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SPECTRAL-REFINER: ACCURATE FINE-TUNING OF SPATIOTEMPORAL FOURIER NEURAL OPERATOR FOR TURBULENT FLOWS

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arXiv 2024年

作者： Cao, Shuhao Brarda, Francesco Li, Rui Peng Xi, Yuanzhe School of Science and Engineering University of Missouri Kansas City United States Department of Mathematics Emory University United States Center for Applied Scientific Computing Lawrence Livermore National Laboratory United States

Recent advancements in operator-type neural networks have shown promising results in approximating the solutions of spatiotemporal Partial Differential Equations (PDEs). However, these neural networks often entail considerable training expenses, and may not always achieve the desired accuracy required in many scientific and engineering disciplines. In this paper, we propose a new learning framework to address these issues. A new spatiotemporal adaptation is proposed to generalize any Fourier Neural Operator (FNO) variant to learn maps between Bochner spaces, which can perform an arbitrary-length temporal super-resolution for the first time. To better exploit this capacity, a new paradigm is proposed to refine the commonly adopted end-to-end neural operator training and evaluations with the help from the wisdom from traditional numerical PDE theory and techniques. Specifically, in the learning problems for the turbulent flow modeled by the Navier-Stokes Equations (NSE), the proposed paradigm trains an FNO only for a few epochs. Then, only the newly proposed spatiotemporal spectral convolution layer is fine-tuned without the frequency truncation. The spectral fine-tuning loss function uses a negative Sobolev norm for the first time in operator learning, defined through a reliable functional-type a posteriori error estimator whose evaluation is exact thanks to the Parseval identity. Moreover, unlike the difficult nonconvex optimization problems in the end-to-end training, this fine-tuning loss is convex. Numerical experiments on commonly used NSE benchmarks demonstrate significant improvements in both computational efficiency and accuracy, compared to end-to-end evaluation and traditional numerical PDE solvers under certain conditions. The source code is publicly available at https://***/scaomath/*** Codes 65M70 (Primary), 35Q30, 76M22, 65M50, 68T07 (Secondary) Copyright © 2024, The Authors. All rights reserved.

关键词： Navier Stokes equations

ANALYSIS OF THE SIMPL METHOD FOR DENSITY-BASED TOPOLOGY OPTIMIZATION

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arXiv 2024年

作者： Keith, Brendan Kim, Dohyun Lazarov, Boyan S. Surowiec, Thomas M. Division of Applied Mathematics Brown University ProvidenceRI02912 United States Lawrence Livermore National Laboratory LivermoreCA94550 United States Department of Numerical Analysis and Scientific Computing Simula Research Laboratory Oslo0164 Norway

We present a rigorous convergence analysis of a new method for density-based topology optimization that provides point-wise bound preserving design updates and faster convergence than other popular first-order topology optimization methods. Due to its strong bound preservation, the method is exceptionally robust, as demonstrated in numerous examples here and in the companion article [31]. Furthermore, it is easy to implement with clear structure and analytical expressions for the updates. Our analysis covers two versions of the method, characterized by the employed line search strategies. We consider a modified Armijo backtracking line search and a Bregman backtracking line search. For both line search algorithms, our algorithm delivers a strict monotone decrease in the objective function and further intuitive convergence properties, e.g., strong and pointwise convergence of the density variables on the active sets, norm convergence to zero of the increments, convergence of the Lagrange multipliers, and more. In addition, the numerical experiments demonstrate apparent mesh-independent convergence of the algorithm. We refer to the new algorithm as the SiMPL method, pronounced like "simple", which stands for Sigmoidal Mirror descent with a Projected Latent *** Codes 68Q25, 68R10, 68U05 Copyright © 2024, The Authors. All rights reserved.

关键词： Lagrange multipliers

Graph Coarsening via Supervised Granular-Ball for Scalable Graph Neural Network Training

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arXiv 2024年

作者： Xia, Shuyin Ma, Xinjun Liu, Zhiyuan Liu, Cheng Zhao, Sen Wang, Guoyin Key Laboratory of Big Data Intelligent Computing China Chongqing Key Laboratory of Computational Intelligence China Chongqing University of Posts and Telecommunications Chongqing China National Center for Applied Mathematics in Chongqing Chongqing Normal University Chongqing401331 China

Graph Neural Networks (GNNs) have demonstrated significant achievements in processing graph data, yet scalability remains a substantial challenge. To address this, numerous graph coarsening methods have been developed. However, most existing coarsening methods are training-dependent, leading to lower efficiency, and they all require a predefined coarsening rate, lacking an adaptive approach. In this paper, we employ granular-ball computing to effectively compress graph data. We construct a coarsened graph network by iteratively splitting the graph into granular-balls based on a purity threshold and using these granular-balls as super vertices. This granulation process significantly reduces the size of the original graph, thereby greatly enhancing the training efficiency and scalability of GNNs. Additionally, our algorithm can adaptively perform splitting without requiring a predefined coarsening rate. Experimental results demonstrate that our method achieves accuracy comparable to training on the original graph. Noise injection experiments further indicate that our method exhibits robust performance. Moreover, our approach can reduce the graph size by up to 20 times without compromising test accuracy, substantially enhancing the scalability of GNNs. Copyright © 2024, The Authors. All rights reserved.

