Lili Ju

Si-Yu Yi, Wei Ju, Yifang Qin et al. · pku

Graph clustering, which learns the node representations for effective cluster assignments, is a fundamental yet challenging task in data analysis and has received considerable attention accompanied by graph neural networks in recent years. However, most existing methods overlook the inherent relational information among the non-independent and non-identically distributed nodes in a graph. Due to the lack of exploration of relational attributes, the semantic information of the graph-structured data fails to be fully exploited which leads to poor clustering performance. In this paper, we propose a novel self-supervised deep graph clustering method named Relational Redundancy-Free Graph Clustering (R$^2$FGC) to tackle the problem. It extracts the attribute- and structure-level relational information from both global and local views based on an autoencoder and a graph autoencoder. To obtain effective representations of the semantic information, we preserve the consistent relation among augmented nodes, whereas the redundant relation is further reduced for learning discriminative embeddings. In addition, a simple yet valid strategy is utilized to alleviate the over-smoothing issue. Extensive experiments are performed on widely used benchmark datasets to validate the superiority of our R$^2$FGC over state-of-the-art baselines. Our codes are available at https://github.com/yisiyu95/R2FGC.

Wei Ju, Yifang Qin, Ziyue Qiao et al.

Point-of-Interest (POI) recommendation, which benefits from the proliferation of GPS-enabled devices and location-based social networks (LBSNs), plays an increasingly important role in recommender systems. It aims to provide users with the convenience to discover their interested places to visit based on previous visits and current status. Most existing methods usually merely leverage recurrent neural networks (RNNs) to explore sequential influences for recommendation. Despite the effectiveness, these methods not only neglect topological geographical influences among POIs, but also fail to model high-order sequential substructures. To tackle the above issues, we propose a Kernel-Based Graph Neural Network (KBGNN) for next POI recommendation, which combines the characteristics of both geographical and sequential influences in a collaborative way. KBGNN consists of a geographical module and a sequential module. On the one hand, we construct a geographical graph and leverage a message passing neural network to capture the topological geographical influences. On the other hand, we explore high-order sequential substructures in the user-aware sequential graph using a graph kernel neural network to capture user preferences. Finally, a consistency learning framework is introduced to jointly incorporate geographical and sequential information extracted from two separate graphs. In this way, the two modules effectively exchange knowledge to mutually enhance each other. Extensive experiments conducted on two real-world LBSN datasets demonstrate the superior performance of our proposed method over the state-of-the-arts. Our codes are available at https://github.com/Fang6ang/KBGNN.

Qiang Du, Lili Ju, Xiao Li et al.

The nonlocal Allen-Cahn (NAC) equation is a generalization of the classic Allen-Cahn equation by replacing the Laplacian with a parameterized nonlocal diffusion operator, and satisfies the maximum principle as its local counterpart. In this paper, we develop and analyze first and second order exponential time differencing (ETD) schemes for solving the NAC equation, which unconditionally preserve the discrete maximum principle. The fully discrete numerical schemes are obtained by applying the stabilized ETD approximations for time integration with the quadrature-based finite difference discretization in space. We derive their respective optimal maximum-norm error estimates and further show that the proposed schemes are asymptotically compatible, i.e., the approximate solutions always converge to the classic Allen-Cahn solution when the horizon, the spatial mesh size and the time step size go to zero. We also prove that the schemes are energy stable in the discrete sense. Various experiments are performed to verify these theoretical results and to investigate numerically the relation between the discontinuities and the nonlocal parameters.

Bin Feng, Tenglong Ao, Zequn Liu et al.

How to automatically synthesize natural-looking dance movements based on a piece of music is an incrementally popular yet challenging task. Most existing data-driven approaches require hard-to-get paired training data and fail to generate long sequences of motion due to error accumulation of autoregressive structure. We present a novel 3D dance synthesis system that only needs unpaired data for training and could generate realistic long-term motions at the same time. For the unpaired data training, we explore the disentanglement of beat and style, and propose a Transformer-based model free of reliance upon paired data. For the synthesis of long-term motions, we devise a new long-history attention strategy. It first queries the long-history embedding through an attention computation and then explicitly fuses this embedding into the generation pipeline via multimodal adaptation gate (MAG). Objective and subjective evaluations show that our results are comparable to strong baseline methods, despite not requiring paired training data, and are robust when inferring long-term music. To our best knowledge, we are the first to achieve unpaired data training - an ability that enables to alleviate data limitations effectively. Our code is released on https://github.com/BFeng14/RobustDancer

Wei Ju, Xiao Luo, Meng Qu et al.

