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Many important problems in science and engineering require solving the so-called parametric partial differential equations (PDEs), i.e., PDEs with different physical parameters, boundary conditions, shapes of computational domains, etc.…

Given a set of solution snapshots of a hyperbolic PDE, we are interested in learning a reduced order model (ROM). To this end, we propose a novel decompose then learn approach. We decompose the solution by expressing it as a composition of…

Variational autoencoders (VAEs) are widely used deep generative models capable of learning unsupervised latent representations of data. Such representations are often difficult to interpret or control. We consider the problem of…

We introduce an information-theoretic framework that uses variational autoencoders (VAEs) to extract compact, physically interpretable manifolds from high-dimensional flow-field data. To this end, the Kullback--Leibler (KL) divergence in…

Motivation: Despite advances in the computational analysis of high-throughput molecular profiling assays (e.g. transcriptomics), a dichotomy exists between methods that are simple and interpretable, and ones that are complex but with lower…

Inferring unknown initial states in shock-dominated compressible flows from sparse and noisy measurements is a challenging ill-posed inverse problem due to nonlinear wave interactions and limited sensing. In this work, we develop a…

We propose a non-intrusive Deep Learning-based Reduced Order Model (DL-ROM) capable of capturing the complex dynamics of mechanical systems showing inertia and geometric nonlinearities. In the first phase, a limited number of high fidelity…

Adapting foundation models for specific purposes has become a standard approach to build machine learning systems for downstream applications. Yet, it is an open question which mechanisms take place during adaptation. Here we develop a new…

With the advancement of neural networks, there has been a notable increase, both in terms of quantity and variety, in research publications concerning the application of autoencoders to reduced-order models. We propose a polytopic…

The modal decomposition techniques of proper orthogonal decomposition (POD) and dynamic mode decomposition (DMD) have become a common method for analysing the spatio-temporal coherence of dynamical systems. In particular, these techniques…

In this paper, we present an in-depth investigation of the convolutional autoencoder (CAE) bottleneck. Autoencoders (AE), and especially their convolutional variants, play a vital role in the current deep learning toolbox. Researchers and…

In this work, a numerical simulation of 1D Burgers' equation is developed using finite difference method and a reduced order model (ROM) of the simulation is developed using proper orthogonal decomposition (POD). The objective of this work…

Analyzing large-scale data from simulations of turbulent flows is memory intensive, requiring significant resources. This major challenge highlights the need for data compression techniques. In this study, we apply a physics-informed Deep…

We present a comparative computational study of two stabilized Reduced Order Models (ROMs) for the simulation of convection-dominated incompressible flow (Reynolds number of the order of a few thousands). Representative solutions in the…

Many physical systems exhibit a low-dimensional structure that varies with external parameters: link lengths in a robot, forcing constants in a fluid, or Reynolds numbers in a flow shift the underlying manifold while preserving its…

Predictive simulations of complex systems are essential for applications ranging from weather forecasting to drug design. The veracity of these predictions hinges on their capacity to capture the effective system dynamics. Massively…

Estimation of riverbed profiles, also known as bathymetry, plays a vital role in many applications, such as safe and efficient inland navigation, prediction of bank erosion, land subsidence, and flood risk management. The high cost and…

We study reduced-order models of three-dimensional perturbations in linearized channel flow using balanced proper orthogonal decomposition (BPOD). The models are obtained from three-dimensional simulations in physical space as opposed to…

A major goal for reduced-order models of unsteady fluid flows is to uncover and exploit latent low-dimensional structure. Proper orthogonal decomposition (POD) provides an energy-optimal linear basis to represent the flow kinematics, but…

End-to-end learning of dynamical systems with black-box models, such as neural ordinary differential equations (ODEs), provides a flexible framework for learning dynamics from data without prescribing a mathematical model for the dynamics.…