关键词： Graph neural networks

A fast and accurate domain decomposition nonlinear manifold reduced order model

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arXiv 2023年

作者： Diaz, Alejandro N. Choi, Youngsoo Heinkenschloss, Matthias Department of Compuational Applied Mathematics and Operations Research Rice University HoustonTX77005 United States Center for Applied Scientific Computing Lawrence Livermore National Laboratory LivermoreCA94550 United States

This paper integrates nonlinear-manifold reduced order models (NM-ROMs) with domain decomposition (DD). NM-ROMs approximate the full order model (FOM) state in a nonlinear-manifold by training a shallow, sparse autoencoder using FOM snapshot data. These NM-ROMs can be advantageous over linear-subspace ROMs (LS-ROMs) for problems with slowly decaying Kolmogorov n-width. However, the number of NM-ROM parameters that need to be trained scales with the size of the FOM. Moreover, for "extreme-scale" problems, the storage of high-dimensional FOM snapshots alone can make ROM training expensive. To alleviate the training cost, this paper applies DD to the FOM, computes NM-ROMs on each subdomain, and couples them to obtain a global NM-ROM. This approach has several advantages: Subdomain NM-ROMs can be trained in parallel, involve fewer parameters to be trained than global NM-ROMs, require smaller subdomain FOM dimensional training data, and can be tailored to subdomain-specific features of the FOM. The shallow, sparse architecture of the autoencoder used in each subdomain NM-ROM allows application of hyper-reduction (HR), reducing the complexity caused by nonlinearity and yielding computational speedup of the NM-ROM. This paper provides the first application of NM-ROM (with HR) to a DD problem. In particular, this paper details an algebraic DD reformulation of the FOM, training a NM-ROM with HR for each subdomain, and a sequential quadratic programming (SQP) solver to evaluate the coupled global NM-ROM. Theoretical convergence results for the SQP method and a priori and a posteriori error estimates for the DD NM-ROM with HR are provided. The proposed DD NM-ROM with HR approach is numerically compared to a DD LS-ROM with HR on the 2D steady-state Burgers’ equation, showing an order of magnitude improvement in accuracy of the proposed DD NM-ROM over the DD LS-ROM. Copyright © 2023, The Authors. All rights reserved.

关键词： Quadratic programming

A Simple Introduction to the SiMPL Method for Density-Based Topology Optimization

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arXiv 2024年

作者： Kim, Dohyun Lazarov, Boyan S. Surowiec, Thomas M. Keith, Brendan Division of Applied Mathematics Brown University ProvidenceRI02912 United States Lawrence Livermore National Laboratory LivermoreCA94550 United States Department of Numerical Analysis and Scientific Computing Simula Research Laboratory Oslo0164 Norway

We introduce a novel method for solving density-based topology optimization problems: Sigmoidal Mirror descent with a Projected Latent variable (SiMPL). The SiMPL method (pronounced as "the simple method") optimizes a design using only first-order derivative information of the objective function. The bound constraints on the density field are enforced with the help of the (negative) Fermi–Dirac entropy, which is also used to define a non-symmetric distance function called a Bregman divergence on the set of admissible designs. This Bregman divergence leads to a simple update rule that is further simplified with the help of a so-called latent variable. Because the SiMPL method involves discretizing the latent variable, it produces a sequence of pointwise-feasible iterates, even when high-order finite elements are used in the discretization. Numerical experiments demonstrate that the method outperforms other popular first-order optimization algorithms. To outline the general applicability of the technique, we include examples with (self-load) compliance minimization and compliant mechanism optimization problems. Copyright © 2024, The Authors. All rights reserved.

关键词： Mirrors

A Kaczmarz-inspired approach to accelerate the optimization of neural network wavefunctions

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arXiv 2024年

作者： Goldshlager, Gil Abrahamsen, Nilin Lin, Lin Department of Mathematics University of California BerkeleyCA94720 United States The Simons Institute for the Theory of Computing BerkeleyCA94720 United States Applied Mathematics and Computational Research Division Lawrence Berkeley National Laboratory BerkeleyCA94720 United States

Neural network wavefunctions optimized using the variational Monte Carlo method have been shown to produce highly accurate results for the electronic structure of atoms and small molecules, but the high cost of optimizing such wavefunctions prevents their application to larger systems. We propose the Subsampled Projected-Increment Natural Gradient Descent (SPRING) optimizer to reduce this bottleneck. SPRING combines ideas from the recently introduced minimum-step stochastic reconfiguration optimizer (MinSR) and the classical randomized Kaczmarz method for solving linear least-squares problems. We demonstrate that SPRING outperforms both MinSR and the popular Kronecker-Factored Approximate Curvature method (KFAC) across a number of small atoms and molecules, given that the learning rates of all methods are optimally tuned. For example, on the oxygen atom, SPRING attains chemical accuracy after forty thousand training iterations, whereas both MinSR and KFAC fail to do so even after one hundred thousand iterations. Copyright © 2024, The Authors. All rights reserved.