This paper studies semi-supervised graph classification, a crucial task with a wide range of applications in social network analysis and bioinformatics. Recent works typically adopt graph neural networks to learn graph-level representations for classification, failing to explicitly leverage features derived from graph topology (e.g., paths). Moreover, when labeled data is scarce, these methods are far from satisfactory due to their insufficient topology exploration of unlabeled data. We address the challenge by proposing a novel semi-supervised framework called Twin Graph Neural Network (TGNN). To explore graph structural information from complementary views, our TGNN has a message passing module and a graph kernel module. To fully utilize unlabeled data, for each module, we calculate the similarity of each unlabeled graph to other labeled graphs in the memory bank and our consistency loss encourages consistency between two similarity distributions in different embedding spaces. The two twin modules collaborate with each other by exchanging instance similarity knowledge to fully explore the structure information of both labeled and unlabeled data. We evaluate our TGNN on various public datasets and show that it achieves strong performance.

Yifang Qin, Yifan Wang, Fang Sun et al.

Point-of-Interest (POI) recommendation plays a vital role in various location-aware services. It has been observed that POI recommendation is driven by both sequential and geographical influences. However, since there is no annotated label of the dominant influence during recommendation, existing methods tend to entangle these two influences, which may lead to sub-optimal recommendation performance and poor interpretability. In this paper, we address the above challenge by proposing DisenPOI, a novel Disentangled dual-graph framework for POI recommendation, which jointly utilizes sequential and geographical relationships on two separate graphs and disentangles the two influences with self-supervision. The key novelty of our model compared with existing approaches is to extract disentangled representations of both sequential and geographical influences with contrastive learning. To be specific, we construct a geographical graph and a sequential graph based on the check-in sequence of a user. We tailor their propagation schemes to become sequence-/geo-aware to better capture the corresponding influences. Preference proxies are extracted from check-in sequence as pseudo labels for the two influences, which supervise the disentanglement via a contrastive loss. Extensive experiments on three datasets demonstrate the superiority of the proposed model.

Wei Ju, Yiyang Gu, Binqi Chen et al.

This paper studies the problem of graph-level clustering, which is a novel yet challenging task. This problem is critical in a variety of real-world applications such as protein clustering and genome analysis in bioinformatics. Recent years have witnessed the success of deep clustering coupled with graph neural networks (GNNs). However, existing methods focus on clustering among nodes given a single graph, while exploring clustering on multiple graphs is still under-explored. In this paper, we propose a general graph-level clustering framework named Graph-Level Contrastive Clustering (GLCC) given multiple graphs. Specifically, GLCC first constructs an adaptive affinity graph to explore instance- and cluster-level contrastive learning (CL). Instance-level CL leverages graph Laplacian based contrastive loss to learn clustering-friendly representations while cluster-level CL captures discriminative cluster representations incorporating neighbor information of each sample. Moreover, we utilize neighbor-aware pseudo-labels to reward the optimization of representation learning. The two steps can be alternatively trained to collaborate and benefit each other. Experiments on a range of well-known datasets demonstrate the superiority of our proposed GLCC over competitive baselines.

Chengwu Liu, Jianhao Shen, Huajian Xin et al.

We present FIMO, an innovative dataset comprising formal mathematical problem statements sourced from the International Mathematical Olympiad (IMO) Shortlisted Problems. Designed to facilitate advanced automated theorem proving at the IMO level, FIMO is currently tailored for the Lean formal language. It comprises 149 formal problem statements, accompanied by both informal problem descriptions and their corresponding LaTeX-based informal proofs. Through initial experiments involving GPT-4, our findings underscore the existing limitations in current methodologies, indicating a substantial journey ahead before achieving satisfactory IMO-level automated theorem proving outcomes.

Jingyang Yuan, Xiao Luo, Yifang Qin et al.

Node classification on graphs is a significant task with a wide range of applications, including social analysis and anomaly detection. Even though graph neural networks (GNNs) have produced promising results on this task, current techniques often presume that label information of nodes is accurate, which may not be the case in real-world applications. To tackle this issue, we investigate the problem of learning on graphs with label noise and develop a novel approach dubbed Consistent Graph Neural Network (CGNN) to solve it. Specifically, we employ graph contrastive learning as a regularization term, which promotes two views of augmented nodes to have consistent representations. Since this regularization term cannot utilize label information, it can enhance the robustness of node representations to label noise. Moreover, to detect noisy labels on the graph, we present a sample selection technique based on the homophily assumption, which identifies noisy nodes by measuring the consistency between the labels with their neighbors. Finally, we purify these confident noisy labels to permit efficient semantic graph learning. Extensive experiments on three well-known benchmark datasets demonstrate the superiority of our CGNN over competing approaches.

