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Water market is a contemporary marketplace for water trading and is deemed to one of the most efficient instruments to improve the social welfare. In modern water markets, the two widely used trading systems are an improved pair-wise trading, and a 'smart market' or common pool method. In comparison with the economic model, this paper constructs a conceptual mathematic model through the HARA utility functions. Mirroring the concepts such as Nash Equilibrium, Pareto optimal and stable matching in economy, three significant propositions are acquired which illustrate the advantages of the common pool method compared with the improved pair-wise trading.

Type I gamma-ray bursts (GRBs) are believed to originate from compact binary merger usually with duration less than 2 seconds for the main emission. However, recent observations of GRB 211211A and GRB 230307A indicate that some merger-origin GRBs could last much longer. Since they show strikingly similar properties (indicating a common mechanism) which are different from the classic "long"-short burst (e.g. GRB 060614), forming an interesting subclass of type I GRBs, we suggest to name them as type IL GRBs. By identifying the first peak of GRB 230307A as a quasi-thermal precursor, we find that the prompt emission of type IL GRB is composed of three episodes: (1) a precursor followed by a short quiescent (or weak emission) period, (2) a long-duration main emission, and (3) an extended emission. With this burst pattern, a good candidate, GRB 170228A, was found in the Fermi/GBM archive data, and subsequent temporal and spectral analyses indeed show that GRB 170228A falls in the same cluster with GRB 211211A and GRB 230307A in many diagnostic figures. Thus this burst pattern could be a good reference for rapidly identifying type IL GRB and conducting low-latency follow-up observation. We estimated the occurrence rate and discussed the physical origins and implications for the three emission episodes of type IL GRBs. Our analysis suggests the pre-merger precursor model, especially the super flare model, is more favored for type IL GRBs.

In recent years, pretraining models have made significant advancements in the fields of natural language processing (NLP), computer vision (CV), and life sciences. The significant advancements in NLP and CV are predominantly driven by the expansion of model parameters and data size, a phenomenon now recognized as the scaling laws. However, research exploring scaling law in molecular pretraining models remains unexplored. In this work, we present Uni-Mol2 , an innovative molecular pretraining model that leverages a two-track transformer to effectively integrate features at the atomic level, graph level, and geometry structure level. Along with this, we systematically investigate the scaling law within molecular pretraining models, characterizing the power-law correlations between validation loss and model size, dataset size, and computational resources. Consequently, we successfully scale Uni-Mol2 to 1.1 billion parameters through pretraining on 800 million conformations, making it the largest molecular pretraining model to date. Extensive experiments show consistent improvement in the downstream tasks as the model size grows. The Uni-Mol2 with 1.1B parameters also outperforms existing methods, achieving an average 27% improvement on the QM9 and 14% on COMPAS-1D dataset.

Molecular simulations are essential tools in computational chemistry, enabling the prediction and understanding of molecular interactions and thermodynamic properties of biomolecules. However, traditional force fields face significant challenges in accurately representing novel molecules and complex chemical environments due to the labor-intensive process of manually setting optimization parameters and the high computational cost of quantum mechanical calculations. To overcome these difficulties, we fine-tuned a high-accuracy DPA-2 pre-trained model and applied it to optimize force field parameters on-the-fly, significantly reducing computational costs. Our method combines this fine-tuned DPA-2 model with a node-embedding-based similarity metric, allowing seamless augmentation to new chemical species without manual intervention. We applied this process to the TYK2 inhibitor and PTP1B systems and demonstrated its effectiveness through the improvement of free energy perturbation calculation results. This advancement contributes valuable insights and tools for the computational chemistry community.

Characterizing the nature of many-body localization transitions (MBLTs) and their potential critical behaviors has remained a challenging problem. In this work, we study the dynamics of the displacement, quantifying the spread of the radial probability distribution in the Fock space, for systems with MBLTs, and perform a finite-size scaling analysis. We find that the scaling exponents satisfy theoretical bounds, and can identify universality classes. We show that reliable extrapolations to the thermodynamic limit for the MBLT induced by quasiperiodic fields is possible even for computationally accessible system sizes. Our work highlights that the displacement is a valuable tool for studying MBLTs, as relevant to ongoing experimental efforts.

