Data-Augmented Predictive Deep Neural Network: Enhancing the extrapolation capabilities of non-intrusive surrogate models

Numerically solving a large parametric nonlinear dynamical system is challenging due to its high complexity and the high computational costs. In recent years, machine-learning-aided surrogates are being actively researched. However, many methods fail in accurately generalizing in the entire time interval $[0, T]$, when the training data is available only in a training time interval $[0, T_0]$, with $T_0<T$. To improve the extrapolation capabilities of the surrogate models in the entire time domain, we propose a new deep learning framework, where kernel dynamic mode decomposition (KDMD) is employed to evolve the dynamics of the latent space generated by the encoder part of a convolutional autoencoder (CAE). After adding the KDMD-decoder-extrapolated data into the original data set, we train the CAE along with a feed-forward deep neural network using the augmented data. The trained network can predict future states outside the training time interval at any out-of-training parameter samples. The proposed method is tested on two numerical examples: a FitzHugh-Nagumo model and a model of incompressible flow past a cylinder. Numerical results show accurate and fast prediction performance in both the time and the parameter domain.

Active-Learning-Driven Surrogate Modeling for Efficient Simulation of Parametric Nonlinear Systems

When repeated evaluations for varying parameter configurations of a high-fidelity physical model are required, surrogate modeling techniques based on model order reduction are desired. In absence of the governing equations describing the dynamics, we need to construct the parametric reduced-order surrogate model in a non-intrusive fashion. In this setting, the usual residual-based error estimate for optimal parameter sampling associated with the reduced basis method is not directly available. Our work provides a non-intrusive optimality criterion to efficiently populate the parameter snapshots, thereby, enabling us to effectively construct a parametric surrogate model. We consider separate parameter-specific proper orthogonal decomposition (POD) subspaces and propose an active-learning-driven surrogate model using kernel-based shallow neural networks, abbreviated as ActLearn-POD-KSNN surrogate model. To demonstrate the validity of our proposed ideas, we present numerical experiments using two physical models, namely Burgers' equation and shallow water equations. Both the models have mixed -- convective and diffusive -- effects within their respective parameter domains, with each of them dominating in certain regions. The proposed ActLearn-POD-KSNN surrogate model efficiently predicts the solution at new parameter locations, even for a setting with multiple interacting shock profiles.