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A hybrid FEM-PINN method for time-dependent partial differential equations

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arXiv 2024年

作者： Feng, Xiaodong Shangguan, Haojiong Tang, Tao Wan, Xiaoliang Zhou, Tao Division of Science and Technology BNU-HKBU United International College Zhuhai519087 China School of Electrical and Computer Engineering Guangzhou Nanfang College Guangzhou510970 China Department of Mathematics Center for Computation and Technology Louisiana State University Baton Rouge70803 United States Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China

In this work, we present a hybrid numerical method for solving evolution partial differential equations (PDEs) by merging the time finite element method with deep neural networks. In contrast to the conventional deep learning-based formulation where the neural network is defined on a spatiotemporal domain, our methodology utilizes finite element basis functions in the time direction where the space-dependent coefficients are defined as the output of a neural network. We then apply the Galerkin or collocation projection in the time direction to obtain a system of PDEs for the space-dependent coefficients which is approximated in the framework of PINN. The advantages of such a hybrid formulation are twofold: statistical errors are avoided for the integral in the time direction, and the neural network’s output can be regarded as a set of reduced spatial basis functions. To further alleviate the difficulties from high dimensionality and low regularity, we have developed an adaptive sampling strategy that refines the training set. More specifically, we use an explicit density model to approximate the distribution induced by the PDE residual and then augment the training set with new time-dependent random samples given by the learned density model. The effectiveness and efficiency of our proposed method have been demonstrated through a series of numerical experiments. © 2024, CC BY.

关键词： Partial differential equations

MC-Nonlocal-PINNs: handling nonlocal operators in PINNs via Monte Carlo sampling

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arXiv 2022年

作者： Feng, Xiaodong Qian, Yue Shen, Wanfang Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China Shandong Key Laboratory of Blockchain Finance Shandong University of Finance and Economics Jinan250014 China

We propose, Monte Carlo Nonlocal physics-informed neural networks (MC-Nonlocal-PINNs), which is a generalization of MC-fPINNs in [1], for solving general nonlocal models such as integral equations and nonlocal PDEs. Similar as in MC-fPINNs, our MC-Nonlocal-PINNs handle the nonlocal operators in a Monte Carlo way, resulting in a very stable approach for high dimensional problems. We present a variety of test problems, including high dimensional Volterra type integral equations, hypersingular integral equations and nonlocal PDEs, to demonstrate the effectiveness of our approach. © 2022, CC BY.

关键词： Deep neural networks

An Augmented Subspace Based Adaptive Proper Orthogonal Decomposition Method for Time Dependent Partial Differential Equations

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arXiv 2023年

作者： Dai, Xiaoying Hu, Miao Xin, Jack Zhou, Aihui LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China Department of Mathematics University of California at Irvine IrvineCA92697 United States

In this paper, we propose an augmented subspace based adaptive proper orthogonal decomposition (POD) method for solving the time dependent partial differential equations. By augmenting the POD subspace with some auxiliary modes, we obtain an augmented subspace. We use the difference between the approximation obtained in this augmented subspace and that obtained in the original POD subspace to construct an error indicator, by which we obtain a general framework for augmented subspace based adaptive POD method. We then provide two strategies to obtain some specific augmented subspaces, the random vector based augmented subspace and the coarse-grid approximations based augmented subspace. We apply our new method to two typical 3D advection-diffusion equations with the advection being the Kolmogorov flow and the ABC flow. Numerical results show that our method is more efficient than the existing adaptive POD methods, especially for the advection dominated models. Copyright © 2023, The Authors. All rights reserved.

关键词： Advection

Detecting and Exploiting Unit Commitment Structures in MIPs

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Detecting and Exploiting Unit Commitment Structures in MIPs

International Conference on Power and Energy Technology (ICPET)

作者： Zhao-Gang Su Sheng-Jie Chen Yu-Fei Li Liang Chen Caixia Kou Zhi Cai Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China Institute of Applied Mathematics Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China College of Sciences Beijing University of Posts and Telecommunications Beijing China Beijing Key Laboratory of Research and System Evaluation of Power Dispatching Automation Technology China Electric Power Research Institute Beijing China

ISBN: (数字)9798350375855

ISBN: (纸本)9798350375862

The unit commitment (UC) problem has been extensively researched in the literature, which is typically formulated as a mixed integer programming (MIP) problem. However, current studies lack effective methods to identify potential UC problem structures within a general MIP problem, making it challenging to leverage well-established valid inequalities for UC problems to accelerate the solution process. In this paper, we present a novel detection method that automatically identifies UC structures within an MIP problem by analyzing the relationships between variables and constraints. Moreover, we enhance the MIP formulation by incorporating three well-studied classes of valid inequalities as cutting planes, based on the identified UC structures. Our experiments demonstrate that our proposed UC detection method greatly improves computational efficacy of a standard MIP solver compared to the default setting.

