arXiv subjects

Eric Darve

Publications and source records attributed to Eric Darve.

At least 55 records · Page 3Linked to original sources

arXiv2020

Solving Inverse Problems in Steady-State Navier-Stokes Equations using Deep Neural Networks

Inverse problems in fluid dynamics are ubiquitous in science and engineering, with applications ranging from electronic cooling system design to ocean modeling. We propose a general and robust approach for solving inverse problems in the steady-state Navier-Stokes equations by combining deep neural networks and numerical partial differential equation (PDE) schemes. Our approach expresses numerical simulation as a computational graph with differentiable operators. We then solve inverse problems by constrained optimization, using gradients calculated from the computational graph with reverse-mode automatic differentiation. This technique enables us to model unknown physical properties using deep neural networks and embed them into the PDE model. We demonstrate the effectiveness of our method by computing spatially-varying viscosity and conductivity fields with deep neural networks (DNNs) and training the DNNs using partial observations of velocity fields. We show that the DNNs are capable of modeling complex spatially-varying physical fields with sparse and noisy data. Our implementation leverages the open access ADCME, a library for solving inverse modeling problems in scientific computing using automatic differentiation.

math.NA↗

arXiv2020

TED: A Pretrained Unsupervised Summarization Model with Theme Modeling and Denoising

Text summarization aims to extract essential information from a piece of text and transform the text into a concise version. Existing unsupervised abstractive summarization models leverage recurrent neural networks framework while the recently proposed transformer exhibits much more capability. Moreover, most of previous summarization models ignore abundant unlabeled corpora resources available for pretraining. In order to address these issues, we propose TED, a transformer-based unsupervised abstractive summarization system with pretraining on large-scale data. We first leverage the lead bias in news articles to pretrain the model on millions of unlabeled corpora. Next, we finetune TED on target domains through theme modeling and a denoising autoencoder to enhance the quality of generated summaries. Notably, TED outperforms all unsupervised abstractive baselines on NYT, CNN/DM and English Gigaword datasets with various document styles. Further analysis shows that the summaries generated by TED are highly abstractive, and each component in the objective function of TED is highly effective.

cs.CL↗

arXiv2020

Hierarchical Orthogonal Factorization: Sparse Square matrices

In this work, we develop a new fast algorithm, spaQR -- sparsified QR, for solving large, sparse linear systems. The key to our approach is using low-rank approximations to sparsify the separators in a Nested Dissection based Householder QR factorization. First, a modified version of Nested Dissection is used to identify interiors/separators and reorder the matrix. Then, classical Householder QR is used to factorize the interiors, going from the leaves to the root to the elimination tree. After every level of interior factorization, we sparsify the remaining separators by using low-rank approximations. This operation reduces the size of the separators without introducing any fill-in in the matrix. However, it introduces a small approximation error which can be controlled by the user. The resulting approximate factorization is stored as a sequence of sparse orthogonal and sparse upper-triangular factors. Hence, it can be applied efficiently to solve linear systems. Additionally, we further improve the algorithm by using a block diagonal scaling. Then, we show a systematic analysis of the approximation error and effectiveness of the algorithm in solving linear systems. Finally, we perform numerical tests on benchmark unsymmetric problems to evaluate the performance of the algorithm. The factorization time scales as $\mathcal{O}(N \log N)$ and the solve time scales as $\mathcal{O}(N)$.

math.NA↗

arXiv2020

TaskTorrent: a Lightweight Distributed Task-Based Runtime System in C++

We present TaskTorrent, a lightweight distributed task-based runtime in C++. TaskTorrent uses a parametrized task graph to express the task DAG, and one-sided active messages to trigger remote tasks asynchronously. As a result the task DAG is completely distributed and discovered in parallel. It is a C++14 library and only depends on MPI. We explain the API and the implementation. We perform a series of benchmarks against StarPU and ScaLAPACK. Micro benchmarks show it has a minimal overhead compared to other solutions. We then apply it to two large linear algebra problems. TaskTorrent scales very well to thousands of cores, exhibiting good weak and strong scalings.

cs.DC↗

arXiv2020

Second Order Accurate Hierarchical Approximate Factorization of Sparse SPD Matrices

