arXiv subjects

Alexandre M. Tartakovsky

Publications and source records attributed to Alexandre M. Tartakovsky.

At least 37 records · Page 2Linked to original sources

arXiv2020

Non-local model for surface tension in fluid-fluid simulations

We propose a non-local model for surface tension obtained in the form of an integral of a molecular-force-like function with support $3.5\varepsilon$ added to the Navier-Stokes momentum conservation equation. We demonstrate analytically and numerically that with the non-local model interfaces with a radius of curvature larger than the support length behave macroscopically and microscopically, otherwise. For static droplets, the pressure difference $P_{\varepsilon, in} - P_{\varepsilon, out}$ satisfies the Young-Laplace law for droplet radius greater than $3.5\varepsilon$ and otherwise deviates from the Young-Laplace law. The latter indicates that the surface tension in the proposed model decreases with decreasing radius of curvature, which agrees with molecular dynamics and experimental studies of nanodroplets. Using the non-local model we perform numerical simulations of droplets under dynamic conditions, including a rising droplet, a droplet in shear flow, and two colliding droplets in shear flow, and compare results with a standard Navier-Stokes model subject to the Young-Laplace boundary condition at the fluid-fluid interface implemented via the Conservative Level Set (CLS) method. We find good agreement with existing numerical methods and analytical results for a rising macroscopic droplet and a droplet in a shear flow. For colliding droplets in shear flow, the non-local model converges (with respect to the grid size) to the correct behavior, including sliding, coalescence, and merging and breaking of two droplets depending on the capillary number. In contrast, we find that the results of the CLS model are highly grid-size dependent.

physics.flu-dyn↗

arXiv2020

Inverse Modeling of Viscoelasticity Materials using Physics Constrained Learning

We propose a novel approach to model viscoelasticity materials using neural networks, which capture rate-dependent and nonlinear constitutive relations. However, inputs and outputs of the neural networks are not directly observable, and therefore common training techniques with input-output pairs for the neural networks are inapplicable. To that end, we develop a novel computational approach to both calibrate parametric and learn neural-network-based constitutive relations of viscoelasticity materials from indirect displacement data in the context of multi-physics interactions. We show that limited displacement data hold sufficient information to quantify the viscoelasticity behavior. We formulate the inverse computation---modeling viscoelasticity properties from observed displacement data---as a PDE-constrained optimization problem and minimize the error functional using a gradient-based optimization method. The gradients are computed by a combination of automatic differentiation and physics constrained learning. The effectiveness of our method is demonstrated through numerous benchmark problems in geomechanics and porous media transport.

math.NA↗

arXiv2019

Physics-Informed Neural Networks for Multiphysics Data Assimilation with Application to Subsurface Transport

Data assimilation for parameter and state estimation in subsurface transport problems remains a significant challenge due to the sparsity of measurements, the heterogeneity of porous media, and the high computational cost of forward numerical models. We present a physics-informed deep neural networks (DNNs) machine learning method for estimating space-dependent hydraulic conductivity, hydraulic head, and concentration fields from sparse measurements. In this approach, we employ individual DNNs to approximate the unknown parameters (e.g., hydraulic conductivity) and states (e.g., hydraulic head and concentration) of a physical system, and jointly train these DNNs by minimizing the loss function that consists of the governing equations residuals in addition to the error with respect to measurement data. We apply this approach to assimilate conductivity, hydraulic head, and concentration measurements for joint inversion of the conductivity, hydraulic head, and concentration fields in a steady-state advection--dispersion problem. We study the accuracy of the physics-informed DNN approach with respect to data size, number of variables (conductivity and head versus conductivity, head, and concentration), DNNs size, and DNN initialization during training. We demonstrate that the physics-informed DNNs are significantly more accurate than standard data-driven DNNs when the training set consists of sparse data. We also show that the accuracy of parameter estimation increases as additional variables are inverted jointly.

