of problems. We present also a novel neural element method (NEM) [9][10] as a typical example of a possible combination of these two types of methods. The key idea in NEM is to use artificial neurons to form elemental units called neural-pulse-units (NPUs), using which 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. Detailed theory, formulation and codes in Python and numerical examples are presented to demonstrate this NEM. For the first time, we have made a clear connection, (in theory, formulation, and coding), between ANN methods and physical-law-based computational methods. We believe that his novel NEM changes fundamentally the way approaching mechanics problems, and opens a possible new window of opportunity for a range of applications. It offers a new direction of research on un-conventional computational methods. It may also have an impact on how the well-established weak and W2 formulations can be introduced to machining learning processes, for example, creating well-behaved loss functions with preferable convexity.
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