In recent years, machine learning (ML) methods have emerged as promising alternatives for molecular docking, offering the potential for high accuracy without incurring prohibitive computational costs. However, recent studies have indicated that these ML models may overfit to quantitative metrics while neglecting the physical constraints inherent in the problem. In this work, we present Uni-Mol Docking V2, which demonstrates a remarkable improvement in performance, accurately predicting the binding poses of 77+% of ligands in the PoseBusters benchmark with an RMSD value of less than 2.0 {\AA}, and 75+% passing all quality checks. This represents a significant increase from the 62% achieved by the previous Uni-Mol Docking model. Notably, our Uni-Mol Docking approach generates chemically accurate predictions, circumventing issues such as chirality inversions and steric clashes that have plagued previous ML models. Furthermore, we observe enhanced performance in terms of high-quality predictions (RMSD values of less than 1.0 {\AA} and 1.5 {\AA}) and physical soundness when Uni-Mol Docking is combined with more physics-based methods like Uni-Dock. Our results represent a significant advancement in the application of artificial intelligence for scientific research, adopting a holistic approach to ligand docking that is well-suited for industrial applications in virtual screening and drug design. The code, data and service for Uni-Mol Docking are publicly available for use and further development in https://github.com/dptech-corp/Uni-Mol.

In the AI-for-science era, scientific computing scenarios such as concurrent learning and high-throughput computing demand a new generation of infrastructure that supports scalable computing resources and automated workflow management on both cloud and high-performance supercomputers. Here we introduce Dflow, an open-source Python toolkit designed for scientists to construct workflows with simple programming interfaces. It enables complex process control and task scheduling across a distributed, heterogeneous infrastructure, leveraging containers and Kubernetes for flexibility. Dflow is highly observable and can scale to thousands of concurrent nodes per workflow, enhancing the efficiency of complex scientific computing tasks. The basic unit in Dflow, known as an Operation (OP), is reusable and independent of the underlying infrastructure or context. Dozens of workflow projects have been developed based on Dflow, spanning a wide range of projects. We anticipate that the reusability of Dflow and its components will encourage more scientists to publish their workflows and OP components. These components, in turn, can be adapted and reused in various contexts, fostering greater collaboration and innovation in the scientific community.

In this paper, we consider a kind of shallow water wave model called the Kadomtsev-Petviashvili-Benjamin-Bona-Mahony (KP-BBM) equation. We firstly consider the unperturbed KP-BBM equation. Then by using the geometric singular perturbation (GSP) theory, especially the invariant manifold theory, method of dynamical system and Melnikov function, the existence of solitary wave solutions of perturbed KP-BBM equation is proved. In other words, we dissuss the equation under different nonlinear terms. Finally, we validate our results with numerical simulations., Comment: We declare that the authors are ranked in alphabetic order of their names and all of them have the same contributions to this paper

Isolated interacting quantum systems generally thermalize, yet there are several counterexamples for the breakdown of ergodicity, such as many-body localization and quantum scars. Recently, ergodicity breaking has been observed in systems subjected to linear potentials, termed Stark many-body localization. This phenomenon is closely associated with Hilbert-space fragmentation, characterized by a strong dependence of dynamics on initial conditions. Here, we experimentally explore initial-state dependent dynamics using a ladder-type superconducting processor with up to 24 qubits, which enables precise control of the qubit frequency and initial state preparation. In systems with linear potentials, we observe distinct non-equilibrium dynamics for initial states with the same quantum numbers and energy, but with varying domain wall numbers. This distinction becomes increasingly pronounced as the system size grows, in contrast with disordered interacting systems. Our results provide convincing experimental evidence of the fragmentation in Stark systems, enriching our understanding of the weak breakdown of ergodicity., Comment: main text: 7 pages, 4 figures; supplementary: 13 pages, 14 figures