cs.LG↗

arXiv2019

Physics-Informed Machine Learning with Conditional Karhunen-Loève Expansions

We present a new physics-informed machine learning approach for the inversion of PDE models with heterogeneous parameters. In our approach, the space-dependent partially-observed parameters and states are approximated via Karhunen-Loève expansions (KLEs). Each of these KLEs is then conditioned on their corresponding measurements, resulting in low-dimensional models of the parameters and states that resolve observed data. Finally, the coefficients of the KLEs are estimated by minimizing the norm of the residual of the PDE model evaluated at a finite set of points in the computational domain, ensuring that the reconstructed parameters and states are consistent with both the observations and the PDE model to an arbitrary level of accuracy. In our approach, KLEs are constructed using the eigendecomposition of covariance models of spatial variability. For the model parameters, we employ a parameterized covariance model calibrated on parameter observations; for the model states, the covariance is estimated from a number of forward simulations of the PDE model corresponding to realizations of the parameters drawn from their KLE. We apply the proposed approach to identifying heterogeneous log-diffusion coefficients in diffusion equations from spatially sparse measurements of the log-diffusion coefficient and the solution of the diffusion equation. We find that the proposed approach compares favorably against state-of-the-art point estimates such as maximum a posteriori estimation and physics-informed neural networks.

math.AP↗

arXiv2019

Electric Load and Power Forecasting Using Ensemble Gaussian Process Regression

We propose a new forecasting method for predicting load demand and generation scheduling. Accurate week-long forecasting of load demand and optimal power generation is critical for efficient operation of power grid systems. In this work, we use a synthetic data set describing a power grid with 700 buses and 134 generators over a 365-days period with data synthetically generated at an hourly rate. The proposed approach for week-long forecasting is based on the Gaussian process regression (GPR) method, with prior covariance matrices of the quantities of interest (QoI) computed from ensembles formed by up to twenty preceding weeks of QoI observations. Then, we use these covariances within the GPR framework to forecast the QoIs for the following week. We demonstrate that the the proposed ensemble GPR (EGPR) method is capable of accurately forecasting weekly total load demand and power generation profiles. The EGPR method is shown to outperform traditional forecasting methods including the standard GPR and autoregressive integrated moving average (ARIMA) methods.

cs.LG↗

arXiv2019

A New Approach For Learning Coarse-Grained Potentials with Application to Immiscible Fluids

Even though atomistic and coarse-grained (CG) models have been used to simulate liquid nanodroplets in vapor, very few rigorous studies of the liquid-liquid interface structure are available, and most of them are limited to planar interfaces. In this work, we evaluate several existing force fields (FF)s, including two atomistic and three CG FFs, with respect to modeling the interface structure and thermodynamic properties of the water-hexane interface. Both atomistic FFs are able to quantitatively reproduce the interfacial tension and the coexisting densities of the experimentally-observed planar interface. We use the atomistic FFs to model water droplets in hexane and use these simulations to test the CG FFs. We find that the tested CG FFs cannot reproduce the interfacial tensions of planar and/or curved interfaces. Finally, we propose a new approach for learning CG potentials within the CG SDK (Shinoda-DeVane-Klein) FF framework from atomistic simulation data. We demonstrate that the new potential significantly improves the prediction of both the interfacial tension and structure of water-hexane planar and curved interfaces.

physics.chem-ph↗

arXiv2019

Enforcing constraints for interpolation and extrapolation in Generative Adversarial Networks

We suggest ways to enforce given constraints in the output of a Generative Adversarial Network (GAN) generator both for interpolation and extrapolation (prediction). For the case of dynamical systems, given a time series, we wish to train GAN generators that can be used to predict trajectories starting from a given initial condition. In this setting, the constraints can be in algebraic and/or differential form. Even though we are predominantly interested in the case of extrapolation, we will see that the tasks of interpolation and extrapolation are related. However, they need to be treated differently. For the case of interpolation, the incorporation of constraints is built into the training of the GAN. The incorporation of the constraints respects the primary game-theoretic setup of a GAN so it can be combined with existing algorithms. However, it can exacerbate the problem of instability during training that is well-known for GANs. We suggest adding small noise to the constraints as a simple remedy that has performed well in our numerical experiments. The case of extrapolation (prediction) is more involved. During training, the GAN generator learns to interpolate a noisy version of the data and we enforce the constraints. This approach has connections with model reduction that we can utilize to improve the efficiency and accuracy of the training. Depending on the form of the constraints, we may enforce them also during prediction through a projection step. We provide examples of linear and nonlinear systems of differential equations to illustrate the various constructions.

