arXiv subjects

Eric Darve

Publications and source records attributed to Eric Darve.

At least 37 records · Page 2Linked to original sources

arXiv2023

Physics-based parameterized neural ordinary differential equations: prediction of laser ignition in a rocket combustor

In this work, we present a novel physics-based data-driven framework for reduced-order modeling of laser ignition in a model rocket combustor based on parameterized neural ordinary differential equations (PNODE). Deep neural networks are embedded as functions of high-dimensional parameters of laser ignition to predict various terms in a 0D flow model including the heat source function, pre-exponential factors, and activation energy. Using the governing equations of a 0D flow model, our PNODE needs only a limited number of training samples and predicts trajectories of various quantities such as temperature, pressure, and mass fractions of species while satisfying physical constraints. We validate our physics-based PNODE on solution snapshots of high-fidelity Computational Fluid Dynamics (CFD) simulations of laser-induced ignition in a prototype rocket combustor. We compare the performance of our physics-based PNODE with that of kernel ridge regression and fully connected neural networks. Our results show that our physics-based PNODE provides solutions with lower mean absolute errors of average temperature over time, thus improving the prediction of successful laser ignition with high-dimensional parameters.

cs.LG↗

arXiv2023

Learning Reduced-Order Models for Cardiovascular Simulations with Graph Neural Networks

Reduced-order models based on physics are a popular choice in cardiovascular modeling due to their efficiency, but they may experience reduced accuracy when working with anatomies that contain numerous junctions or pathological conditions. We develop one-dimensional reduced-order models that simulate blood flow dynamics using a graph neural network trained on three-dimensional hemodynamic simulation data. Given the initial condition of the system, the network iteratively predicts the pressure and flow rate at the vessel centerline nodes. Our numerical results demonstrate the accuracy and generalizability of our method in physiological geometries comprising a variety of anatomies and boundary conditions. Our findings demonstrate that our approach can achieve errors below 2% and 3% for pressure and flow rate, respectively, provided there is adequate training data. As a result, our method exhibits superior performance compared to physics-based one-dimensional models, while maintaining high efficiency at inference time.

cs.LG↗

arXiv2022

Beam-based RF Station Fault Identification at the SLAC Linac Coherent Light Source

Accelerators produce too many signals for a small operations team to monitor in real time. In addition, many of these signals are only interpretable by subject matter experts with years of experience. As a result, changes in accelerator performance can require time-intensive consultations with experts to identify the underlying problem. Herein, we focus on a particular anomaly detection task for radio-frequency (RF) stations at the Linac Coherent Light Source (LCLS). The existing RF station diagnostics are bandwidth limited, resulting in slow, unreliable signals. As a result, anomaly detection is currently a manual process. We propose a beam-based method, identifying changes in the accelerator status using shot-to-shot data from the beam position monitoring system; by comparing the beam-based anomalies to data from RF stations, we identify the source of the change. We find that our proposed method can be fully automated while identifying more events with fewer false positives than the RF station diagnostics alone. Our automated fault identification system has been used to create a new data set for investigating the interaction between the RF stations and accelerator performance.

physics.acc-ph↗

arXiv2021

A Hybrid Direct-Iterative Method for Solving KKT Linear Systems

We propose a solution strategy for linear systems arising in interior method optimization, which is suitable for implementation on hardware accelerators such as graphical processing units (GPUs). The current gold standard for solving these systems is the LDL^T factorization. However, LDL^T requires pivoting during factorization, which substantially increases communication cost and degrades performance on GPUs. Our novel approach solves a large indefinite system by solving multiple smaller positive definite systems, using an iterative solve for the Schur complement and an inner direct solve (via Cholesky factorization) within each iteration. Cholesky is stable without pivoting, thereby reducing communication and allowing reuse of the symbolic factorization. We demonstrate the practicality of our approach and show that on large systems it can efficiently utilize GPUs and outperform LDL^T factorization of the full system.

math.OC↗

arXiv2021

Integrating Deep Neural Networks with Full-waveform Inversion: Reparametrization, Regularization, and Uncertainty Quantification

