Deep Learning for the Approximation of a Shape Functional

TL;DR

This paper explores the use of convolutional neural networks to approximate the torsional rigidity of planar domains, demonstrating good accuracy and potential for high-dimensional shape functional problems.

Contribution

It introduces a CNN-based approach to approximate shape functionals like torsional rigidity from digital domain images, combining PDE solutions with deep learning.

Findings

CNN accurately predicts torsional rigidity from images

The method respects known properties of the torsion functional

Good performance on classical conjectures

Abstract

Artificial Neuronal Networks are models widely used for many scientific tasks. One of the well-known field of application is the approximation of high-dimensional problems via Deep Learning. In the present paper we investigate the Deep Learning techniques applied to Shape Functionals, and we start from the so--called Torsional Rigidity. Our aim is to feed the Neuronal Network with digital approximations of the planar domains where the Torsion problem (a partial differential equation problem) is defined, and look for a prediction of the value of Torsion. Dealing with images, our choice fell on Convolutional Neural Network (CNN), and we train such a network using reference solutions obtained via Finite Element Method. Then, we tested the network against some well-known properties involving the Torsion as well as an old standing conjecture. In all cases, good approximation properties and accuracies occurred.