cs.LG↗

arXiv2019

Analytical steady-state solutions for pressure with a multiscale non-local model for two-fluid systems

We consider the nonlocal multiscale model for surface tension \citep{Tartakovsky2018} as an alternative to the (macroscale) Young-Laplace law. The nonlocal model is obtained in the form of an integral of a molecular-force-like function with support $\varepsilon$ added to the Navier-Stokes momentum conservation equation. Using this model, we calculate analytical forms for the steady-state equilibrium pressure gradient and pressure profile for circular and spherical bubbles and flat interfaces in two and three dimensions. According to the analytical solutions, the pressure changes continuously across the interface in a way that is quantitatively similar to what is observed in MD simulations. Furthermore, the pressure difference $P_{\varepsilon, in} - P_{\varepsilon, out}$ satisfies the Young-Laplace law for the radius of curvature greater than $3\varepsilon$ and deviates from the Young-Laplace law otherwise (i.e., $P_{\varepsilon, in} - P_{\varepsilon, out}$ goes to zero as the radius of the curvature goes to zero, where $P_{\varepsilon, out}$ is the pressure outside of the bubble at the distance greater than $3\varepsilon$ from the interface and $P_{\varepsilon, in}$ is the pressure at the center of the bubble). The latter indicates that the surface tension in the proposed model decreases with the decreasing radius of curvature, which agrees with molecular dynamics simulations and laboratory experiments with nanobubbles. Therefore, our results demonstrate that the nonlocal model behaves microscopically at scales smaller than $\varepsilon$ and macroscopically, otherwise.

physics.flu-dyn↗

arXiv2019

Conditional Karhunen-Loève expansion for uncertainty quantification and active learning in partial differential equation models

We use a conditional Karhunen-Loève (KL) model to quantify and reduce uncertainty in a stochastic partial differential equation (SPDE) problem with partially-known space-dependent coefficient, $Y(x)$. We assume that a small number of $Y(x)$ measurements are available and model $Y(x)$ with a KL expansion. We achieve reduction in uncertainty by conditioning the KL expansion coefficients on measurements. We consider two approaches for conditioning the KL expansion: In Approach 1, we condition the KL model first and then truncate it. In Approach 2, we first truncate the KL expansion and then condition it. We employ the conditional KL expansion together with Monte Carlo and sparse grid collocation methods to compute the moments of the solution of the SPDE problem. Uncertainty of the problem is further reduced by adaptively selecting additional observation locations using two active learning methods. Method 1 minimizes the variance of the PDE coefficient, while Method 2 minimizes the variance of the solution of the PDE. We demonstrate that conditioning leads to dimension reduction of the KL representation of $Y(x)$. For a linear diffusion SPDE with uncertain log-normal coefficient, we show that Approach 1 provides a more accurate approximation of the conditional log-normal coefficient and solution of the SPDE than Approach 2 for the same number of random dimensions in a conditional KL expansion. Furthermore, Approach 2 provides a good estimate for the number of terms of the truncated KL expansion of the conditional field of Approach 1. Finally, we demonstrate that active learning based on Method 2 is more efficient for uncertainty reduction in the SPDE's states (i.e., it leads to a larger reduction of the variance) than active learning using Method 2.

math.PR↗

arXiv2019

Approximate Bayesian Model Inversion for PDEs with Heterogeneous and State-Dependent Coefficients