关键词： Integer programming Production Generators Time factors Standards

POINTWISE ESTIMATES FOR THE FUNDAMENTAL SOLUTIONS OF HIGHER ORDER SCHRÖDINGER EQUATIONS IN ODD DIMENSIONS II: HIGH DIMENSIONAL CASE

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arXiv 2024年

作者： Cheng, Han Huang, Shanlin Huang, Tianxiao Zheng, Quan Institute of Applied Physics and Computational Mathematics Beijing100088 China School of Mathematics and Statistics Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Hubei Wuhan430074 China Sun Yat-sen University Guangdong Zhuhai519082 China School of Mathematics and Statistics Huazhong University of Science and Technology Hubei Wuhan430074 China

In this paper, for any odd n and any integer m ≥ 1 with n > 4m, we study the fundamental solution of the higher order Schrödinger equation i∂tu(x, t) = ((−∆)m + V(x))u(x, t), t ∈ R, x ∈ Rn, where V is a real-valued C n+21 −2m potential with certain decay. Let Pac(H) denote the projection onto the absolutely continuous spectrum space of H = (−∆)m+V, and assume that H has no positive embedded eigenvalue. Our main result says that e−itHPac(H) has integral kernel K(t, x, y) satisfying n(m−1) |K(t, x, y)| ≤ C(1 + |t|)−(2nm −σ)(1 + |t|− 2nm ) (1 + |t|− 21m |x − y|)− 2m−1 , t , 0, x, y ∈ Rn, where σ = 2 if 0 is an eigenvalue of H, and σ = 0 otherwise. A similar result for smoothing operators H 2αm e−itHPac(H) is also given. The regularity condition V ∈ C n+21 −2m is optimal in the second order case, and it also seems optimal when m > 1. © 2024, CC0.

关键词： Schrodinger equation

Energy-Efficient Beamforming Design for Integrated Sensing and Communications Systems

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arXiv 2023年

作者： Zou, Jiaqi Sun, Songlin Masouros, Christos Cui, Yuanhao Liu, Ya-Feng Ng, Derrick Wing Kwan Beijing100876 China The Department of Electrical and Electronic Engineering University College London LondonWC1E 7JE United Kingdom Beijing China The Department of Electrical and Electronic Engineering University College London WC1E 7JE United Kingdom The State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China The School of Electrical Engineering and Telecommunications University of New South Wales SydneyNSW2052 Australia

In this paper, we investigate the design of energy-efficient beamforming for an ISAC system, where the transmitted waveform is optimized for joint multi-user communication and target estimation simultaneously. We aim to maximize the system energy efficiency (EE), taking into account the constraints of a maximum transmit power budget, a minimum required signal-to-interference-plus-noise ratio (SINR) for communication, and a maximum tolerable Cramér-Rao bound (CRB) for target estimation. We first consider communication-centric EE maximization. To handle the non-convex fractional objective function, we propose an iterative quadratic-transform-Dinkelbach method, where Schur complement and semi-definite relaxation (SDR) techniques are leveraged to solve the subproblem in each iteration. For the scenarios where sensing is critical, we propose a novel performance metric for characterizing the sensing-centric EE and optimize the metric adopted in the scenario of sensing a point-like target and an extended target. To handle the nonconvexity, we employ the successive convex approximation (SCA) technique to develop an efficient algorithm for approximating the nonconvex problem as a sequence of convex ones. Furthermore, we adopt a Pareto optimization mechanism to articulate the tradeoff between the communication-centric EE and sensing-centric EE. We formulate the search of the Pareto boundary as a constrained optimization problem and propose a computationally efficient algorithm to handle it. Numerical results validate the effectiveness of our proposed algorithms compared with the baseline schemes and the obtained approximate Pareto boundary shows that there is a non-trivial tradeoff between communication-centric EE and sensing-centric EE, where the number of communication users and EE requirements have serious effects on the achievable tradeoff. © 2023, CC BY-NC-ND.