Starting from more than 11,200 short-period (less than 0.5 days) EW-type eclipsing binary candidates with the All-Sky Automated Survey for Supernovae (ASAS-SN) V-band light curves, we use MCMC and neural networks (NNs) to obtain the mass ratio ($q$), orbital inclination ($incl$), fill-out factor ($f$) and temperature ratio ($T_s/T_p$). After cross-matching with the Gaia DR3 database, the final sample contains parameters of 2,399 A-type and 8,712 W-type contact binaries (CBs). We present the distributions of parameters of these 11,111 short-period CBs. The mass ratio ($q$) and fill-out factor ($f$) are found to obey log-normal distributions, and the remaining parameters obey normal distributions. There is a significant period-temperature correlation of these CBs. Additionally, the temperature ratio (${T_s}$/${T_p}$) tends to increase as the orbital period decreases for W-type CBs. There is no significant correlation between them for A-type CBs. The mass ratio and fill-out factor ($q-f$) diagram suggest there is no significant correlation between these two parameters. A clear correlation exists between the mass ratio and radius ratio. The radius ratio increases with the mass ratio. Moreover, the deep fill-out CBs tend to fall on the upper boundary of the $q$$-$${R_s}$/${R_p}$ distribution, while the shallow fill-out CBs fall on the lower boundary., Comment: 15 pages, 8 figures, 1 table.Accepted for publication in ApJS. The full dataset will be available with the ApJS article

Characterizing the nature of hydrodynamical transport properties in quantum dynamics provides valuable insights into the fundamental understanding of exotic non-equilibrium phases of matter. Simulating infinite-temperature transport on large-scale complex quantum systems remains an outstanding challenge. Here, using a controllable and coherent superconducting quantum simulator, we experimentally realize the analog quantum circuit, which can efficiently prepare the Haar-random states, and probe spin transport at infinite temperature. We observe diffusive spin transport during the unitary evolution of the ladder-type quantum simulator with ergodic dynamics. Moreover, we explore the transport properties of the systems subjected to strong disorder or a titled potential, revealing signatures of anomalous subdiffusion in accompany with the breakdown of thermalization. Our work demonstrates a scalable method of probing infinite-temperature spin transport on analog quantum simulators, which paves the way to study other intriguing out-of-equilibrium phenomena from the perspective of transport., Comment: Main text: 12 pages, 7 figures; Supplementary: 12 pages, 8 figures

Multi-sensor modal fusion has demonstrated strong advantages in 3D object detection tasks. However, existing methods that fuse multi-modal features require transforming features into the bird's eye view space and may lose certain information on Z-axis, thus leading to inferior performance. To this end, we propose a novel end-to-end multi-modal fusion transformer-based framework, dubbed FusionFormer, that incorporates deformable attention and residual structures within the fusion encoding module. Specifically, by developing a uniform sampling strategy, our method can easily sample from 2D image and 3D voxel features spontaneously, thus exploiting flexible adaptability and avoiding explicit transformation to the bird's eye view space during the feature concatenation process. We further implement a residual structure in our feature encoder to ensure the model's robustness in case of missing an input modality. Through extensive experiments on a popular autonomous driving benchmark dataset, nuScenes, our method achieves state-of-the-art single model performance of 72.6% mAP and 75.1% NDS in the 3D object detection task without test time augmentation.

Building a multi-modality multi-task neural network toward accurate and robust performance is a de-facto standard in perception task of autonomous driving. However, leveraging such data from multiple sensors to jointly optimize the prediction and planning tasks remains largely unexplored. In this paper, we present FusionAD, to the best of our knowledge, the first unified framework that fuse the information from two most critical sensors, camera and LiDAR, goes beyond perception task. Concretely, we first build a transformer based multi-modality fusion network to effectively produce fusion based features. In constrast to camera-based end-to-end method UniAD, we then establish a fusion aided modality-aware prediction and status-aware planning modules, dubbed FMSPnP that take advantages of multi-modality features. We conduct extensive experiments on commonly used benchmark nuScenes dataset, our FusionAD achieves state-of-the-art performance and surpassing baselines on average 15% on perception tasks like detection and tracking, 10% on occupancy prediction accuracy, reducing prediction error from 0.708 to 0.389 in ADE score and reduces the collision rate from 0.31% to only 0.12%.

2D RGB images and 3D LIDAR point clouds provide complementary knowledge for the perception system of autonomous vehicles. Several 2D and 3D fusion methods have been explored for the LIDAR semantic segmentation task, but they suffer from different problems. 2D-to-3D fusion methods require strictly paired data during inference, which may not be available in real-world scenarios, while 3D-to-2D fusion methods cannot explicitly make full use of the 2D information. Therefore, we propose a Bidirectional Fusion Network with Cross-Modality Knowledge Distillation (CMDFusion) in this work. Our method has two contributions. First, our bidirectional fusion scheme explicitly and implicitly enhances the 3D feature via 2D-to-3D fusion and 3D-to-2D fusion, respectively, which surpasses either one of the single fusion schemes. Second, we distillate the 2D knowledge from a 2D network (Camera branch) to a 3D network (2D knowledge branch) so that the 3D network can generate 2D information even for those points not in the FOV (field of view) of the camera. In this way, RGB images are not required during inference anymore since the 2D knowledge branch provides 2D information according to the 3D LIDAR input. We show that our CMDFusion achieves the best performance among all fusion-based methods on SemanticKITTI and nuScenes datasets. The code will be released at https://github.com/Jun-CEN/CMDFusion.