Qingshan Chen, Lili Ju, Roger Temam

An energy-conserving and an energy-and-enstrophy conserving numerical schemes are derived, by approximating the Hamiltonian formulation, based on the Poisson brackets and the vorticity-divergence variables, of the inviscid shallow water flows. The conservation of the energy and/or enstrophy stems from skew-symmetry of the Poisson brackets, which is retained in the discrete approximations. These schemes operate on unstructured orthogonal dual meshes, over bounded or unbounded domains, and they are also shown to possess the same optimal dispersive wave relations as those of the Z-grid scheme.

Thi-Thao-Phuong Hoang, Lili Ju, Zhu Wang

The paper is concerned with overlapping domain decomposition and exponential time differencing for the diffusion equation discretized in space by cell-centered finite differences. Two localized exponential time differencing methods are proposed to solve the fully discrete problem: the first method is based on Schwarz iteration applied at each time step and involves solving stationary problems in the subdomains at each iteration, while the second method is based on the Schwarz waveform relaxation algorithm in which time-dependent subdomain problems are solved at each iteration. The convergence of the associated iterative solutions to the corresponding fully discrete multidomain solution and to the exact semi-discrete solution is rigorously proved. Numerical experiments are carried out to confirm theoretical results and to compare the performance of the two methods.

Thi-Thao-Phuong Hoang, Lili Ju, Wei Leng et al.

In this paper we present and analyze a general framework for constructing high order explicit local time stepping (LTS) methods for hyperbolic conservation laws. In particular, we consider the model problem discretized by Runge-Kutta discontinuous Galerkin (RKDG) methods and design LTS algorithms based on strong stability preserving Runge-Kutta (SSP-RK) schemes, that allow spatially variable time step sizes to be used for time integrations in different regions. The proposed algorithms are of predictor-corrector type, in which the interface information along the time direction is first predicted based on the SSP-RK approximations and Taylor expansions, and then the fluxes over the region of interface are corrected to conserve mass exactly at each time step. Following the proposed framework, we detail the corresponding LTS schemes with accuracy up to the fourth order, and prove their conservation property and nonlinear stability for the scalar conservation laws. Numerical experiments are also presented to demonstrate excellent performance of the proposed LTS algorithms.

Zezhong Zhang, Feng Bao, Lili Ju et al.

Transfer learning for partial differential equations (PDEs) is to develop a pre-trained neural network that can be used to solve a wide class of PDEs. Existing transfer learning approaches require much information of the target PDEs such as its formulation and/or data of its solution for pre-training. In this work, we propose to construct transferable neural feature spaces from purely function approximation perspectives without using PDE information. The construction of the feature space involves re-parameterization of the hidden neurons and uses auxiliary functions to tune the resulting feature space. Theoretical analysis shows the high quality of the produced feature space, i.e., uniformly distributed neurons. Extensive numerical experiments verify the outstanding performance of our method, including significantly improved transferability, e.g., using the same feature space for various PDEs with different domains and boundary conditions, and the superior accuracy, e.g., several orders of magnitude smaller mean squared error than the state of the art methods.

Wenkai Liu, Tao Guan, Bin Zhu et al.

In the domain of 3D scene representation, 3D Gaussian Splatting (3DGS) has emerged as a pivotal technology. However, its application to large-scale, high-resolution scenes (exceeding 4k$\times$4k pixels) is hindered by the excessive computational requirements for managing a large number of Gaussians. Addressing this, we introduce 'EfficientGS', an advanced approach that optimizes 3DGS for high-resolution, large-scale scenes. We analyze the densification process in 3DGS and identify areas of Gaussian over-proliferation. We propose a selective strategy, limiting Gaussian increase to key primitives, thereby enhancing the representational efficiency. Additionally, we develop a pruning mechanism to remove redundant Gaussians, those that are merely auxiliary to adjacent ones. For further enhancement, we integrate a sparse order increment for Spherical Harmonics (SH), designed to alleviate storage constraints and reduce training overhead. Our empirical evaluations, conducted on a range of datasets including extensive 4K+ aerial images, demonstrate that 'EfficientGS' not only expedites training and rendering times but also achieves this with a model size approximately tenfold smaller than conventional 3DGS while maintaining high rendering fidelity.