We describe a second-order accurate approach to sparsifying the off-diagonal blocks in the hierarchical approximate factorizations of sparse symmetric positive definite matrices. The norm of the error made by the new approach depends quadratically, not linearly, on the error in the low-rank approximation of the given block. The analysis of the resulting two-level preconditioner shows that the preconditioner is second-order accurate as well. We incorporate the new approach into the recent Sparsified Nested Dissection algorithm [SIAM J. Matrix Anal. Appl., 41 (2020), pp. 715-746], and test it on a wide range of problems. The new approach halves the number of Conjugate Gradient iterations needed for convergence, with almost the same factorization complexity, improving the total runtimes of the algorithm. Our approach can be incorporated into other rank-structured methods for solving sparse linear systems.

math.NA↗

arXiv2020

Out-of-Vocabulary Embedding Imputation with Grounded Language Information by Graph Convolutional Networks

Due to the ubiquitous use of embeddings as input representations for a wide range of natural language tasks, imputation of embeddings for rare and unseen words is a critical problem in language processing. Embedding imputation involves learning representations for rare or unseen words during the training of an embedding model, often in a post-hoc manner. In this paper, we propose an approach for embedding imputation which uses grounded information in the form of a knowledge graph. This is in contrast to existing approaches which typically make use of vector space properties or subword information. We propose an online method to construct a graph from grounded information and design an algorithm to map from the resulting graphical structure to the space of the pre-trained embeddings. Finally, we evaluate our approach on a range of rare and unseen word tasks across various domains and show that our model can learn better representations. For example, on the Card-660 task our method improves Pearson's and Spearman's correlation coefficients upon the state-of-the-art by 11% and 17.8% respectively using GloVe embeddings.

cs.CL↗

arXiv2020

Anomaly Detection with Domain Adaptation

We study the problem of semi-supervised anomaly detection with domain adaptation. Given a set of normal data from a source domain and a limited amount of normal examples from a target domain, the goal is to have a well-performing anomaly detector in the target domain. We propose the Invariant Representation Anomaly Detection (IRAD) to solve this problem where we first learn to extract a domain-invariant representation. The extraction is achieved by an across-domain encoder trained together with source-specific encoders and generators by adversarial learning. An anomaly detector is then trained using the learnt representations. We evaluate IRAD extensively on digits images datasets (MNIST, USPS and SVHN) and object recognition datasets (Office-Home). Experimental results show that IRAD outperforms baseline models by a wide margin across different datasets. We derive a theoretical lower bound for the joint error that explains the performance decay from overtraining and also an upper bound for the generalization error.

cs.LG↗

arXiv2020

Regularized Cycle Consistent Generative Adversarial Network for Anomaly Detection

In this paper, we investigate algorithms for anomaly detection. Previous anomaly detection methods focus on modeling the distribution of non-anomalous data provided during training. However, this does not necessarily ensure the correct detection of anomalous data. We propose a new Regularized Cycle Consistent Generative Adversarial Network (RCGAN) in which deep neural networks are adversarially trained to better recognize anomalous samples. This approach is based on leveraging a penalty distribution with a new definition of the loss function and novel use of discriminator networks. It is based on a solid mathematical foundation, and proofs show that our approach has stronger guarantees for detecting anomalous examples compared to the current state-of-the-art. Experimental results on both real-world and synthetic data show that our model leads to significant and consistent improvements on previous anomaly detection benchmarks. Notably, RCGAN improves on the state-of-the-art on the KDDCUP, Arrhythmia, Thyroid, Musk and CIFAR10 datasets.

cs.LG↗

arXiv2020

Isogeometric Collocation Method for the Fractional Laplacian in the 2D Bounded Domain

We consider the isogeometric analysis for fractional PDEs involving the fractional Laplacian in two dimensions. An isogeometric collocation method is developed to discretize the fractional Laplacian and applied to the fractional Poisson problem and the time-dependent fractional porous media equation. Numerical studies exhibit monotonous convergence with a rate of $\mathcal{O}(N^{-1})$, where $N$ is the degrees of freedom. A comparison with finite element analysis shows that the method enjoys higher accuracy per degree of freedom and has a better convergence rate. We demonstrate that isogeometric analysis offers a novel and promising computational tool for nonlocal problems.

