  This work presents a non-intrusive surrogate modeling scheme based on machine
learning technology for predictive modeling of complex systems, described by
parametrized time-dependent PDEs. For these problems, typical finite element
approaches involve the spatiotemporal discretization of the PDE and the
solution of the corresponding linear system of equations at each time step.
Instead, the proposed method utilizes a convolutional autoencoder in
conjunction with a feed forward neural network to establish a low-cost and
accurate mapping from the problem's parametric space to its solution space. For
this purpose, time history response data are collected by solving the
high-fidelity model via FEM for a reduced set of parameter values. Then, by
applying the convolutional autoencoder to this data set, a low-dimensional
representation of the high-dimensional solution matrices is provided by the
encoder, while the reconstruction map is obtained by the decoder. Using the
latent representation given by the encoder, a feed-forward neural network is
efficiently trained to map points from the problem's parametric space to the
compressed version of the respective solution matrices. This way, the encoded
response of the system at new parameter values is given by the neural network,
while the entire response is delivered by the decoder. This approach
effectively bypasses the need to serially formulate and solve the system's
governing equations at each time increment, thus resulting in a significant
cost reduction and rendering the method ideal for problems requiring repeated
model evaluations or 'real-time' computations. The elaborated methodology is
demonstrated on the stochastic analysis of time-dependent PDEs solved with the
Monte Carlo method, however, it can be straightforwardly applied to other
similar-type problems, such as sensitivity analysis, design optimization, etc.