Zhiyuan Ning, Zaitian Wang, Ran Zhang et al.

Graph clustering aims to divide the graph into different clusters. The recently emerging deep graph clustering approaches are largely built on graph neural networks (GNN). However, GNN is designed for general graph encoding and there is a common issue of representation collapse in existing GNN-based deep graph clustering algorithms. We attribute two main reasons for such issues: (i) the inductive bias of GNN models: GNNs tend to generate similar representations for proximal nodes. Since graphs often contain a non-negligible amount of inter-cluster links, the bias results in error message passing and leads to biased clustering; (ii) the clustering guided loss function: most traditional approaches strive to make all samples closer to pre-learned cluster centers, which causes a degenerate solution assigning all data points to a single label thus making all samples similar and less discriminative. To address these challenges, we investigate graph clustering from a graph cut perspective and propose an innovative and non-GNN-based Deep Cut-informed Graph embedding and Clustering framework, namely DCGC. This framework includes two modules: (i) cut-informed graph encoding; (ii) self-supervised graph clustering via optimal transport. For the encoding module, we derive a cut-informed graph embedding objective to fuse graph structure and attributes by minimizing their joint normalized cut. For the clustering module, we utilize the optimal transport theory to obtain the clustering assignments, which can balance the guidance of "proximity to the pre-learned cluster center". With the above two tailored designs, DCGC is more suitable for the graph clustering task, which can effectively alleviate the problem of representation collapse and achieve better performance. We conduct extensive experiments to demonstrate that our method is simple but effective compared with benchmarks.

Tianzheng Lu, Lili Ju, Liyong Zhu

The transferable neural network (TransNet) is a two-layer shallow neural network with pre-determined and uniformly distributed neurons in the hidden layer, and the least-squares solvers can be particularly used to compute the parameters of its output layer when applied to the solution of partial differential equations. In this paper, we integrate the TransNet technique with the nonoverlapping domain decomposition and the interface conditions to develop a novel multiple transferable neural network (Multi-TransNet) method for solving elliptic interface problems, which typically contain discontinuities in both solutions and their derivatives across interfaces. We first propose an empirical formula for the TransNet to characterize the relationship between the radius of the domain-covering ball, the number of hidden-layer neurons, and the optimal neuron shape. In the Multi-TransNet method, we assign each subdomain one distinct TransNet with an adaptively determined number of hidden-layer neurons to maintain the globally uniform neuron distribution across the entire computational domain, and then unite all the subdomain TransNets together by incorporating the interface condition terms into the loss function. The empirical formula is also extended to the Multi-TransNet and further employed to estimate appropriate neuron shapes for the subdomain TransNets, greatly reducing the parameter tuning cost. Additionally, we propose a normalization approach to adaptively select the weighting parameters for the terms in the loss function. Ablation studies and extensive experiments with comparison tests on different types of elliptic interface problems with low to high contrast diffusion coefficients in two and three dimensions are carried out to numerically demonstrate the superior accuracy, efficiency, and robustness of the proposed Multi-TransNet method.

Qi Sun, Shengyan Li, Bowen Zheng et al.

Green's function provides an inherent connection between theoretical analysis and numerical methods for elliptic partial differential equations, and general absence of its closed-form expression necessitates surrogate modeling to guide the design of effective solvers. Unfortunately, numerical computation of Green's function remains challenging due to its doubled dimensionality and intrinsic singularity. In this paper, we present a novel singularity-encoded learning approach to resolve these problems in an unsupervised fashion. Our method embeds the Green's function within a one-order higher-dimensional space by encoding its prior estimate as an augmented variable, followed by a neural network parametrization to manage the increased dimensionality. By projecting the trained neural network solution back onto the original domain, our deep surrogate model exploits its spectral bias to accelerate conventional iterative schemes, serving either as a preconditioner or as part of a hybrid solver. The effectiveness of our proposed method is empirically verified through numerical experiments with two and four dimensional Green's functions, achieving satisfactory resolution of singularities and acceleration of iterative solvers.

Toan Huynh, Ruth Lopez Fajardo, Guannan Zhang et al.