math.NA↗

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↗

arXiv2020

Coupled Time-lapse Full Waveform Inversion for Subsurface Flow Problems using Intrusive Automatic Differentiation

We describe a novel framework for estimating subsurface properties, such as rock permeability and porosity, from time-lapse observed seismic data by coupling full-waveform inversion, subsurface flow processes, and rock physics models. For the inverse modeling, we handle the back-propagation of gradients by an intrusive automatic differentiation strategy that offers three levels of user control: (1) at the wave physics level, we adopted the discrete adjoint method in order to use our existing high-performance FWI code; (2) at the rock physics level, we used built-in operators from the $\texttt{TensorFlow}$ backend; (3) at the flow physics level, we implemented customized PDE operators for the potential and nonlinear saturation equations. These three levels of gradient computation strike a good balance between computational efficiency and programming efficiency, and when chained together, constitute a coupled inverse system. We use numerical experiments to demonstrate that (1) the three-level coupled inverse problem is superior in terms of accuracy to a traditional decoupled inversion strategy; (2) it is able to simultaneously invert for parameters in empirical relationships such as the rock physics models; and (3) the inverted model can be used for reservoir performance prediction and reservoir management/optimization purposes.

physics.geo-ph↗

arXiv2020

Learning Constitutive Relations using Symmetric Positive Definite Neural Networks

We present the Cholesky-factored symmetric positive definite neural network (SPD-NN) for modeling constitutive relations in dynamical equations. Instead of directly predicting the stress, the SPD-NN trains a neural network to predict the Cholesky factor of a tangent stiffness matrix, based on which the stress is calculated in the incremental form. As a result of the special structure, SPD-NN weakly imposes convexity on the strain energy function, satisfies time consistency for path-dependent materials, and therefore improves numerical stability, especially when the SPD-NN is used in finite element simulations. Depending on the types of available data, we propose two training methods, namely direct training for strain and stress pairs and indirect training for loads and displacement pairs. We demonstrate the effectiveness of SPD-NN on hyperelastic, elasto-plastic, and multiscale fiber-reinforced plate problems from solid mechanics. The generality and robustness of the SPD-NN make it a promising tool for a wide range of constitutive modeling applications.

math.NA↗

arXiv2020

A General Approach to Seismic Inversion with Automatic Differentiation

Imaging Earth structure or seismic sources from seismic data involves minimizing a target misfit function, and is commonly solved through gradient-based optimization. The adjoint-state method has been developed to compute the gradient efficiently; however, its implementation can be time-consuming and difficult. We develop a general seismic inversion framework to calculate gradients using reverse-mode automatic differentiation. The central idea is that adjoint-state methods and reverse-mode automatic differentiation are mathematically equivalent. The mapping between numerical PDE simulation and deep learning allows us to build a seismic inverse modeling library, ADSeismic, based on deep learning frameworks, which supports high performance reverse-mode automatic differentiation on CPUs and GPUs. We demonstrate the performance of ADSeismic on inverse problems related to velocity model estimation, rupture imaging, earthquake location, and source time function retrieval. ADSeismic has the potential to solve a wide variety of inverse modeling applications within a unified framework.

physics.comp-ph↗

arXiv2020

Learning Constitutive Relations from Indirect Observations Using Deep Neural Networks

We present a new approach for predictive modeling and its uncertainty quantification for mechanical systems, where coarse-grained models such as constitutive relations are derived directly from observation data. We explore the use of a neural network to represent the unknown constitutive relations, compare the neural networks with piecewise linear functions, radial basis functions, and radial basis function networks, and show that the neural network outperforms the others in certain cases. We analyze the approximation error of the neural networks using a scaling argument. The training and predicting processes in our framework combine the finite element method, automatic differentiation, and neural networks (or other function approximators). Our framework also allows uncertainty quantification in the form of confidence intervals. Numerical examples on a multiscale fiber-reinforced plate problem and a nonlinear rubbery membrane problem from solid mechanics demonstrate the effectiveness of our framework.

math.NA↗

arXiv2020

Physics Constrained Learning for Data-driven Inverse Modeling from Sparse Observations

