Non-intrusive Surrogate Modeling for Parametrized Time-dependent PDEs using Convolutional Autoencoders

Abstract: 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.