Harshit Kapadia

Harshit Kapadia, Lihong Feng, Peter Benner

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.

Ali Forootani, Harshit Kapadia, Sridhar Chellappa et al.

The sparse identification of nonlinear dynamical systems (SINDy) is a data-driven technique employed for uncovering and representing the fundamental dynamics of intricate systems based on observational data. However, a primary obstacle in the discovery of models for nonlinear partial differential equations (PDEs) lies in addressing the challenges posed by the curse of dimensionality and large datasets. Consequently, the strategic selection of the most informative samples within a given dataset plays a crucial role in reducing computational costs and enhancing the effectiveness of SINDy-based algorithms. To this aim, we employ a greedy sampling approach to the snapshot matrix of a PDE to obtain its valuable samples, which are suitable to train a deep neural network (DNN) in a SINDy framework. SINDy based algorithms often consist of a data collection unit, constructing a dictionary of basis functions, computing the time derivative, and solving a sparse identification problem which ends to regularised least squares minimization. In this paper, we extend the results of a SINDy based deep learning model discovery (DeePyMoD) approach by integrating greedy sampling technique in its data collection unit and new sparsity promoting algorithms in the least squares minimization unit. In this regard we introduce the greedy sampling neural network in sparse identification of nonlinear partial differential equations (GN-SINDy) which blends a greedy sampling method, the DNN, and the SINDy algorithm. In the implementation phase, to show the effectiveness of GN-SINDy, we compare its results with DeePyMoD by using a Python package that is prepared for this purpose on numerous PDE discovery

Harshit Kapadia, Peter Benner, Lihong Feng

In situations where the solution of a high-fidelity dynamical system needs to be evaluated repeatedly, over a vast pool of parametric configurations and in absence of access to the underlying governing equations, data-driven model reduction techniques are preferable. We propose a novel active learning approach to build a parametric data-driven reduced-order model (ROM) by greedily picking the most important parameter samples from the parameter domain. As a result, during the ROM construction phase, the number of high-fidelity solutions dynamically grow in a principled fashion. The high-fidelity solution snapshots are expressed in several parameter-specific linear subspaces, with the help of proper orthogonal decomposition (POD), and the relative distance between these subspaces is used as a guiding mechanism to perform active learning. For successfully achieving this, we provide a distance measure to evaluate the similarity between pairs of linear subspaces with different dimensions, and also show that this distance measure is a metric. The usability of the proposed subspace-distance-enabled active learning (SDE-AL) framework is demonstrated by augmenting two existing non-intrusive reduced-order modeling approaches, and providing their active-learning-driven (ActLearn) extensions, namely, SDE-ActLearn-POD-KSNN, and SDE-ActLearn-POD-NN. Furthermore, we report positive results for two parametric physical models, highlighting the efficiency of the proposed SDE-AL approach.