Efficient preparation of spin-squeezed states is important for quantum-enhanced metrology. Current protocols for generating strong spin squeezing rely on either high dimensionality or long-range interactions. A key challenge is how to generate considerable spin squeezing in one-dimensional systems with only nearest-neighbor interactions. Here, we develop variational spin-squeezing algorithms to solve this problem. We consider both digital and analog quantum circuits for these variational algorithms. After the closed optimization loop of the variational spin-squeezing algorithms, the generated squeezing can be comparable to the strongest squeezing created from two-axis twisting. By analyzing the experimental imperfections, the variational spin-squeezing algorithms proposed in this work are feasible in recent developed noisy intermediate-scale quantum computers.

Recently deep learning based quantitative structure-activity relationship (QSAR) models has shown surpassing performance than traditional methods for property prediction tasks in drug discovery. However, most DL based QSAR models are restricted to limited labeled data to achieve better performance, and also are sensitive to model scale and hyper-parameters. In this paper, we propose Uni-QSAR, a powerful Auto-ML tool for molecule property prediction tasks. Uni-QSAR combines molecular representation learning (MRL) of 1D sequential tokens, 2D topology graphs, and 3D conformers with pretraining models to leverage rich representation from large-scale unlabeled data. Without any manual fine-tuning or model selection, Uni-QSAR outperforms SOTA in 21/22 tasks of the Therapeutic Data Commons (TDC) benchmark under designed parallel workflow, with an average performance improvement of 6.09\%. Furthermore, we demonstrate the practical usefulness of Uni-QSAR in drug discovery domains.

We study the resonance and dynamics of a qubit exposed to a strong aperiodic bichromatic field by using a periodic counter-rotating hybridized rotating wave (CHRW) Hamiltonian, which is derived from the original Hamiltonian with the unitary transformations under a reasonable approximation and enables the application of the Floquet theory. It is found that the consistency between the CHRW results and numerically exact generalized-Floquet-theory (GFT) results in the valid regime of the former while the widely used rotating-wave approximation (RWA) breaks down. We illustrate that the resonance exhibits band structure and the Bloch-Siegert shifts induced by the counter-rotating couplings of the bichromatic field become notable at the multiphoton resonance band. In addition, the CHRW method is found to have a great advantage of efficiency over the GFT approach particularly in the low beat-frequency case where the latter converges very slowly. The present CHRW method provides a highly efficient way to calculate the resonance frequency incorporating the Bloch-Siegert shift and provides insights into the effects of the counter-rotating couplings of the bichromatic field in the strong-driving regimes., Comment: 10 pages, 6 figures

Molecular docking, given a ligand molecule and a ligand binding site (called ``pocket'') on a protein, predicting the binding mode of the protein-ligand complex, is a widely used technique in drug design. Many deep learning models have been developed for molecular docking, while most existing deep learning models perform docking on the whole protein, rather than on a given pocket as the traditional molecular docking approaches, which does not match common needs. What's more, they claim to perform better than traditional molecular docking, but the approach of comparison is not fair, since traditional methods are not designed for docking on the whole protein without a given pocket. In this paper, we design a series of experiments to examine the actual performance of these deep learning models and traditional methods. For a fair comparison, we decompose the docking on the whole protein into two steps, pocket searching and docking on a given pocket, and build pipelines to evaluate traditional methods and deep learning methods respectively. We find that deep learning models are actually good at pocket searching, but traditional methods are better than deep learning models at docking on given pockets. Overall, our work explicitly reveals some potential problems in current deep learning models for molecular docking and provides several suggestions for future works.

