What are the computational limitations of neural networks in FEM?

I started to think the computational power of neural networks and why they have not been applied widely in non-adaptive finite element methods (FEM). In industry, I have noticed that most systems use non-adaptive FEMs. I would like to understand what benefit neural networks could provide in such an application. I am particularly interested in differential computation.

How can you evaluate the pros and cons of neural networks in FEM?

A good place to start would be to conduct a literature review.

• Ramuhalli, L. Udpa, S.S. Udpa SS. "Finite-element neural networks for solving differential equations." P. IEEE Trans Neural Netw. 2005 Nov;16(6):1381-92.

The solution of partial differential equations (PDE) arises in a wide variety of engineering problems. Solutions to most practical problems use numerical analysis techniques such as finite-element or finite-difference methods. The drawbacks of these approaches include computational costs associated with the modeling of complex geometries. This paper proposes a finite-element neural network (FENN) obtained by embedding a finite-element model in a neural network architecture that enables fast and accurate solution of the forward problem. Results of applying the FENN to several simple electromagnetic forward and inverse problems are presented. Initial results indicate that the FENN performance as a forward model is comparable to that of the conventional finite-element method (FEM). The FENN can also be used in an iterative approach to solve inverse problems associated with the PDE. Results showing the ability of the FENN to solve the inverse problem given the measured signal are also presented. The parallel nature of the FENN also makes it an attractive solution for parallel implementation in hardware and software.

• "Neural network for constitutive modelling in finite element analysis" A. A. Javadi, T. P. Tan

Finite element method has, in recent years, been widely used as a powerful tool in analysis of engineering problems. In this numerical analysis, the behavior of the actual material is approximated with that of an idealized material that deforms in accordance with some constitutive relationships. Therefore, the choice of an appropriate constitutive model, which adequately describes the behavior of the material, plays a significant role in the accuracy and reliability of the numerical predictions. Several constitutive models have been developed for various materials. Most of these models involve determination of material parameters, many of which have no physical meaning [1, 2].

In this paper a neural network-based finite element analysis will be presented for modeling engineering problems. The methodology involves incorporation of neural network in a finite element program as a substitute to conventional constitutive material model. Capabilities of the presented methodology will be illustrated by application to practical engineering problems. The results of the analyses will be compared to those obtained from conventional constitutive models.

• Jun Takeuchi & Yukio Kosugi, "Neural network representation of finite element method" Neural Networks Volume 7, Issue 2, 1994, Pages 389-395

A finite element method (FEM)-based neural network for boundary value problems is considered. The neural network consists of node-units and element-subnets whose synaptic weights are predetermined using FEM's formulation procedure. Using the network inversion technique, unknown inputs of the network are updated to satisfy both the governing law and the boundary conditions, and consequently reach the solution of the problem. We applied the network to the electric field problem governed by Poisson's equation. The numerical simulation shows the validity of this network.