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Entropy conditions play a crucial role in the extraction of a physically relevant solution for a system of conservation laws, thus motivating the construction of entropy stable schemes that satisfy a discrete analogue of such conditions. TeCNO schemes (Fjordholm et al. 2012) form a class of arbitrary high-order entropy stable finite difference solvers, which require specialized reconstruction algorithms satisfying the sign property at each cell interface. Recently, third-order WENO schemes called SP-WENO (Fjordholm and Ray, 2016) and SP-WENOc (Ray, 2018) have been designed to satisfy the sign property. However, these WENO algorithms can perform poorly near shocks, with the numerical solutions exhibiting large spurious oscillations. In the present work, we propose a variant of the SP-WENO, termed as Deep Sign-Preserving WENO (DSP-WENO), where a neural network is trained to learn the WENO weighting strategy. The sign property and third-order accuracy are strongly imposed in the algorithm, which constrains the WENO weight selection region to a convex polygon. Thereafter, a neural network is trained to select the WENO weights from this convex region with the goal of improving the shock-capturing capabilities without sacrificing the rate of convergence in smooth regions. The proposed synergistic approach retains the mathematical framework of the TeCNO scheme while integrating deep learning to remedy the computational issues of the WENO-based reconstruction. We present several numerical experiments to demonstrate the significant improvement with DSP-WENO over the existing variants of WENO satisfying the sign property., Comment: 38 pages, 16 figures, 5 tables

We propose a novel modular inference approach combining two different generative models -- generative adversarial networks (GAN) and normalizing flows -- to approximate the posterior distribution of physics-based Bayesian inverse problems framed in high-dimensional ambient spaces. We dub the proposed framework GAN-Flow. The proposed method leverages the intrinsic dimension reduction and superior sample generation capabilities of GANs to define a low-dimensional data-driven prior distribution. Once a trained GAN-prior is available, the inverse problem is solved entirely in the latent space of the GAN using variational Bayesian inference with normalizing flow-based variational distribution, which approximates low-dimensional posterior distribution by transforming realizations from the low-dimensional latent prior (Gaussian) to corresponding realizations of a low-dimensional variational posterior distribution. The trained GAN generator then maps realizations from this approximate posterior distribution in the latent space back to the high-dimensional ambient space. We also propose a two-stage training strategy for GAN-Flow wherein we train the two generative models sequentially. Thereafter, GAN-Flow can estimate the statistics of posterior-predictive quantities of interest at virtually no additional computational cost. The synergy between the two types of generative models allows us to overcome many challenges associated with the application of Bayesian inference to large-scale inverse problems, chief among which are describing an informative prior and sampling from the high-dimensional posterior. We demonstrate the efficacy and flexibility of GAN-Flow on various physics-based inverse problems of varying ambient dimensionality and prior knowledge using different types of GANs and normalizing flows.

Increases in wildfire activity and the resulting impacts have prompted the development of high-resolution wildfire behavior models for forecasting fire spread. Recent progress in using satellites to detect fire locations further provides the opportunity to use measurements to improve fire spread forecasts from numerical models through data assimilation. This work develops a method for inferring the history of a wildfire from satellite measurements, providing the necessary information to initialize coupled atmosphere-wildfire models from a measured wildfire state in a physics-informed approach. The fire arrival time, which is the time the fire reaches a given spatial location, acts as a succinct representation of the history of a wildfire. In this work, a conditional Wasserstein Generative Adversarial Network (cWGAN), trained with WRF-SFIRE simulations, is used to infer the fire arrival time from satellite active fire data. The cWGAN is used to produce samples of likely fire arrival times from the conditional distribution of arrival times given satellite active fire detections. Samples produced by the cWGAN are further used to assess the uncertainty of predictions. The cWGAN is tested on four California wildfires occurring between 2020 and 2022, and predictions for fire extent are compared against high resolution airborne infrared measurements. Further, the predicted ignition times are compared with reported ignition times. An average Sorensen's coefficient of 0.81 for the fire perimeters and an average ignition time error of 32 minutes suggest that the method is highly accurate.

We present a deep learning-based computational algorithm for inversion of circular Radon transforms in the partial radial setup, arising in photoacoustic tomography. We first demonstrate that the truncated singular value decomposition-based method, which is the only traditional algorithm available to solve this problem, leads to severe artifacts which renders the reconstructed field as unusable. With the objective of overcoming this computational bottleneck, we train a ResBlock based U-Net to recover the inferred field that directly operates on the measured data. Numerical results with augmented Shepp-Logan phantoms, in the presence of noisy full and limited view data, demonstrate the superiority of the proposed algorithm.

