# Deep RayUniversity of Southern California | USC · Department of Aerospace and Mechanical Engineering

Deep Ray

Doctor of Philosophy

## About

33

Publications

7,046

Reads

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600

Citations

Introduction

My research is focused on designing robust methods for solving fluid flow problems. My primary interests are: i) integrating data-driven tools to enhance existing methods in scientific computing; ii) designing high-order numerical methods for hyperbolic conservation laws, which provably satisfy physical constraints; iii) uncertainty quantification; iv) design of schemes for two-phase flows through porous media.
NOTE: For pre-prints unavailable here, please visit: deepray.github.io

Additional affiliations

July 2017 - June 2019

Position

- PostDoc Position

Description

- My research involved the use of deep learning tools to enhance the performance existing numerical frameworks. Special focus was given to develop artificial networks capable of detecting and controlling spurious oscillations generated by high-order methods used to approximate discontinuous solutions.

Education

July 2012 - July 2017

**Tata Institute of Fundamental Research (CAM)**

Field of study

- Mathematics

July 2010 - June 2012

**Tata Institute of Fundamental Research (CAM)**

Field of study

- Mathematics

July 2007 - June 2010

**Hindu College - University of Delhi**

Field of study

- Mathematics

## Publications

Publications (33)

The ability to impute missing images from a sequence of medical images plays an important role in enabling the detection, diagnosis and treatment of disease. Motivated by this, in this manuscript we propose a novel, probabilistic deep-learning algorithm for imputing images. Within this approach, given a sequence of contrast enhanced CT images, we t...

The essentially non-oscillatory (ENO) procedure and its variant, the ENO-SR procedure, are very efficient algorithms for interpolating (reconstructing) rough functions. We prove that the ENO (and ENO-SR) procedure are equivalent to deep ReLU neural networks. This demonstrates the ability of deep ReLU neural networks to approximate rough functions t...

Brain extraction, which refers to the task of segmenting brain tissue in an MR image of a subject, forms an essential first step for many quantitative neuroimaging applications. These include quantifying grey and white matter volumes, monitoring neurological diseases like multiple sclerosis (MS) and Alzheimer's disease, and estimating brain atrophy...

In this work, we train conditional Wasserstein generative adversarial networks to effectively sample from the posterior of physics-based Bayesian inference problems. The generator is constructed using a U-Net architecture, with the latent information injected using conditional instance normalization. The former facilitates a multiscale inverse map,...

A numerical method using discontinuous polynomial approximations is formulated for solving a phase-field model of two immiscible fluids with a soluble surfactant. The proposed scheme is shown to decay the total free Helmholtz energy at the discrete level, which is consistent with the continuous model dynamics. The scheme recovers the Langmuir adsor...

In this paper, we present an efficient numerical algorithm for solving the time-dependent Cahn–Hilliard–Navier–Stokes equations that model the flow of two phases with different densities. The pressure-correction step in the projection method consists of a Poisson problem with a modified right-hand side. Spatial discretization is based on discontinu...

Inverse problems are notoriously difficult to solve because they can have no solutions, multiple solutions, or have solutions that vary significantly in response to small perturbations in measurements. Bayesian inference, which poses an inverse problem as a stochastic inference problem, addresses these difficulties and provides quantitative estimat...

We present a novel active learning algorithm, termed as iterative surrogate model optimization (ISMO), for robust and efficient numerical approximation of PDE constrained optimization problems. This algorithm is based on deep neural networks and its key feature is the iterative selection of training data through a feedback loop between deep neural...

While a nonlinear viscosity is used widely to control oscillations when solving conservation laws using high-order elements based methods, such techniques are less straightforward to apply in global spectral methods as a local estimate of solution regularity generally is required. In this work we demonstrate how to train and use a local artificial...

Generative adversarial networks (GANs) have found multiple applications in the solution of inverse problems in science and engineering. These applications are driven by the ability of these networks to learn complex distributions and to map the original feature space to a low-dimensional latent space. In this manuscript we consider the use of GANs...

In this paper, we present an efficient numerical algorithm for solving the time-dependent Cahn--Hilliard--Navier--Stokes equations that model the flow of two phases with different densities. The pressure-correction step in the projection method consists of a Poisson problem with a modified right-hand side. Spatial discretization is based on discont...

A numerical method using discontinuous polynomial approximations is formulated for solving a phase-field model of two immiscible fluids with a soluble surfactant. The scheme recovers the Langmuir adsorption isotherms at equilibrium. Simulations of spinodal decomposition, flow through a cylinder and flow through a sequence of pore throats show the d...

We establish a notion of random entropy solution for degenerate fractional conservation laws incorporating randomness in the initial data, convective flux and diffusive flux. In order to quantify the solution uncertainty, we design a multi-level Monte Carlo Finite Difference Method (MLMC-FDM) to approximate the ensemble average of the random entrop...

