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

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 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 the feasibility of the methodology