A hybrid MGA-MSGD ANN training approach for approximate solution of linear elliptic PDEs

by   Hamidreza Dehghani, et al.

We introduce a hybrid "Modified Genetic Algorithm-Multilevel Stochastic Gradient Descent" (MGA-MSGD) training algorithm that considerably improves accuracy and efficiency of solving 3D mechanical problems described, in strong-form, by PDEs via ANNs (Artificial Neural Networks). This presented approach allows the selection of a number of locations of interest at which the state variables are expected to fulfil the governing equations associated with a physical problem. Unlike classical PDE approximation methods such as finite differences or the finite element method, there is no need to establish and reconstruct the physical field quantity throughout the computational domain in order to predict the mechanical response at specific locations of interest. The basic idea of MGA-MSGD is the manipulation of the learnable parameters' components responsible for the error explosion so that we can train the network with relatively larger learning rates which avoids trapping in local minima. The proposed training approach is less sensitive to the learning rate value, training points density and distribution, and the random initial parameters. The distance function to minimise is where we introduce the PDEs including any physical laws and conditions (so-called, Physics Informed ANN). The Genetic algorithm is modified to be suitable for this type of ANN in which a Coarse-level Stochastic Gradient Descent (CSGD) is exploited to make the decision of the offspring qualification. Employing the presented approach, a considerable improvement in both accuracy and efficiency, compared with standard training algorithms such as classical SGD and Adam optimiser, is observed. The local displacement accuracy is studied and ensured by introducing the results of Finite Element Method (FEM) at sufficiently fine mesh as the reference displacements. A slightly more complex problem is solved ensuring its feasibility.



There are no comments yet.


page 21

page 23

page 24

page 25


Hybrid FEM-NN models: Combining artificial neural networks with the finite element method

We present a methodology combining neural networks with physical princip...

MeshingNet: A New Mesh Generation Method based on Deep Learning

We introduce a novel approach to automatic unstructured mesh generation ...

H^1-norm error estimate for a nonstandard finite element approximation of second-order linear elliptic PDEs in non-divergence form

This paper establishes the optimal H^1-norm error estimate for a nonstan...

A Machine Learning approach to enhance the SUPG stabilization method for advection-dominated differential problems

We propose using Machine Learning and Artificial Neural Networks (ANN) t...

Evolving Neural Networks in Reinforcement Learning by means of UMDAc

Neural networks are gaining popularity in the reinforcement learning fie...

A comparison of reduced-order modeling approaches for PDEs with bifurcating solutions

This paper focuses on reduced-order models (ROMs) built for the efficien...
This week in AI

Get the week's most popular data science and artificial intelligence research sent straight to your inbox every Saturday.