Molecular conformation generation (MCG) is a fundamental and important problem in drug discovery. Many traditional methods have been developed to solve the MCG problem, such as systematic searching, model-building, random searching, distance geometry, molecular dynamics, Monte Carlo methods, etc. However, they have some limitations depending on the molecular structures. Recently, there are plenty of deep learning based MCG methods, which claim they largely outperform the traditional methods. However, to our surprise, we design a simple and cheap algorithm (parameter-free) based on the traditional methods and find it is comparable to or even outperforms deep learning based MCG methods in the widely used GEOM-QM9 and GEOM-Drugs benchmarks. In particular, our design algorithm is simply the clustering of the RDKIT-generated conformations. We hope our findings can help the community to revise the deep learning methods for MCG. The code of the proposed algorithm could be found at https://gist.github.com/ZhouGengmo/5b565f51adafcd911c0bc115b2ef027c.

Structure-based drug design, i.e., finding molecules with high affinities to the target protein pocket, is one of the most critical tasks in drug discovery. Traditional solutions, like virtual screening, require exhaustively searching on a large molecular database, which are inefficient and cannot return novel molecules beyond the database. The pocket-based 3D molecular generation model, i.e., directly generating a molecule with a 3D structure and binding position in the pocket, is a new promising way to address this issue. Herein, we propose VD-Gen, a novel pocket-based 3D molecular generation pipeline. VD-Gen consists of several carefully designed stages to generate fine-grained 3D molecules with binding positions in the pocket cavity end-to-end. Rather than directly generating or sampling atoms with 3D positions in the pocket like in early attempts, in VD-Gen, we first randomly initialize many virtual particles in the pocket; then iteratively move these virtual particles, making the distribution of virtual particles approximate the distribution of molecular atoms. After virtual particles are stabilized in 3D space, we extract a 3D molecule from them. Finally, we further refine atoms in the extracted molecule by iterative movement again, to get a high-quality 3D molecule, and predict a confidence score for it. Extensive experiment results on pocket-based molecular generation demonstrate that VD-Gen can generate novel 3D molecules to fill the target pocket cavity with high binding affinities, significantly outperforming previous baselines.

Variable stars play a key role in understanding the Milky Way and the universe. The era of astronomical big data presents new challenges for quick identification of interesting and important variable stars. Accurately estimating the periods is the most important step to distinguish different types of variable stars. Here, we propose a new method of determining the variability periods. By combining the statistical parameters of the light curves, the colors of the variables, the window function and the GLS algorithm, the aperiodic variables are excluded and the periodic variables are divided into eclipsing binaries and NEB variables (other types of periodic variable stars other than eclipsing binaries), the periods of the two main types of variables are derived. We construct a random forest classifier based on 241,154 periodic variables from the ASAS-SN and OGLE datasets of variables. The random forest classifier is trained on 17 features, among which 11 are extracted from the light curves and 6 are from the Gaia Early DR3, ALLWISE and 2MASS catalogs. The variables are classified into 7 superclasses and 17 subclasses. In comparison with the ASAS-SN and OGLE catalogs, the classification accuracy is generally above approximately 82% and the period accuracy is 70%-99%. To further test the reliability of the new method and classifier, we compare our results with the results of Chen et al. (2020) for ZTF DR2. The classification accuracy is generally above 70%. The period accuracy of the EW and SR variables is 50% and 53%, respectively. And the period accuracy of other types of variables is 65%-98%., Comment: 23 pages, 10 figures

In this paper, persistence of solitary wave solutions of the regularized long wave equation with small perturbations are investigated by the geometric singular perturbation theory. Two different kinds of the perturbations are considered in this paper: one is the weak backward diffusion and dissipation, the other is the Marangoni effects. Indeed, the solitary wave persists under small perturbations. Furthermore, the different perturbations do affect the proper wave speed ensuring the persistence of the solitary waves. Finally, numerical simulations are utilized to confirm the theoretical results.

The variational preparation of complex quantum states using the quantum approximate optimization algorithm (QAOA) is of fundamental interest, and becomes a promising application of quantum computers. Here, we systematically study the performance of QAOA for preparing ground states of target Hamiltonians near the critical points of their quantum phase transitions, and generating Greenberger-Horne-Zeilinger (GHZ) states. We reveal that the performance of QAOA is related to the translational invariance of the target Hamiltonian: Without the translational symmetry, for instance due to the open boundary condition (OBC) or randomness in the system, the QAOA becomes less efficient. We then propose a generalized QAOA assisted by the parameterized resource Hamiltonian (PRH-QAOA), to achieve a better performance. In addition, based on the PRH-QAOA, we design a low-depth quantum circuit beyond one-dimensional geometry, to generate GHZ states with perfect fidelity. The experimental realization of the proposed scheme for generating GHZ states on Rydberg-dressed atoms is discussed. Our work paves the way for performing QAOA on programmable quantum processors without translational symmetry, especially for recently developed two-dimensional quantum processors with OBC., Comment: 21 pages, 15 figures

