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

Paulo Akira F. Enabe, Rodrigo Provasi

TL;DR

The paper addresses efficient prediction of displacement fields for one-dimensional Euler–Bernoulli beams across varying materials and geometries by integrating the Virtual Element Method (VEM) with a neural network surrogate. The main approach combines a general-order VEM discretization, including axial effects, with a neural network that processes node-based and material-based inputs through separate sub-networks, trained using Sobolev loss and GradNorm for balanced, physics-informed learning. Key contributions include (i) a general-order VEM formulation for Euler–Bernoulli beams, (ii) a modular neural-architecture that decouples node and material data, (iii) a Sobolev training framework that augments loss with derivatives, and (iv) GradNorm for dynamic loss balancing, demonstrated on a portico geometry with both order-4 and order-5 formulations. The results indicate that the hybrid model can yield accurate displacement predictions with limited data and enable fast inference, suggesting significant potential for real-time or design-optimization workflows in structural mechanics. This work advances hybrid physics-driven and data-driven strategies, offering a scalable surrogate that preserves physics while delivering computational efficiency for beam simulations.

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.

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

TL;DR

The paper addresses efficient prediction of displacement fields for one-dimensional Euler–Bernoulli beams across varying materials and geometries by integrating the Virtual Element Method (VEM) with a neural network surrogate. The main approach combines a general-order VEM discretization, including axial effects, with a neural network that processes node-based and material-based inputs through separate sub-networks, trained using Sobolev loss and GradNorm for balanced, physics-informed learning. Key contributions include (i) a general-order VEM formulation for Euler–Bernoulli beams, (ii) a modular neural-architecture that decouples node and material data, (iii) a Sobolev training framework that augments loss with derivatives, and (iv) GradNorm for dynamic loss balancing, demonstrated on a portico geometry with both order-4 and order-5 formulations. The results indicate that the hybrid model can yield accurate displacement predictions with limited data and enable fast inference, suggesting significant potential for real-time or design-optimization workflows in structural mechanics. This work advances hybrid physics-driven and data-driven strategies, offering a scalable surrogate that preserves physics while delivering computational efficiency for beam simulations.

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.
Paper Structure (13 sections, 2 theorems, 55 equations, 3 figures, 1 algorithm)

This paper contains 13 sections, 2 theorems, 55 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. General-order virtual element formulation for Euler-Bernoulli beams
  3. Virtual element formulation components
  4. Virtual element formulation
  5. Architecture of the neural network
  6. Sobolev training
  7. GradNorm
  8. Training the neural network
  9. Numerical results
  10. A brief discussion about nonlinearity in deep learning
  11. Surrogate model
  12. Conclusion
  13. Functional analysis definitions and results

Key Result

Theorem A.1

Let $g: \mathbb{R} \to \mathbb{R}$ be a continuous, nonconstant activation function. Then, for any compact set $K \subset \mathbb{R}^n$ and any continuous function $f \in C(K)$, the set of functions $\mathcal{G}(g)$, consisting of finite linear combinations of $g$ applied to affine transformations o

Figures (3)

  • Figure 1: Sample figure caption.
  • Figure 2: Portico geometry.
  • Figure 3: $H^1$-error comparison of order 4 and order 5 with standard deviation.

Theorems & Definitions (3)

  • Definition A.1
  • Theorem A.1
  • Theorem A.2: Johnson-Lindenstrauss Lemma