A Hybrid Virtual Element Method and Deep Learning Approach for Solving One-Dimensional Euler-Bernoulli Beams

TL;DR

This paper introduces a hybrid approach combining Virtual Element Method and deep learning to efficiently predict displacement fields in one-dimensional Euler-Bernoulli beams, aiming to enhance computational efficiency and flexibility.

Contribution

It presents a novel integration of VEM with neural networks, incorporating Sobolev training and GradNorm to improve accuracy and generalization in structural beam modeling.

Findings

Demonstrates potential for data-driven surrogate modeling of beams.

Incorporates advanced training techniques for better generalization.

Lays groundwork for scalable numerical and deep learning methods.

Abstract

A hybrid framework integrating the Virtual Element Method (VEM) with deep learning is presented as an initial step toward developing efficient and flexible numerical models for one-dimensional Euler-Bernoulli beams. The primary aim is to explore a data-driven surrogate model capable of predicting displacement fields across varying material and geometric parameters while maintaining computational efficiency. Building upon VEM's ability to handle higher-order polynomials and non-conforming discretizations, the method offers a robust numerical foundation for structural mechanics. A neural network architecture is introduced to separately process nodal and material-specific data, effectively capturing complex interactions with minimal reliance on large datasets. To address challenges in training, the model incorporates Sobolev training and GradNorm techniques, ensuring balanced loss contributions and enhanced generalization. While this framework is in its early stages, it demonstrates the potential for further refinement and development into a scalable alternative to traditional methods. The proposed approach lays the groundwork for advancing numerical and data-driven techniques in beam modeling, offering a foundation for future research in structural mechanics.