Full-waveform inversion (FWI) is an accurate imaging approach for modeling velocity structure by minimizing the misfit between recorded and predicted seismic waveforms. However, the strong non-linearity of FWI resulting from fitting oscillatory waveforms can trap the optimization in local minima. We propose a neural-network-based full waveform inversion method (NNFWI) that integrates deep neural networks with FWI by representing the velocity model with a generative neural network. Neural networks can naturally introduce spatial correlations as regularization to the generated velocity model, which suppresses noise in the gradients and mitigates local minima. The velocity model generated by neural networks is input to the same partial differential equation (PDE) solvers used in conventional FWI. The gradients of both the neural networks and PDEs are calculated using automatic differentiation, which back-propagates gradients through the acoustic PDEs and neural network layers to update the weights of the generative neural network. Experiments on 1D velocity models, the Marmousi model, and the 2004 BP model demonstrate that NNFWI can mitigate local minima, especially for imaging high-contrast features like salt bodies, and significantly improves the inversion in the presence of noise. Adding dropout layers to the neural network model also allows analyzing the uncertainty of the inversion results through Monte Carlo dropout. NNFWI opens a new pathway to combine deep learning and FWI for exploiting both the characteristics of deep neural networks and the high accuracy of PDE solvers. Because NNFWI does not require extra training data and optimization loops, it provides an attractive and straightforward alternative to conventional FWI.

physics.geo-ph↗

arXiv2021

Autonomous Inversion of In Situ Deformation Measurement Data for CO2 Storage Decision Support

Current methods of estimating the change in stress caused by injecting fluid into subsurface formations require choosing the type of constitutive model and the model parameters based on core, log, and geophysical data during the characterization phase, with little feedback from operational observations to validate or refine these choices. It is shown that errors in the assumed constitutive response, even when informed by laboratory tests on core samples, are likely to be common, large, and underestimate the magnitude of stress change caused by injection. Recent advances in borehole-based strain instruments and borehole and surface-based tilt and displacement instruments have now enabled monitoring of the deformation of the storage system throughout its operational lifespan. This data can enable validation and refinement of the knowledge of the geomechanical properties and state of the system, but brings with it a challenge to transform the raw data into actionable knowledge. We demonstrate a method to perform a gradient-based deterministic inversion of geomechanical monitoring data. This approach allows autonomous integration of the instrument data without the need for time consuming manual interpretation and selection of updated model parameters. The approach presented is very flexible as to what type of geomechanical constitutive response can be used. The approach is easily adaptable to nonlinear physics-based constitutive models to account for common rock behaviors such as creep and plasticity. The approach also enables training of machine learning-based constitutive models by allowing back propagation of errors through the finite element calculations. This enables strongly enforcing known physics, such as conservation of momentum and continuity, while allowing data-driven models to learn the truly unknown physics such as the constitutive or petrophysical responses.

physics.geo-ph↗

arXiv2021

Universal Sentence Representation Learning with Conditional Masked Language Model

This paper presents a novel training method, Conditional Masked Language Modeling (CMLM), to effectively learn sentence representations on large scale unlabeled corpora. CMLM integrates sentence representation learning into MLM training by conditioning on the encoded vectors of adjacent sentences. Our English CMLM model achieves state-of-the-art performance on SentEval, even outperforming models learned using supervised signals. As a fully unsupervised learning method, CMLM can be conveniently extended to a broad range of languages and domains. We find that a multilingual CMLM model co-trained with bitext retrieval (BR) and natural language inference (NLI) tasks outperforms the previous state-of-the-art multilingual models by a large margin, e.g. 10% improvement upon baseline models on cross-lingual semantic search. We explore the same language bias of the learned representations, and propose a simple, post-training and model agnostic approach to remove the language identifying information from the representation while still retaining sentence semantics.

cs.CL↗

arXiv2021

A Simple and Effective Method To Eliminate the Self Language Bias in Multilingual Representations

Language agnostic and semantic-language information isolation is an emerging research direction for multilingual representations models. We explore this problem from a novel angle of geometric algebra and semantic space. A simple but highly effective method "Language Information Removal (LIR)" factors out language identity information from semantic related components in multilingual representations pre-trained on multi-monolingual data. A post-training and model-agnostic method, LIR only uses simple linear operations, e.g. matrix factorization and orthogonal projection. LIR reveals that for weak-alignment multilingual systems, the principal components of semantic spaces primarily encodes language identity information. We first evaluate the LIR on a cross-lingual question answer retrieval task (LAReQA), which requires the strong alignment for the multilingual embedding space. Experiment shows that LIR is highly effectively on this task, yielding almost 100% relative improvement in MAP for weak-alignment models. We then evaluate the LIR on Amazon Reviews and XEVAL dataset, with the observation that removing language information is able to improve the cross-lingual transfer performance.

