Neural Networks For The Approximation Of Euler's Elastica
Please login to view abstract download link
In this work, the planar Euler's elastica is described by a constrained second-order Lagrangian, and the static equilibrium equations are derived via a variational principle. The deformed configurations of the elastica, obtained under fixed boundary conditions and subject to the inextensibility constraint, are used for the training of neural networks. The supervised training enables the networks to generate approximation of the solutions for previously unseen boundary conditions. In [1], two approaches are proposed for this purpose. We present a discrete approach learning discrete solutions from the discrete data. Based on the continuous nature of the solution curves, we consider a continuous approach which provides a continuous approximation of the solution curves. We present numerical evidence that the proposed neural networks can effectively approximate configurations of the elastica for a range of different boundary conditions. Furthermore, different ways to assure the inextensibility constraint of the predicted trajectories are considered in the continuous approaches. The structure of the problem is preserved by construction of the neural network or incorporated into the loss function. Although the discrete network does not include structural information, results show accurate satisfaction of the inextensibility constraint. While the planar Euler's elastica is a relatively simple beam model, it is relevant in many fields for simulation of industrial applications. Also, as a constrained higher-order Lagrangian system, it represents an interesting study case to investigate through neural networks. The proposed methods can be generalised to more complex beam models or other complex boundary value problems investigating static and dynamic simulations.
