Tailoring wavelets for chaos control

Algebraic Graph-Based Machine Learning Model for Li-Cluster Prediction

In cluster research, determining the ground-state structure of medium-sized clusters is hindered by a large number of local minimum on potential energy surfaces. The global optimization heuristic algorithm is time-consuming due to the use of DFT to determine the relative size of the cluster energy. Although machine learning (ML) is proved to be a promising way to reduce the DFT computational costs, a suitable method to represent clusters as input vectors is one of the bottlenecks in the application of ML to cluster research. In this work, we proposed a multiscale weighted spectral subgraph (MWSS) as an effective low-dimension representation of clusters and build an MWSS-based ML model to discover the structure-energy relationships in lithium clusters. We combine this model with the particle swarm optimization algorithm and DFT calculations to search for globally stable structures of clusters. We have successfully predicted the ground-state structure of Li20.

Evolutionary homology on coupled dynamical systems with applications to protein flexibility analysis

While the spatial topological persistence is naturally constructed from a radius-based filtration, it has hardly been derived from a temporal filtration. Most topological models are designed for the global topology of a given object as a whole. There is no method reported in the literature for the topology of an individual component in an object to the best of our knowledge. For many problems in science and engineering, the topology of an individual component is important for describing its properties. We propose evolutionary homology (EH) constructed via a time evolution-based filtration and topological persistence. Our approach couples a set of dynamical systems or chaotic oscillators by the interactions of a physical system, such as a macromolecule. The interactions are approximated by weighted graph Laplacians. Simplices, simplicial complexes, algebraic groups and topological persistence are defined on the coupled trajectories of the chaotic oscillators. The resulting EH gives rise to time-dependent topological invariants or evolutionary barcodes for an individual component of the physical system, revealing its topology-function relationship. In conjunction with Wasserstein metrics, the proposed EH is applied to protein flexibility analysis, an important problem in computational biophysics. Numerical results for the B-factor prediction of a benchmark set of 364 proteins indicate that the proposed EH outperforms all the other state-of-the-art methods in the field.

MathDL: mathematical deep learning for D3R Grand Challenge 4

We present the performances of our mathematical deep learning (MathDL) models for D3R Grand Challenge 4 (GC4). This challenge involves pose prediction, affinity ranking, and free energy estimation for beta secretase 1 (BACE) as well as affinity ranking and free energy estimation for Cathepsin S (CatS). We have developed advanced mathematics, namely differential geometry, algebraic graph, and/or algebraic topology, to accurately and efficiently encode high dimensional physical/chemical interactions into scalable low-dimensional rotational and translational invariant representations. These representations are integrated with deep learning models, such as generative adversarial networks (GAN) and convolutional neural networks (CNN) for pose prediction and energy evaluation, respectively. Overall, our MathDL models achieved the top place in pose prediction for BACE ligands in Stage 1a. Moreover, our submissions obtained the highest Spearman correlation coefficient on the affinity ranking of 460 CatS compounds, and the smallest centered root mean square error on the free energy set of 39 CatS molecules. It is worthy to mention that our method on docking pose predictions has significantly improved from our previous ones.

AGL-Score: Algebraic Graph Learning Score for Protein-Ligand Binding Scoring, Ranking, Docking, and Screening

Although algebraic graph theory-based models have been widely applied in physical modeling and molecular studies, they are typically incompetent in the analysis and prediction of biomolecular properties, confirming the common belief that "one cannot hear the shape of a drum". A new development in the century-old question about the spectrum-geometry relationship is provided. Novel algebraic graph learning score (AGL-Score) models are proposed to encode high-dimensional physical and biological information into intrinsically low-dimensional representations. The proposed AGL-Score models employ multiscale weighted colored subgraphs to describe crucial molecular and biomolecular interactions in terms of graph invariants derived from graph Laplacian, its pseudo-inverse, and adjacency matrices. Additionally, AGL-Score models are integrated with an advanced machine learning algorithm to predict biomolecular macroscopic properties from the low-dimensional graph representation of biomolecular structures. The proposed AGL-Score models are extensively validated for their scoring power, ranking power, docking power, and screening power via a number of benchmark datasets, namely CASF-2007, CASF-2013, and CASF-2016. Numerical results indicate that the proposed AGL-Score models are able to outperform other state-of-the-art scoring functions in protein-ligand binding scoring, ranking, docking, and screening. This study indicates that machine learning methods are powerful tools for molecular docking and virtual screening. It also indicates that spectral geometry or spectral graph theory has the ability to infer geometric properties.

Perturbation analysis and comparison of network synchronization methods

In many networked systems, synchronization is important and useful, and how to enhance synchronizability is an interesting problem. Based on the matrix perturbation theory, we analyze five methods of network synchronization enhancement, including the link removal, node removal, dividing hub node, pull control, and pinning control methods, and obtain explicit expressions for eigenvalue changes. By these comparisons, we find that, among all these methods, the pull control method is remarkable, as it can extend the synchronization (coupling strength) region from both the left and right sides, for any controlled node. Extensive simulation results are given to support the accuracy of the perturbation-based analysis.