We present a novel active learning algorithm, termed as iterative surrogate model optimization (ISMO), for robust and efficient numerical approximation of PDE constrained optimization problems. This algorithm is based on deep neural networks and its key feature is the iterative selection of training data through a feedback loop between deep neural...

While a nonlinear viscosity is used widely to control oscillations when solving conservation laws using high-order elements based methods, such techniques are less straightforward to apply in global spectral methods as a local estimate of solution regularity generally is required. In this work we demonstrate how to train and use a local artificial...

Many large scale problems in computational fluid dynamics such as uncertainty quantification, Bayesian inversion, data assimilation and PDE constrained optimization are considered very challenging computationally as they require a large number of expensive (forward) numerical solutions of the corresponding PDEs. We propose a machine learning algori...

High-order numerical solvers for conservation laws suffer from Gibbs phenomenon close to discontinuities, leading to spurious oscillations and a detrimental effect on the solution accuracy. A possible strategy to reduce it comprises adding a suitable amount of artificial dissipation. Although several viscosity models have been proposed in the liter...

Neural networks are increasingly used in complex (data-driven) simulations as surrogates or for accelerating the computation of classical surrogates. In many applications physical constraints, such as mass or energy conservation, must be satisfied to obtain reliable results. However, standard machine learning algorithms are generally not tailored t...

Deep neural networks and the ENO procedure are both efficient frameworks for approximating rough functions. We prove that at any order, the ENO interpolation procedure can be cast as a deep ReLU neural network. This surprising fact enables the transfer of several desirable properties of the ENO procedure to deep neural networks, including its high-...

In a recent paper (Ray and Hesthaven, 2018) [38], we proposed a new type of troubled-cell indicator to detect discontinuities in the numerical solutions of one-dimensional conservation laws. This was achieved by suitably training an artificial neural network on canonical local solution structures for conservation laws. The proposed indicator was in...

A non-intrusive reduced-basis (RB) method is proposed for parametrized unsteady flows. A set of reduced basis functions are extracted from a collection of high-fidelity solutions via a proper orthogonal decomposition (POD), and the coefficients of the reduced basis functions are recovered by a feedforward neural network (NN). As a regression model...

Neural networks are increasingly used in complex (data-driven) simulations as surrogates or for accelerating the computation of classical surrogates. In many applications physical constraints, such as mass or energy conservation, must be satisfied to obtain reliable results. However, standard machine learning algorithms are generally not tailored t...

Many large scale problems in computational fluid dynamics such as uncertainty quantification, Bayesian inversion, data assimilation and PDE constrained optimization are considered very challenging computationally as they require a large number of expensive (forward) numerical solutions of the corresponding PDEs. We propose a machine learning algori...

A third-order WENO reconstruction has been recently proposed (Fjordholm and Ray, J Sci Comput, 68(1):42–63, 2016, [5]) in the context of finite difference schemes for conservation laws and tested for scalar conservation laws. The method, which is called SP-WENO, satisfies the sign property required for constructing high-order finite difference sche...

A non-intrusive reduced-basis (RB) method is proposed for parametrized unsteady flows. A set of reduced basis functions are extracted from a collection of high-fidelity solutions via a proper orthogonal decomposition (POD), and the coefficients of the reduced basis functions are recovered by a feedforward neural network (NN). As a regression model...

High-resolution schemes for conservation laws need to suitably limit the numerical solution near discontinuities, in order to avoid Gibbs oscillations. The solution quality and the computational cost of such schemes strongly depend on their ability to correctly identify troubled-cells, namely, cells where the solution loses regularity. Motivated by...

We construct a finite volume scheme for the compressible Navier–Stokes equations on triangular grids which are entropy stable at the semi-discrete level. This is achieved by using entropy stable inviscid fluxes constructed in the recently published work titled Entropy Stable Scheme on Two-Dimensional Unstructured Grids for Euler Equations by Ray, C...

High-resolution schemes for conservation laws need to suitably limit the numerical solution near discontinuities, in order to avoid Gibbs oscillations. The solution quality and the computational cost of such schemes strongly depend on their ability to correctly identify troubled-cells, namely, cells where the solution loses regularity. Motivated by...

We propose a third-order WENO reconstruction which satisfies the sign
property, required for constructing high resolution entropy stable finite
difference scheme for conservation laws. The reconstruction technique, which is
termed as SP-WENO, is endowed with additional properties making it a more
robust option compared to ENO schemes of the same or...

We propose an entropy stable high-resolution finite volume scheme to approximate systems of two-dimensional symmetrizable conservation laws on unstructured grids. In particular we consider Euler equations governing compressible flows. The scheme is constructed using a combination of entropy conservative fluxes and entropy-stable numerical dissipati...

We are concerned with fully-discrete schemes for the numerical approximation
of diffusive-dispersive hyperbolic conservation laws with a discontinuous flux
function in one-space dimension. More precisely, we show the convergence of
approximate solutions, generated by the scheme corresponding to vanishing
diffusive-dispersive scalar conservation law...