We present two approximate Bayesian inference methods for parameter estimation in partial differential equation (PDE) models with space-dependent and state-dependent parameters. We demonstrate that these methods provide accurate and cost-effective alternatives to Markov Chain Monte Carlo simulation. We assume a parameterized Gaussian prior on the unknown functions, and approximate the posterior density by a parameterized multivariate Gaussian density. The parameters of the prior and posterior are estimated from sparse observations of the PDE model's states and the unknown functions themselves by maximizing the evidence lower bound (ELBO), a lower bound on the log marginal likelihood of the observations. The first method, Laplace-EM, employs the expectation maximization algorithm to maximize the ELBO, with a Laplace approximation of the posterior on the E-step, and minimization of a Kullback-Leibler divergence on the M-step. The second method, DSVI-EB, employs the doubly stochastic variational inference (DSVI) algorithm, in which the ELBO is maximized via gradient-based stochastic optimization, with nosiy gradients computed via simple Monte Carlo sampling and Gaussian backpropagation. We apply these methods to identifying diffusion coefficients in linear and nonlinear diffusion equations, and we find that both methods provide accurate estimates of posterior densities and the hyperparameters of Gaussian priors. While the Laplace-EM method is more accurate, it requires computing Hessians of the physics model. The DSVI-EB method is found to be less accurate but only requires gradients of the physics model.

stat.ME↗

arXiv2018

Learning Parameters and Constitutive Relationships with Physics Informed Deep Neural Networks

We present a physics informed deep neural network (DNN) method for estimating parameters and unknown physics (constitutive relationships) in partial differential equation (PDE) models. We use PDEs in addition to measurements to train DNNs to approximate unknown parameters and constitutive relationships as well as states. The proposed approach increases the accuracy of DNN approximations of partially known functions when a limited number of measurements is available and allows for training DNNs when no direct measurements of the functions of interest are available. We employ physics informed DNNs to estimate the unknown space-dependent diffusion coefficient in a linear diffusion equation and an unknown constitutive relationship in a non-linear diffusion equation. For the parameter estimation problem, we assume that partial measurements of the coefficient and states are available and demonstrate that under these conditions, the proposed method is more accurate than state-of-the-art methods. For the non-linear diffusion PDE model with a fully unknown constitutive relationship (i.e., no measurements of constitutive relationship are available), the physics informed DNN method can accurately estimate the non-linear constitutive relationship based on state measurements only. Finally, we demonstrate that the proposed method remains accurate in the presence of measurement noise.

math.AP↗

arXiv2018

Discrete-element model for the interaction between ocean waves and sea ice

We present a discrete element method (DEM) model to simulate the mechanical behavior of sea ice in response to ocean waves. The interaction of ocean waves and sea ice can potentially lead to the fracture and fragmentation of sea ice depending on the wave amplitude and period. The fracture behavior of sea ice is explicitly modeled by a DEM method, where sea ice is modeled by densely packed spherical particles with finite size. These particles are bonded together at their contact points through mechanical bonds that can sustain both tensile and compressive forces and moments. Fracturing can be naturally represented by the sequential breaking of mechanical bonds. For a given amplitude and period of incident ocean wave, the model provides information for the spatial distribution and time evolution of stress and micro-fractures and the fragment size distribution. We demonstrate that the fraction of broken bonds, , increases with increasing wave amplitude. In contrast, the ice fragment size l decreases with increasing amplitude. This information is important for the understanding of breakup of individual ice floes and floe fragment size.

physics.comp-ph↗

arXiv2018

Method of model reduction and multifidelity models for solute transport in random layered porous media

This work presents a hierarchical model for solute transport in bounded layered porous media with random permeability. The model generalizes the Taylor-Aris dispersion theory to stochastic transport in random layered porous media with a known velocity covariance function. In the hierarchical model, we represent (random) concentration in terms of its cross-sectional average and a variation function. We derive a one-dimensional stochastic advection-dispersion-type equation for the average concentration and a stochastic Poisson equation for the variation function, as well as expressions for the effective velocity and dispersion coefficient. We observe that velocity fluctuations enhance dispersion in a non-monotonic fashion: the dispersion initially increases with correlation length λ, reaches a maximum, and decreases to zero at infinity. Maximum enhancement can be obtained at the correlation length about 0.25 the size of the porous media perpendicular to flow.

physics.flu-dyn↗

arXiv2018

Continuum Model for Nanoscale Multiphase Flows

We propose a nonlocal model for surface tension. This model, in combination with the Landau-Lifshitz-Navier-Stokes equations, describes mesoscale features of the multiphase flow, including the static (pressure) tensor and curvature dependence of surface tension. The nonlocal model is obtained in the form of an integral of a molecular-force-like function added into the momentum conservation equation. We present an analytical steady-state solution for fluid pressure at the fluid-fluid interface and numerical Smoothed Particle Hydrodynamics solutions that reveal the mesoscopic features of the proposed model.