This chapter reports on a language teaching practicum course of a Master of Teaching Chinese to Speakers of Other Languages (MTCSOL) program in China in Fall 2020. Driven by the challenges of the COVID-19 pandemic, this report presents a model to guide practice in online language teacher training via virtual learning communities. In the course, native and nonnative students in the MTCSOL program formed study groups in which they conducted discussions regarding the lectures and completed teaching demonstrations through videoconferences. All students provided online feedback on their peers' teaching demonstrations. The content analysis of post-task interviews revealed that both native and nonnative students benefited from online interactions. Nonnative students had opportunities to practice language and professional skills, while native students developed intercultural communication competence. Peer feedback was also highly rated, as feedback receivers had different perspectives on how to improve their teaching, and feedback providers learned from their peers' strengths and weaknesses. We argue that the model can (1) promote learning autonomy and cultural exchange and (2) boost teachers' self-confidence, which could thus become a desirable new model for language teacher training. [For the complete volume, "The World Universities' Response to COVID-19: Remote Online Language Teaching," see ED614006.]

We investigate the steering dissipative dynamics of a two-level system (qubit) by means of the modulation of an assisted tunneling degree of freedom which is described by a quantum-oscillator spin-boson model. Our results reveal that the decoherence rate of the qubit can be significantly suppressed and simultaneously its quality factor is enhanced. Moreover, the modulated dynamical susceptibility exhibits a multi-peak feature which is indicative of the underlying structure and measurable in experiment. Our findings demonstrate that the interplay between the combined degrees of freedom and the qubit is crucial for reducing the dissipation of qubit and expanding the coherent regime of quantum operation much large. The strategy might be used to fight against deterioration of quantum coherence in quantum information processing., Comment: 5 pages, 2 figures

Choosing ${\kappa}$ (horizontal ordinate of the saddle point associated to the homoclinic orbit) as bifurcation parameter, bifurcations of the travelling wave solutions is studied in a perturbed $(1 + 1)$-dimensional dispersive long wave equation. The solitary wave solution exists at a suitable wave speed $c$ for the bifurcation parameter ${\kappa}\in (0,1-\frac{\sqrt3}{3})\cup (1+\frac{\sqrt3}{3},2)$, while the kink and anti-kink wave solutions exist at a unique wave speed $c^*=\sqrt{15}/3$ for $\kappa=0$ or $\kappa=2$. The methods are based on the geometric singular perturbation (GSP, for short) approach, Melnikov method and invariant manifolds theory. Interestingly, not only the explicit analytical expression of the complicated homoclinic Melnikov integral is directly obtained for the perturbed long wave equation, but also the explicit analytical expression of the limit wave speed is directly given. Numerical simulations are utilized to verify our mathematical results.

We investigate the effects of counterrotating terms on geometric phase and its relation to the resonance of the Rabi model. We apply the unitary transformation with a single parameter to the Rabi model and obtain the transformed Hamiltonian involving multiple harmonic terms. By combining the counter-rotating-hybridized rotating-wave method with time-dependent perturbation theory, we solve systematically time evolution operator and then obtain the geometric phase of the two-level system. Our results are beyond adiabatic approximation and rotating-wave approximation (RWA). Higher-order harmonic resonance happens when driving frequency is equal to higher-order subharmonic of the Rabi frequency. In comparison with numerically exact results, our calculated results are accurate over a wide range of parameters space, especially in higher-order harmonic resonance regimes. In these regimes we demonstrate geometric phases change dramatically while those of the RWA are smooth. The RWA is thoroughly invalid even if the driving strength is extremely weak. We find it is the higher-order harmonic terms that play an important role on the cyclic state and demonstrate the characters of geometric phase in higher-order harmonic resonance regime. We also present analytical formalism of the change rate of geometric phase and quasienergies, which agree well with numerically exact ones even in the strong driving case. The developed method can be applied to explore the dynamics of strongly driven qubits and physical properties of higher-order harmonic processes., Comment: 22 pages, 5 figures