  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.
