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High-fidelity direct numerical simulation of turbulent flows for most real-world applications remains an outstanding computational challenge. Several machine learning approaches have recently been proposed to alleviate the computational…

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…

There exists an increasing interest for using immersed boundary methods (IBMs) (Peskin 2000) to model moving objects in computational fluid dynamics. Indeed, this approach is particularly efficient, because the fluid mesh does not require…

We consider the approximation of initial/boundary value problems involving, possibly high-dimensional, dissipative evolution partial differential equations (PDEs) using a deep neural network framework. More specifically, we first propose…

The numerical simulation and optimization of technical systems described by partial differential equations is expensive, especially in multi-query scenarios in which the underlying equations have to be solved for different parameters. A…

Quantum dynamics, typically expressed in the form of a time-dependent Schr\"odinger equation with a Hermitian Hamiltonian, is a natural application for quantum computing. However, when simulating quantum dynamics that involves the emission…

In this note, we revisit the problem of flow approximation properties of neural ordinary differential equations (NODEs). The approximation properties have been considered as a flow controllability problem in recent literature. The neural…

Although neural networks are widely used, it remains challenging to formally verify the safety and robustness of neural networks in real-world applications. Existing methods are designed to verify the network before deployment, which are…

We investigate conditional neural fields (CNFs), mesh-agnostic, coordinate-based decoders conditioned on a low-dimensional latent, for spatial dimensionality reduction of turbulent flows. CNFs are benchmarked against Proper Orthogonal…

The paper presents numerical methods for unsteady flows of a viscous incompressible fluid in internal domains with many inlet/outlet sections. The novel variants of dissipative boundary conditions augmented by the inertia terms are used at…

In this work, the Immersed Boundary Method (IBM) with feedback forcing introduced by Goldstein et al. (1993) and often referred in the literature as the Virtual Boundary Method (VBM), is addressed. The VBM has been extensively applied both…

We present a simple modification of the direct-forcing immersed boundary method (IBM) proposed by Uhlmann [J. Comput. Phys, 2005] in order to enable it to be applied to particulate flows with solid-to-fluid density ratios around unity. The…

Modeling complex systems using standard neural ordinary differential equations (NODEs) often faces some essential challenges, including high computational costs and susceptibility to local optima. To address these challenges, we propose a…

Advances in visual navigation methods have led to intelligent embodied navigation agents capable of learning meaningful representations from raw RGB images and perform a wide variety of tasks involving structural and semantic reasoning.…

Boundary Vector Cells (BVCs) are a class of neurons in the brains of vertebrates that encode environmental boundaries at specific distances and allocentric directions, playing a central role in forming place fields in the hippocampus. Most…

Computational Fluid Dynamics (CFD) simulation by the numerical solution of the Navier-Stokes equations is an essential tool in a wide range of applications from engineering design to climate modeling. However, the computational cost and…

We investigate uncertainty estimation and multimodality via the non-deterministic predictions of Bayesian neural networks (BNNs) in fluid simulations. To this end, we deploy BNNs in three challenging experimental test-cases of increasing…

The rigorous justification of the hydrodynamic limits of kinetic equations in bounded domains has been actively investigated in recent years. In spite of the progress for the diffuse-reflection boundary case, the more challenging in-flow…

Neural ordinary differential equations (NODE) have been proposed as a continuous depth generalization to popular deep learning models such as Residual networks (ResNets). They provide parameter efficiency and automate the model selection…

We derive new boundary conditions and implementation procedures for nonlinear initial boundary value problems that lead to energy and entropy bounded solutions. A step-by-step procedure for general nonlinear hyperbolic problems on…