cs.CL↗

arXiv2021

On the fractional Laplacian of variable order

We present a novel definition of variable-order fractional Laplacian on Rn based on a natural generalization of the standard Riesz potential. Our definition holds for values of the fractional parameter spanning the entire open set (0, n/2). We then discuss some properties of the fractional Poisson's equation involving this operator and we compute the corresponding Green function, for which we provide some instructive examples for specific problems.

math.AP↗

arXiv2021

Linear solvers for power grid optimization problems: a review of GPU-accelerated linear solvers

The linear equations that arise in interior methods for constrained optimization are sparse symmetric indefinite and become extremely ill-conditioned as the interior method converges. These linear systems present a challenge for existing solver frameworks based on sparse LU or LDL^T decompositions. We benchmark five well known direct linear solver packages using matrices extracted from power grid optimization problems. The achieved solution accuracy varies greatly among the packages. None of the tested packages delivers significant GPU acceleration for our test cases.

math.NA↗

arXiv2021

Trust Region Method for Coupled Systems of PDE Solvers and Deep Neural Networks

Physics-informed machine learning and inverse modeling require the solution of ill-conditioned non-convex optimization problems. First-order methods, such as SGD and ADAM, and quasi-Newton methods, such as BFGS and L-BFGS, have been applied with some success to optimization problems involving deep neural networks in computational engineering inverse problems. However, empirical evidence shows that convergence and accuracy for these methods remain a challenge. Our study unveiled at least two intrinsic defects of these methods when applied to coupled systems of partial differential equations (PDEs) and deep neural networks (DNNs): (1) convergence is often slow with long plateaus that make it difficult to determine whether the method has converged or not; (2) quasi-Newton methods do not provide a sufficiently accurate approximation of the Hessian matrix; this typically leads to early termination (one of the stopping criteria of the optimizer is satisfied although the achieved error is far from minimal). Based on these observations, we propose to use trust region methods for optimizing coupled systems of PDEs and DNNs. Specifically, we developed an algorithm for second-order physics constrained learning, an efficient technique to calculate Hessian matrices based on computational graphs. We show that trust region methods overcome many of the defects and exhibit remarkable fast convergence and superior accuracy compared to ADAM, BFGS, and L-BFGS.

math.NA↗

arXiv2021

Block Basis Factorization for Scalable Kernel Matrix Evaluation

Kernel methods are widespread in machine learning; however, they are limited by the quadratic complexity of the construction, application, and storage of kernel matrices. Low-rank matrix approximation algorithms are widely used to address this problem and reduce the arithmetic and storage cost. However, we observed that for some datasets with wide intra-class variability, the optimal kernel parameter for smaller classes yields a matrix that is less well approximated by low-rank methods. In this paper, we propose an efficient structured low-rank approximation method -- the Block Basis Factorization (BBF) -- and its fast construction algorithm to approximate radial basis function (RBF) kernel matrices. Our approach has linear memory cost and floating-point operations for many machine learning kernels. BBF works for a wide range of kernel bandwidth parameters and extends the domain of applicability of low-rank approximation methods significantly. Our empirical results demonstrate the stability and superiority over the state-of-art kernel approximation algorithms.

stat.ML↗

arXiv2021

PBBFMM3D: a parallel black-box algorithm for kernel matrix-vector multiplication

Kernel matrix-vector product is ubiquitous in many science and engineering applications. However, a naive method requires $O(N^2)$ operations, which becomes prohibitive for large-scale problems. We introduce a parallel method that provably requires $O(N)$ operations to reduce the computation cost. The distinct feature of our method is that it requires only the ability to evaluate the kernel function, offering a black-box interface to users. Our parallel approach targets multi-core shared-memory machines and is implemented using OpenMP. Numerical results demonstrate up to $19\times$ speedup on 32 cores. We also present a real-world application in geostatistics, where our parallel method was used to deliver fast principle component analysis of covariance matrices.

cs.MS↗

arXiv2021

Hierarchical Orthogonal Factorization: Sparse Least Squares Problems

In this work, we develop a fast hierarchical solver for solving large, sparse least squares problems. We build upon the algorithm, spaQR (sparsified QR), that was developed by the authors to solve large sparse linear systems. Our algorithm is built on top of a Nested Dissection based multifrontal QR approach. We use low-rank approximations on the frontal matrices to sparsify the vertex separators at every level in the elimination tree. Using a two-step sparsification scheme, we reduce the number of columns and maintain the ratio of rows to columns in each front without introducing any additional fill-in. With this improvised scheme, we show that the runtime of the algorithm scales as $\mathcal{O}(M \log N)$ and uses $\mathcal{O}(M)$ memory to store the factorization. This is achieved at the expense of a small and controllable approximation error. The end result is an approximate factorization of the matrix stored as a sequence of sparse orthogonal and upper-triangular factors and hence easy to apply/solve with a vector. Finally, we compare the performance of the spaQR algorithm in solving sparse least squares problems with direct multifrontal QR and CGLS iterative method with a standard diagonal preconditioner.