Deep neural networks (DNN) have been used to model nonlinear relations between physical quantities. Those DNNs are embedded in physical systems described by partial differential equations (PDE) and trained by minimizing a loss function that measures the discrepancy between predictions and observations in some chosen norm. This loss function often includes the PDE constraints as a penalty term when only sparse observations are available. As a result, the PDE is only satisfied approximately by the solution. However, the penalty term typically slows down the convergence of the optimizer for stiff problems. We present a new approach that trains the embedded DNNs while numerically satisfying the PDE constraints. We develop an algorithm that enables differentiating both explicit and implicit numerical solvers in reverse-mode automatic differentiation. This allows the gradients of the DNNs and the PDE solvers to be computed in a unified framework. We demonstrate that our approach enjoys faster convergence and better stability in relatively stiff problems compared to the penalty method. Our approach allows for the potential to solve and accelerate a wide range of data-driven inverse modeling, where the physical constraints are described by PDEs and need to be satisfied accurately.

math.NA↗

arXiv2020

Sparse Hierarchical Preconditioners Using Piecewise Smooth Approximations of Eigenvectors

When solving linear systems arising from PDE discretizations, iterative methods (such as Conjugate Gradient, GMRES, or MINRES) are often the only practical choice. To converge in a small number of iterations, however, they have to be coupled with an efficient preconditioner. The efficiency of the preconditioner depends largely on its accuracy on the eigenvectors corresponding to small eigenvalues, and unfortunately, black-box methods typically cannot guarantee sufficient accuracy on these eigenvectors. Thus, constructing the preconditioner becomes a problem-dependent task. However, for a large class of problems, including many elliptic equations, the eigenvectors corresponding to small eigenvalues are smooth functions of the PDE grid. In this paper, we describe a hierarchical approximate factorization approach which focuses on improving accuracy on the smooth eigenvectors. The improved accuracy is achieved by preserving the action of the factorized matrix on piecewise polynomial functions of the grid. Based on the factorization, we propose a family of sparse preconditioners with $O(n)$ or $O(n \log{n})$ construction complexities. Our methods exhibit the optimal $O(n)$ solution times in benchmarks run on large elliptic problems of different types, arising for example in flow or mechanical simulations. In the case of the linear elasticity equation the preconditioners are exact on the near-kernel rigid body modes.

math.NA↗

arXiv2020

Memory Augmented Generative Adversarial Networks for Anomaly Detection

In this paper, we present a memory-augmented algorithm for anomaly detection. Classical anomaly detection algorithms focus on learning to model and generate normal data, but typically guarantees for detecting anomalous data are weak. The proposed Memory Augmented Generative Adversarial Networks (MEMGAN) interacts with a memory module for both the encoding and generation processes. Our algorithm is such that most of the \textit{encoded} normal data are inside the convex hull of the memory units, while the abnormal data are isolated outside. Such a remarkable property leads to good (resp.\ poor) reconstruction for normal (resp.\ abnormal) data and therefore provides a strong guarantee for anomaly detection. Decoded memory units in MEMGAN are more interpretable and disentangled than previous methods, which further demonstrates the effectiveness of the memory mechanism. Experimental results on twenty anomaly detection datasets of CIFAR-10 and MNIST show that MEMGAN demonstrates significant improvements over previous anomaly detection methods.

cs.LG↗

arXiv2020

An Algebraic Sparsified Nested Dissection Algorithm Using Low-Rank Approximations

We propose a new algorithm for the fast solution of large, sparse, symmetric positive-definite linear systems, spaND -- sparsified Nested Dissection. It is based on nested dissection, sparsification and low-rank compression. After eliminating all interiors at a given level of the elimination tree, the algorithm sparsifies all separators corresponding to the interiors. This operation reduces the size of the separators by eliminating some degrees of freedom but without introducing any fill-in. This is done at the expense of a small and controllable approximation error. The result is an approximate factorization that can be used as an efficient preconditioner. We then perform several numerical experiments to evaluate this algorithm. We demonstrate that a version using orthogonal factorization and block-diagonal scaling takes fewer CG iterations to converge than previous similar algorithms on various kinds of problems. Furthermore, this algorithm is provably guaranteed to never break down and the matrix stays symmetric positive-definite throughout the process. We evaluate the algorithm on some large problems and show it exhibits near-linear scaling. The factorization time is roughly O(N) and the number of iterations grows slowly with N.

math.NA↗