The solution of probabilistic inverse problems for which the corresponding forward problem is constrained by physical principles is challenging. This is especially true if the dimension of the inferred vector is large and the prior information about it is in the form of a collection of samples. In this work, a novel deep learning based approach is developed and applied to solving these types of problems. The approach utilizes samples of the inferred vector drawn from the prior distribution and a physics-based forward model to generate training data for a conditional Wasserstein generative adversarial network (cWGAN). The cWGAN learns the probability distribution for the inferred vector conditioned on the measurement and produces samples from this distribution. The cWGAN developed in this work differs from earlier versions in that its critic is required to be 1-Lipschitz with respect to both the inferred and the measurement vectors and not just the former. This leads to a loss term with the full (and not partial) gradient penalty. It is shown that this rather simple change leads to a stronger notion of convergence for the conditional density learned by the cWGAN and a more robust and accurate sampling strategy. Through numerical examples it is shown that this change also translates to better accuracy when solving inverse problems. The numerical examples considered include illustrative problems where the true distribution and/or statistics are known, and a more complex inverse problem motivated by applications in biomechanics., Comment: 34 pages, 9 figures, 3 tables, 1 appendix

These notes were compiled as lecture notes for a course developed and taught at the University of the Southern California. They should be accessible to a typical engineering graduate student with a strong background in Applied Mathematics. The main objective of these notes is to introduce a student who is familiar with concepts in linear algebra and partial differential equations to select topics in deep learning. These lecture notes exploit the strong connections between deep learning algorithms and the more conventional techniques of computational physics to achieve two goals. First, they use concepts from computational physics to develop an understanding of deep learning algorithms. Not surprisingly, many concepts in deep learning can be connected to similar concepts in computational physics, and one can utilize this connection to better understand these algorithms. Second, several novel deep learning algorithms can be used to solve challenging problems in computational physics. Thus, they offer someone who is interested in modeling a physical phenomena with a complementary set of tools., Comment: 7 chapters

In recent years operator networks have emerged as promising deep learning tools for approximating the solution to partial differential equations (PDEs). These networks map input functions that describe material properties, forcing functions and boundary data to the solution of a PDE. This work describes a new architecture for operator networks that mimics the form of the numerical solution obtained from an approximate variational or weak formulation of the problem. The application of these ideas to a generic elliptic PDE leads to a variationally mimetic operator network (VarMiON). Like the conventional Deep Operator Network (DeepONet) the VarMiON is also composed of a sub-network that constructs the basis functions for the output and another that constructs the coefficients for these basis functions. However, in contrast to the DeepONet, the architecture of these sub-networks in the VarMiON is precisely determined. An analysis of the error in the VarMiON solution reveals that it contains contributions from the error in the training data, the training error, the quadrature error in sampling input and output functions, and a "covering error" that measures the distance between the test input functions and the nearest functions in the training dataset. It also depends on the stability constants for the exact solution operator and its VarMiON approximation. The application of the VarMiON to a canonical elliptic PDE and a nonlinear PDE reveals that for approximately the same number of network parameters, on average the VarMiON incurs smaller errors than a standard DeepONet and a recently proposed multiple-input operator network (MIONet). Further, its performance is more robust to variations in input functions, the techniques used to sample the input and output functions, the techniques used to construct the basis functions, and the number of input functions., Comment: 49 pages, 18 figures, 1 Appendix

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, while the latter enables the decoupling of the latent space dimension from the dimension of the measurement, and introduces stochasticity at all scales of the U-Net. We solve PDE-based inverse problems to demonstrate the performance of our approach in quantifying the uncertainty in the inferred field. Further, we show the generator can learn inverse maps which are local in nature, which in turn promotes generalizability when testing with out-of-distribution samples.