关键词： Energy efficiency

An Inexact Augmented Lagrangian Algorithm for Training Leaky ReLU Neural Network with Group Sparsity

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arXiv 2022年

作者： Liu, Wei Liu, Xin Chen, Xiaojun Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China Department of Applied Mathematics The Hong Kong Polytechnic University Hung Hom Kowloon Hong Kong

The leaky ReLU network with a group sparse regularization term has been widely used in the recent years. However, training such network yields a nonsmooth nonconvex optimization problem and there exists a lack of approaches to compute a stationary point deterministically. In this paper, we first resolve the multi-layer composite term in the original optimization problem by introducing auxiliary variables and additional constraints. We show the new model has a nonempty and bounded solution set and its feasible set satisfies the Mangasarian-Fromovitz constraint qualification. Moreover, we show the relationship between the new model and the original problem. Remarkably, we propose an inexact augmented Lagrangian algorithm for solving the new model, and show the convergence of the algorithm to a KKT point. Numerical experiments demonstrate that our algorithm is more efficient for training sparse leaky ReLU neural networks than some well-known algorithms. Copyright © 2022, The Authors. All rights reserved.

关键词： Constrained optimization

POINTWISE ESTIMATES FOR THE FUNDAMENTAL SOLUTIONS OF HIGHER ORDER SCHRÖDINGER EQUATIONS IN ODD DIMENSIONS I: LOW DIMENSIONAL CASE

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arXiv 2024年

In this paper, for any odd n and any integer m ≥ 1 with n tu(x, t) = ((−∆)m + V(x))u(x, t), t ∈ , x ∈ n, where V is a real-valued potential with certain decay. Let Pac(H) denote the projection onto the absolutely continuous spectrum space of H = (−∆)m + V, and assume that H has no positive embedded eigenvalue. Our main result says that e−itHPac(H) has integral kernel K(t, x, y) satisfying (Equation presented), where the constants C, h > 0, and h can be specified by m, n and the zero energy resonances of H. A similar result for smoothing operators like (Equation presented) is also given. One of the main novelties lies in a new and unified approach to obtain the asymptotic expansions of perturbed resolvents under various types of zero energy resonances. © 2024, CC0.

关键词： Schrodinger equation

IB-UQ: Information bottleneck based uncertainty quantification for neural function regression and neural operator learning

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arXiv 2023年

作者： Guo, Ling Wu, Hao Zhou, Wenwen Wang, Yan Zhou, Tao Department of Mathematics Shanghai Normal University Shanghai China School of Mathematical Sciences Institute of Natural Sciences MOE-LSC Shanghai Jiaotong University Shanghai China School of Mathematical Sciences Tongji University Shanghai China LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China

We propose a novel framework for uncertainty quantification via information bottleneck (IB-UQ) for scientific machine learning tasks, including deep neural network (DNN) regression and neural operator learning (DeepONet). Specifically, we incorporate the bottleneck by a confidence-aware encoder, which encodes inputs into latent representations according to the confidence of the input data belonging to the region where training data is located, and utilize a Gaussian decoder to predict means and variances of outputs conditional on representation variables. Furthermore, we propose a data augmentation based information bottleneck objective which can enhance the quantification quality of the extrapolation uncertainty, and the encoder and decoder can be both trained by minimizing a tractable variational bound of the objective. In comparison to uncertainty quantification (UQ) methods for scientific learning tasks that rely on Bayesian neural networks with Hamiltonian Monte Carlo posterior estimators, the model we propose is computationally efficient, particularly when dealing with large-scale data sets. The effectiveness of the IB-UQ model has been demonstrated through several representative examples, such as regression for discontinuous functions, real-world data set regression, learning nonlinear operators for partial differential equations, and a large-scale climate model. The experimental results indicate that the IB-UQ model can handle noisy data, generate robust predictions, and provide confident uncertainty evaluation for out-of-distribution data. © 2023, CC BY-NC-ND.

关键词： Monte Carlo methods