math.NA↗

arXiv2021

Towards a Scalable Hierarchical High-order CFD Solver

Development of highly scalable and robust algorithms for large-scale CFD simulations has been identified as one of the key ingredients to achieve NASA's CFD Vision 2030 goals. In order to improve simulation capability and to effectively leverage new high-performance computing hardware, the most computationally intensive parts of CFD solution algorithms -- namely, linear solvers and preconditioners -- need to achieve asymptotic behavior on massively parallel and heterogeneous architectures and preserve convergence rates as the meshes are refined further. In this work, we present a scalable high-order implicit Discontinuous Galerkin solver from the SU2 framework using a promising preconditioning technique based on algebraic sparsified nested dissection algorithm with low-rank approximations, and communication-avoiding Krylov subspace methods to enable scalability with very large processor counts. The overall approach is tested on a canonical 2D NACA0012 test case of increasing size to demonstrate its scalability on multiple processing cores. Both the preconditioner and the linear solver are shown to exhibit near-linear weak scaling up to 2,048 cores with no significant degradation of the convergence rate.

math.NA↗

arXiv2020

Distributed Machine Learning for Computational Engineering using MPI

We propose a framework for training neural networks that are coupled with partial differential equations (PDEs) in a parallel computing environment. Unlike most distributed computing frameworks for deep neural networks, our focus is to parallelize both numerical solvers and deep neural networks in forward and adjoint computations. Our parallel computing model views data communication as a node in the computational graph for numerical simulations. The advantage of our model is that data communication and computing are cleanly separated and thus provide better flexibility, modularity, and testability. We demonstrate using various large-scale problems that we can achieve substantial acceleration by using parallel solvers for PDEs in training deep neural networks that are coupled with PDEs.

cs.DC↗

arXiv2020

ADCME: Learning Spatially-varying Physical Fields using Deep Neural Networks

ADCME is a novel computational framework to solve inverse problems involving physical simulations and deep neural networks (DNNs). This paper benchmarks its capability to learn spatially-varying physical fields using DNNs. We demonstrate that our approach has superior accuracy compared to the discretization approach on a variety of problems, linear or nonlinear, static or dynamic. Technically, we formulate our inverse problem as a PDE-constrained optimization problem. We express both the numerical simulations and DNNs using computational graphs and therefore, we can calculate the gradients using reverse-mode automatic differentiation. We apply a physics constrained learning algorithm (PCL) to efficiently back-propagate gradients through iterative solvers for nonlinear equations. The open-source software which accompanies the present paper can be found at https://github.com/kailaix/ADCME.jl.

math.NA↗

arXiv2020

Application of Deep Learning-based Interpolation Methods to Nearshore Bathymetry

Nearshore bathymetry, the topography of the ocean floor in coastal zones, is vital for predicting the surf zone hydrodynamics and for route planning to avoid subsurface features. Hence, it is increasingly important for a wide variety of applications, including shipping operations, coastal management, and risk assessment. However, direct high resolution surveys of nearshore bathymetry are rarely performed due to budget constraints and logistical restrictions. Another option when only sparse observations are available is to use Gaussian Process regression (GPR), also called Kriging. But GPR has difficulties recognizing patterns with sharp gradients, like those found around sand bars and submerged objects, especially when observations are sparse. In this work, we present several deep learning-based techniques to estimate nearshore bathymetry with sparse, multi-scale measurements. We propose a Deep Neural Network (DNN) to compute posterior estimates of the nearshore bathymetry, as well as a conditional Generative Adversarial Network (cGAN) that samples from the posterior distribution. We train our neural networks based on synthetic data generated from nearshore surveys provided by the U.S.\ Army Corps of Engineer Field Research Facility (FRF) in Duck, North Carolina. We compare our methods with Kriging on real surveys as well as surveys with artificially added sharp gradients. Results show that direct estimation by DNN gives better predictions than Kriging in this application. We use bootstrapping with DNN for uncertainty quantification. We also propose a method, named DNN-Kriging, that combines deep learning with Kriging and shows further improvement of the posterior estimates.

stat.ML↗