Inverse problems are ubiquitous in nature, arising in almost all areas of science and engineering ranging from geophysics and climate science to astrophysics and biomechanics. One of the central challenges in solving inverse problems is tackling their ill-posed nature. Bayesian inference provides a principled approach for overcoming this by formulating the inverse problem into a statistical framework. However, it is challenging to apply when inferring fields that have discrete representations of large dimensions (the so-called "curse of dimensionality") and/or when prior information is available only in the form of previously acquired solutions. In this work, we present a novel method for efficient and accurate Bayesian inversion using deep generative models. Specifically, we demonstrate how using the approximate distribution learned by a Generative Adversarial Network (GAN) as a prior in a Bayesian update and reformulating the resulting inference problem in the low-dimensional latent space of the GAN, enables the efficient solution of large-scale Bayesian inverse problems. Our statistical framework preserves the underlying physics and is demonstrated to yield accurate results with reliable uncertainty estimates, even in the absence of information about underlying noise model, which is a significant challenge with many existing methods. We demonstrate the effectiveness of proposed method on a variety of inverse problems which include both synthetic as well as experimentally observed data., Comment: Paper: 38 pages, 12 figures, 3 Tables

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 discontinuous Galerkin methods with piecewise linear or piecewise quadratic polynomials. Flux and slope limiting techniques successfully eliminate the bulk shift, overshoot and undershoot in the order parameter, which is shown to be bound-preserving. Several numerical results demonstrate that the proposed numerical algorithm is effective and robust for modeling two-component immiscible flows in porous structures and digital rocks.

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 dynamics of the flow with and without surfactant. Finally the numerical method is used to simulate fluid flows in the pore space of Berea sandstone obtained by micro-CT imaging., Comment: 25 pages, 22 figures

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 entropy solutions. Furthermore, we analyze the convergence rates for MLMC-FDM and compare it with the convergence rates for the deterministic case. Additionally, we formulate error vs. work estimates for the multi-level estimator. Finally, we present several numerical experiments to demonstrate the efficiency of these schemes and validate the theoretical estimates obtained in this work., Comment: To be published in SIAM/ASA Journal on Uncertainty Quantification

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 networks and any underlying standard optimization algorithm. Under suitable hypotheses, we show that the resulting optimizers converge exponentially fast (and with exponentially decaying variance), with respect to increasing number of training samples. Numerical examples for optimal control, parameter identification and shape optimization problems for PDEs are provided to validate the proposed theory and to illustrate that ISMO significantly outperforms a standard deep neural network based surrogate optimization algorithm.

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-order accuracy at approximating Lipschitz functions. Numerical tests for the resulting neural networks show excellent performance for approximating solutions of nonlinear conservation laws and at data compression., Comment: 39 pages

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 to respect such constraints. We propose two different strategies to generate constraint-aware neural networks. We test their performance in the context of front-capturing schemes for strongly nonlinear wave motion in compressible fluid flow. Precisely, in this context so-called Riemann problems have to be solved as surrogates. Their solution describes the local dynamics of the captured wave front in numerical simulations. Three model problems are considered: a cubic flux model problem, an isothermal two-phase flow model, and the Euler equations. We demonstrate that a decrease in the constraint deviation correlates with low discretization errors for all model problems, in addition to the structural advantage of fulfilling the constraint.

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 algorithm, based on deep artificial neural networks, that predicts the underlying \emph{input parameters to observable} map from a few training samples (computed realizations of this map). By a judicious combination of theoretical arguments and empirical observations, we find suitable network architectures and training hyperparameters that result in robust and efficient neural network approximations of the parameters to observable map. Numerical experiments are presented to demonstrate low prediction errors for the trained network networks, even when the network has been trained with a few samples, at a computational cost which is several orders of magnitude lower than the underlying PDE solver. Moreover, we combine the proposed deep learning algorithm with Monte Carlo (MC) and Quasi-Monte Carlo (QMC) methods to efficiently compute uncertainty propagation for nonlinear PDEs. Under the assumption that the underlying neural networks generalize well, we prove that the deep learning MC and QMC algorithms are guaranteed to be faster than the baseline (quasi-) Monte Carlo methods. Numerical experiments demonstrating one to two orders of magnitude speed up over baseline QMC and MC algorithms, for the intricate problem of computing probability distributions of the observable, are also presented.

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 order. The performance of the proposed reconstruction is demonstrated via a series of numerical experiments for linear and nonlinear scalar conservation laws. The scheme is easily extended to multi-dimensional conservation laws., Comment: Accepted for publication by Springer Journal of Scientific Computing

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 laws with a discontinuous coefficient, to the corresponding scalar conservation law with discontinuous coefficient. Finally, the convergence is illustrated by several examples. In particular, it is delineated that the limiting solutions generated by the scheme need not coincide, depending on the relation between diffusion and the dispersion coefficients, with the classical Kruzkov-Oleinik entropy solutions, but contain nonclassical undercompressive shock waves., Comment: 38 Pages, 6 figures