physics.flu-dyn↗

arXiv2018

Multivariate Gaussian Process Regression for Multiscale Data Assimilation and Uncertainty Reduction

We present a multivariate Gaussian process regression approach for parameter field reconstruction based on the field's measurements collected at two different scales, the coarse and fine scales. The proposed approach treats the parameter field defined at fine and coarse scales as a bivariate Gaussian process with a parameterized multiscale covariance model. We employ a full bivariate Matérn kernel as multiscale covariance model, with shape and smoothness hyperparameters that account for the coarsening relation between fine and coarse fields. In contrast to similar multiscale kriging approaches that assume a known coarsening relation between scales, the hyperparameters of the multiscale covariance model are estimated directly from data via pseudo-likelihood maximization. We illustrate the proposed approach with a predictive simulation application for saturated flow in porous media. Multiscale Gaussian process regression is employed to estimate two-dimensional log-saturated hydraulic conductivity distributions from synthetic multiscale measurements. The resulting stochastic model for coarse saturated conductivity is employed to quantify and reduce uncertainty in pressure predictions.

stat.ME↗

arXiv2018

Persistent incomplete mixing in reactive flows

We present an effective stochastic advection-diffusion-reaction (SADR) model that explains incomplete mixing typically observed in transport with bimolecular reactions. Unlike traditional advection-dispersion-reaction models, the SADR model describes mechanical and diffusive mixing as two separate processes. In the SADR model, mechanical mixing is driven by random advective velocity with the variance given by the coefficient of mechanical dispersion. The diffusive mixing is modeled as a Fickian diffusion with the effective diffusion coefficient. We demonstrate that the sum of the two coefficients is equal to the dispersion coefficients, but only the effective diffusion coefficient contributes to the mixing-controlled reactions, indicating that such systems do not get fully mixed at the Representative Elementary Volume scale where the deterministic equations and dispersion coefficient are defined. We use the experimental results of Gramling et al. \cite{Gramling} to show that for transport and bimolecular reactions in porous media, the SADR model is significantly more accurate than the traditional dispersion model, which overestimates the concentration of the reaction product by as much as 60\%.

physics.flu-dyn↗

arXiv2017

Solving differential equations with unknown constitutive relations as recurrent neural networks

We solve a system of ordinary differential equations with an unknown functional form of a sink (reaction rate) term. We assume that the measurements (time series) of state variables are partially available, and we use recurrent neural network to "learn" the reaction rate from this data. This is achieved by including a discretized ordinary differential equations as part of a recurrent neural network training problem. We extend TensorFlow's recurrent neural network architecture to create a simple but scalable and effective solver for the unknown functions, and apply it to a fedbatch bioreactor simulation problem. Use of techniques from recent deep learning literature enables training of functions with behavior manifesting over thousands of time steps. Our networks are structurally similar to recurrent neural networks, but differences in design and function require modifications to the conventional wisdom about training such networks.

cs.LG↗

arXiv2017

PDF estimation for power grid systems via sparse regression

We present a numerical approach for estimating the probability density function (PDF) of quantities of interest (QoIs) of power grid systems subject to uncertain power generation and load fluctuations. In our approach, generation and load fluctuations are modeled by means of autocorrelated-in-time random processes, which are approximated in terms of a finite set of random parameters by means of Karhunen-Loève approximations. The map from random parameters to QoIs is approximated by means of Hermite polynomial expansions. We propose a new approach based on compressive sensing to estimate the coefficients in the Hermite expansions from a small number of realizations (sampling points). Linear transforms identified by iterative rotations are introduced to improve the sparsity of the Hermite representations, exploiting the intrinsic low-dimensional structure of the map. As such, the proposed approach significantly reduces the required number of sampling points to achieve a given accuracy compared to the standard least squares method. The proposed approach is employed to estimate the PDF of relative angular velocities and bus voltages of systems of classical machines driven by autocorrelated random generation. More accurate PDF estimates, as measured by the Kullback-Leibler divergence, are achieved using fewer realizations than required by basic Monte Carlo sampling.

stat.AP↗