We propose a novel framework for adaptively learning the time-evolving solutions of stochastic partial differential equations (SPDEs) using score-based diffusion models within a recursive Bayesian inference setting. SPDEs play a central role in modeling complex physical systems under uncertainty, but their numerical solutions often suffer from model errors and reduced accuracy due to incomplete physical knowledge and environmental variability. To address these challenges, we encode the governing physics into the score function of a diffusion model using simulation data and incorporate observational information via a likelihood-based correction in a reverse-time stochastic differential equation. This enables adaptive learning through iterative refinement of the solution as new data becomes available. To improve computational efficiency in high-dimensional settings, we introduce the ensemble score filter, a training-free approximation of the score function designed for real-time inference. Numerical experiments on benchmark SPDEs demonstrate the accuracy and robustness of the proposed method under sparse and noisy observations.

Wei Ju, Wenxin Jiang

We comment on a recent TKDE paper "Linear Approximation of F-measure for the Performance Evaluation of Classification Algorithms on Imbalanced Data Sets", and make two improvements related to comparison of F-measures for two prediction rules.

Xinyi Wu, Zhenyao Wu, Yuhang Lu et al.

In this paper, we tackle the problem of one-shot unsupervised domain adaptation (OSUDA) for semantic segmentation where the segmentors only see one unlabeled target image during training. In this case, traditional unsupervised domain adaptation models usually fail since they cannot adapt to the target domain with over-fitting to one (or few) target samples. To address this problem, existing OSUDA methods usually integrate a style-transfer module to perform domain randomization based on the unlabeled target sample, with which multiple domains around the target sample can be explored during training. However, such a style-transfer module relies on an additional set of images as style reference for pre-training and also increases the memory demand for domain adaptation. Here we propose a new OSUDA method that can effectively relieve such computational burden. Specifically, we integrate several style-mixing layers into the segmentor which play the role of style-transfer module to stylize the source images without introducing any learned parameters. Moreover, we propose a patchwise prototypical matching (PPM) method to weighted consider the importance of source pixels during the supervised training to relieve the negative adaptation. Experimental results show that our method achieves new state-of-the-art performance on two commonly used benchmarks for domain adaptive semantic segmentation under the one-shot setting and is more efficient than all comparison approaches.

Yuankai Teng, Zhu Wang, Lili Ju et al.

Due to the curse of dimensionality and the limitation on training data, approximating high-dimensional functions is a very challenging task even for powerful deep neural networks. Inspired by the Nonlinear Level set Learning (NLL) method that uses the reversible residual network (RevNet), in this paper we propose a new method of Dimension Reduction via Learning Level Sets (DRiLLS) for function approximation. Our method contains two major components: one is the pseudo-reversible neural network (PRNN) module that effectively transforms high-dimensional input variables to low-dimensional active variables, and the other is the synthesized regression module for approximating function values based on the transformed data in the low-dimensional space. The PRNN not only relaxes the invertibility constraint of the nonlinear transformation present in the NLL method due to the use of RevNet, but also adaptively weights the influence of each sample and controls the sensitivity of the function to the learned active variables. The synthesized regression uses Euclidean distance in the input space to select neighboring samples, whose projections on the space of active variables are used to perform local least-squares polynomial fitting. This helps to resolve numerical oscillation issues present in traditional local and global regressions. Extensive experimental results demonstrate that our DRiLLS method outperforms both the NLL and Active Subspace methods, especially when the target function possesses critical points in the interior of its input domain.

Anthony Gruber, Max Gunzburger, Lili Ju et al.

The popularity of deep convolutional autoencoders (CAEs) has engendered new and effective reduced-order models (ROMs) for the simulation of large-scale dynamical systems. Despite this, it is still unknown whether deep CAEs provide superior performance over established linear techniques or other network-based methods in all modeling scenarios. To elucidate this, the effect of autoencoder architecture on its associated ROM is studied through the comparison of deep CAEs against two alternatives: a simple fully connected autoencoder, and a novel graph convolutional autoencoder. Through benchmark experiments, it is shown that the superior autoencoder architecture for a given ROM application is highly dependent on the size of the latent space and the structure of the snapshot data, with the proposed architecture demonstrating benefits on data with irregular connectivity when the latent space is sufficiently large.

Yuankai Teng, Xiaoping Zhang, Zhu Wang et al.

