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

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

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

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 Mat\'{e}rn 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

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

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.Comment: 29 pages; v2 has major text revision/restructuring, includes results for the Lorenz system and has several more reference

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.Comment: 19 pages, 8 figure

Conditional Karhunen-Lo\`eve expansion for uncertainty quantification and active learning in partial differential equation models

We use a conditional Karhunen-Lo\`eve (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

Gaussian Process Regression and Conditional Polynomial Chaos for Parameter Estimation

We present a new approach for constructing a data-driven surrogate model and using it for Bayesian parameter estimation in partial differential equation (PDE) models. We first use parameter observations and Gaussian Process regression to condition the Karhunen-Lo\'{e}ve (KL) expansion of the unknown space-dependent parameters and then build the conditional generalized Polynomial Chaos (gPC) surrogate model of the PDE states. Next, we estimate the unknown parameters by computing coefficients in the KL expansion minimizing the square difference between the gPC predictions and measurements of the states using the Markov Chain Monte Carlo method. Our approach addresses two major challenges in the Bayesian parameter estimation. First, it reduces dimensionality of the parameter space and replaces expensive direct solutions of PDEs with the conditional gPC surrogates. Second, the estimated parameter field exactly matches the parameter measurements. In addition, we show that the conditional gPC surrogate can be used to estimate the states variance, which, in turn, can be used to guide data acquisition. We demonstrate that our approach improves its accuracy with application to one- and two-dimensional Darcy equation with (unknown) space-dependent hydraulic conductivity. We also discuss the effect of hydraulic conductivity and head locations on the accuracy of the hydraulic conductivity estimations

Physics-Informed Neural Network Method for Forward and Backward Advection-Dispersion Equations

We propose a discretization-free approach based on the physics-informed neural network (PINN) method for solving coupled advection-dispersion and Darcy flow equations with space-dependent hydraulic conductivity. In this approach, the hydraulic conductivity, hydraulic head, and concentration fields are approximated with deep neural networks (DNNs). We assume that the conductivity field is given by its values on a grid, and we use these values to train the conductivity DNN. The head and concentration DNNs are trained by minimizing the residuals of the flow equation and ADE and using the initial and boundary conditions as additional constraints. The PINN method is applied to one- and two-dimensional forward advection-dispersion equations (ADEs), where its performance for various P\'{e}clet numbers ($Pe$) is compared with the analytical and numerical solutions. We find that the PINN method is accurate with errors of less than 1% and outperforms some conventional discretization-based methods for $Pe$ larger than 100. Next, we demonstrate that the PINN method remains accurate for the backward ADEs, with the relative errors in most cases staying under 5% compared to the reference concentration field. Finally, we show that when available, the concentration measurements can be easily incorporated in the PINN method and significantly improve (by more than 50% in the considered cases) the accuracy of the PINN solution of the backward ADE.Comment: 31 pages, 15 figure