关键词： Monte Carlo methods

Moment-Based Multi-Resolution HWENO Scheme for Hyperbolic Conservation Laws

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Communications in Computational Physics 2022年第7期32卷 364-400页

作者： Jiayin Li Chi-Wang Shu Jianxian Qiu School of Mathematical Sciences Xiamen UniversityXiamenFujian 361005P.R.China Division of Applied Mathematics Brown UniversityProvidenceRI 02912USA School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathe-matical Modeling and High-Performance Scientific Computing Xiamen UniversityXiamenFujian 361005P.R.China

In this paper,a high-order moment-based multi-resolution Hermite weighted essentially non-oscillatory(HWENO)scheme is designed for hyperbolic conservation *** main idea of this scheme is derived from our previous work[***.,446(2021)110653],in which the integral averages of the function and its first order derivative are used to reconstruct both the function and its first order derivative values at the ***,in this paper,only the function values at the Gauss-Lobatto points in the one or two dimensional case need to be reconstructed by using the information of the zeroth and first order *** addition,an extra modification procedure is used to modify those first order moments in the troubledcells,which leads to an improvement of stability and an enhancement of resolution near *** obtain the same order of accuracy,the size of the stencil required by this moment-based multi-resolution HWENO scheme is still the same as the general HWENO scheme and is more compact than the generalWENO ***,the linear weights are not unique and are independent of the node position,and the CFL number can still be 0.6whether for the one or two dimensional case,which has to be 0.2 in the two dimensional case for other HWENO *** numerical examples are given to demonstrate the stability and resolution of such moment-based multi-resolution HWENO scheme.

关键词： Moment-based scheme multi-resolution scheme HWENO scheme hyperbolic conservation laws KXRCF troubled-cell indicator HLLC-flux

Efficient Quantum Counting and Quantum Content-Addressable Memory for DNA Similarity

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Efficient Quantum Counting and Quantum Content-Addressable M...

Quantum computing and Engineering (QCE), IEEE International Conference on

作者： Jan Balewski Daan Camps Katherine Klymko Andrew Tritt Lawrence Berkeley National Laboratory National Energy Research Scientific Computing Center Berkeley CA USA Applied Mathematics and Computational Research Division Lawrence Berkeley National Laboratory Berkeley CA USA

We present QCAM, a quantum analogue of Content-Addressable Memory (CAM), useful for finding matches in two sequences of bit-strings. Our QCAM implementation takes advantage of Grover's search algorithm and proposes a highly-optimized quantum circuit implementation of the QCAM oracle. Our circuit construction uses the parallel uniformly controlled rotation gates, which were used in previous work to generate QBArt encodings. These circuits have a high degree of quantum parallelism which reduces their critical depth. The optimal number of repetitions of the Grover iterator used in QCAM depends on the number of true matches and hence is input dependent. We additionally propose a hardware-efficient implementation of the quantum counting algorithm (HEQC) that can infer the optimal number of Grover iterations from the measurement of a single observable. We demonstrate the QCAM application for computing the Jaccard similarity between two sets of k-mers obtained from two DNA sequences.

关键词：

IMPLICIT-EXPLICIT MULTIRATE INFINITESIMAL STAGE-RESTART METHODS

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arXiv 2023年

作者： Fish, Alex C. Reynolds, Daniel R. Roberts, Steven B. Department of Mathematics Southern Methodist University DallasTX United States Center for Applied Scientific Computing Lawrence Livermore National Laboratory LivermoreCA United States

Implicit-Explicit (IMEX) methods are flexible numerical time integration methods which solve an initial-value problem (IVP) that is partitioned into stiff and nonstiff processes with the goal of lower computational costs than a purely implicit or explicit approach. A complementary form of flexible IVP solvers are multirate infinitesimal methods for problems partitioned into fast- and slow-changing dynamics, that solve a multirate IVP by evolving a sequence of "fast" IVPs using any suitably accurate algorithm. This article introduces a new class of high-order implicit-explicit multirate methods that are designed for multirate IVPs in which the slow-changing dynamics are further partitioned in an IMEX fashion. This new class, which we call implicit-explicit multirate stage-restart (IMEX-MRI-SR), both improves upon the previous implicit-explicit multirate generalized-structure additive Runge Kutta (IMEX-MRI-GARK) methods, and extends multirate exponential Runge Kutta (MERK) methods into the IMEX context. We leverage GARK theory to derive conditions guaranteeing orders of accuracy up to four. We provide second-, third-, and fourth-order accurate example methods and perform numerical simulations demonstrating convergence rates and computational performance in both fixed-step and adaptive-step *** Codes 65L20, 65L05, 65L06 Copyright © 2023, The Authors. All rights reserved.

关键词： Initial value problems