Partial differential equations are often used to model various physical phenomena, such as heat diffusion, wave propagation, fluid dynamics, elasticity, electrodynamics and image processing, and many analytic approaches or traditional numerical methods have been developed and widely used for their solutions. Inspired by rapidly growing impact of deep learning on scientific and engineering research, in this paper we propose a novel neural network, GF-Net, for learning the Green's functions of linear reaction-diffusion equations in an unsupervised fashion. The proposed method overcomes the challenges for finding the Green's functions of the equations on arbitrary domains by utilizing physics-informed approach and the symmetry of the Green's function. As a consequence, it particularly leads to an efficient way for solving the target equations under different boundary conditions and sources. We also demonstrate the effectiveness of the proposed approach by experiments in square, annular and L-shape domains.

Anthony Gruber, Max Gunzburger, Lili Ju et al.

A dimension reduction method based on the "Nonlinear Level set Learning" (NLL) approach is presented for the pointwise prediction of functions which have been sparsely sampled. Leveraging geometric information provided by the Implicit Function Theorem, the proposed algorithm effectively reduces the input dimension to the theoretical lower bound with minor accuracy loss, providing a one-dimensional representation of the function which can be used for regression and sensitivity analysis. Experiments and applications are presented which compare this modified NLL with the original NLL and the Active Subspaces (AS) method. While accommodating sparse input data, the proposed algorithm is shown to train quickly and provide a much more accurate and informative reduction than either AS or the original NLL on two example functions with high-dimensional domains, as well as two state-dependent quantities depending on the solutions to parametric differential equations.

Xinyi Wu, Zhenyao Wu, Hao Guo et al.

Semantic segmentation of nighttime images plays an equally important role as that of daytime images in autonomous driving, but the former is much more challenging due to poor illuminations and arduous human annotations. In this paper, we propose a novel domain adaptation network (DANNet) for nighttime semantic segmentation without using labeled nighttime image data. It employs an adversarial training with a labeled daytime dataset and an unlabeled dataset that contains coarsely aligned day-night image pairs. Specifically, for the unlabeled day-night image pairs, we use the pixel-level predictions of static object categories on a daytime image as a pseudo supervision to segment its counterpart nighttime image. We further design a re-weighting strategy to handle the inaccuracy caused by misalignment between day-night image pairs and wrong predictions of daytime images, as well as boost the prediction accuracy of small objects. The proposed DANNet is the first one stage adaptation framework for nighttime semantic segmentation, which does not train additional day-night image transfer models as a separate pre-processing stage. Extensive experiments on Dark Zurich and Nighttime Driving datasets show that our method achieves state-of-the-art performance for nighttime semantic segmentation.

Jianfeng Zhang, Liezhuo Zhang, Yuankai Teng et al.

Binary image segmentation plays an important role in computer vision and has been widely used in many applications such as image and video editing, object extraction, and photo composition. In this paper, we propose a novel interactive binary image segmentation method based on the Markov Random Field (MRF) framework and the fast bilateral solver (FBS) technique. Specifically, we employ the geodesic distance component to build the unary term. To ensure both computation efficiency and effective responsiveness for interactive segmentation, superpixels are used in computing geodesic distances instead of pixels. Furthermore, we take a bilateral affinity approach for the pairwise term in order to preserve edge information and denoise. Through the alternating direction strategy, the MRF energy minimization problem is divided into two subproblems, which then can be easily solved by steepest gradient descent (SGD) and FBS respectively. Experimental results on the VGG interactive image segmentation dataset show that the proposed algorithm outperforms several state-of-the-art ones, and in particular, it can achieve satisfactory edge-smooth segmentation results even when the foreground and background color appearances are quite indistinctive.

Wei Leng, Lili Ju

In this paper, based on the overlapping domain decomposition method (DDM) proposed in \cite{Leng2015}, an one step preconditioner is proposed to solve 2D high frequency Helmholtz equation. The computation domain is decomposed in both $x$ and $y$ directions, and the local solution on each subdomain is updated simultaneously in one iteration, thus there is no sweeping along certain directions. In these ways, the overlapping DDM is similar to the popular DDM for Poisson problem. The one step preconditioner simply take the restricted source on each subdomain, solve the local problems and summarize the local solutions on all subdomains including their PML area. The complexity of solving the problem with the preconditioner is $O(N n_{\text{iter}})$, where $n_{\text{iter}}$ is the number of iteration, and it is shown numerically that $n_{\text{iter}}$ is proportional to the number of subdomains in one direction. 2D Helmholtz problem with nearly a billion unknowns are solved efficiently with the preconditioner on massively parallel machines.