Wavelet transformed Gaussian network model

Accurate prediction of the Debye–Waller temperature factor of proteins is of significant importance in the study of protein dynamics and function. This work explores the utility of wavelets for improving the performance of Gaussian network model (GNM). We propose two wavelet transformed Gaussian network models (wtGNM), namely a scale-one wtGNM and a scale-two wtGNM. Based on a set of 113 protein structures, it shows that the mean correlation with experimental results for the scale-one wtGNM is 0.714 and that for the scale-two wtGNM is 0.738. In contrast, the mean correlation for the original GNM is 0.594. Therefore, the wtGNM is a potential algorithm for improving the GNM prediction of protein B-factors.

Control for a synchronization-desynchronization switch

How to freely enhance or suppress synchronization of networked dynamical systems is of great importance in many disciplines. A unified precise control method for a synchronization-desynchronization switch, called the pull-push control method, is suggested. Namely, synchronization can be achieved when the original systems are desynchronous by pulling (or protecting) one node or a certain subset of nodes, whereas desynchronization can be accomplished when the systems are already synchronous by pushing (or kicking) one node or a certain subset of nodes. With this method, the controlled nodes should be chosen by the generalized eigenvector centrality of the critical synchronization mode of the Laplacian matrix. Compared with existing control methods for synchronization, it displays high efficiency, flexibility, and precision as well.

Network approach to the pinning control of drift-wave turbulence

Network of coupled oscillators has long been employed as an important approach to explore the complicated dynamics in spatially extended systems. Here we show how this approach can be used to the analysis of turbulence pinning control. Specifically, by use of a model of two-dimensional drift-wave plasma turbulence, we investigate how the performance of the turbulence control is influenced by the spatial distribution of the pinning strength. It is found that the dynamics of pinned turbulence can be well captured by a simple model of networked modes, based on which the dependence of the control performance on the pinning distribution can be analytically obtained. In particular, the model predicts that as the distribution of the pinning strength becomes more nonuniform, the performance of turbulence control will be gradually decreased. This theoretical prediction is in good agreement with the results of numerical simulations, including the sinusoidal and localized pinning distributions. Our studies provide a new viewpoint to the mechanism of mode couplings in drift-wave turbulence, as well as be constructive to the design of new schemes for controlling turbulence in realistic systems.

Molecular nonlinear dynamics and protein thermal uncertainty quantification

This work introduces molecular nonlinear dynamics (MND) as a new approach for describing protein folding and aggregation. By using a mode system, we show that the MND of disordered proteins is chaotic while that of folded proteins exhibits intrinsically low dimensional manifolds (ILDMs). The stability of ILDMs is found to strongly correlate with protein energies. We propose a novel method for protein thermal uncertainty quantification based on persistently invariant ILDMs. Extensive comparison with experimental data and the state-of-the-art methods in the field validate the proposed new method for protein B-factor prediction.

Matching rules for collective behaviors on complex networks: optimal configurations for vibration frequencies of networked harmonic oscillators

The structure-dynamics-function has become one of central problems in modern sciences, and it is a great challenge to unveil the organization rules for different dynamical processes on networks. In this work, we study the vibration spectra of the classical mass spring model with different masses on complex networks, and pay our attention to how the mass spatial configuration influences the second-smallest vibrational frequency (ω2) and the largest one (ωN). For random networks, we find that ω2 becomes maximal and ωN becomes minimal if the node degrees are point-to-point-positively correlated with the masses. In these cases, we call it point-to-point matching. Moreover, ω2 becomes minimal under the condition that the heaviest mass is placed on the lowest-degree vertex, and ωN is maximal as long as the lightest mass is placed on the highest-degree vertex, and in both cases all other masses can be arbitrarily settled. Correspondingly, we call it single-point matching. These findings indicate that the matchings between the node dynamics (parameter) and the node position rule the global systems dynamics, and sometimes only one node is enough to control the collective behaviors of the whole system. Therefore, the matching rules might be the common organization rules for collective behaviors on networks.

Intermittent and sustained periodic windows in networked chaotic Rössler oscillators

Route to chaos (or periodicity) in dynamical systems is one of fundamental problems. Here, dynamical behaviors of coupled chaotic Rössler oscillators on complex networks are investigated and two different types of periodic windows with the variation of coupling strength are found. Under a moderate coupling, the periodic window is intermittent, and the attractors within the window extremely sensitively depend on the initial conditions, coupling parameter, and topology of the network. Therefore, after adding or removing one edge of network, the periodic attractor can be destroyed and substituted by a chaotic one, or vice versa. In contrast, under an extremely weak coupling, another type of periodic window appears, which insensitively depends on the initial conditions, coupling parameter, and network. It is sustained and unchanged for different types of network structure. It is also found that the phase differences of the oscillators are almost discrete and randomly distributed except that directly linked oscillators more likely have different phases. These dynamical behaviors have also been generally observed in other networked chaotic oscillators.

Quasiperiodic, periodic, and slowing-down states of coupled heteroclinic cycles

We investigate two coupled oscillators, each of which shows an attracting heteroclinic cycle in the absence of coupling. The two heteroclinic cycles are nonidentical. Weak coupling can lead to the elimination of the slowing-down state that asymptotically approaches the heteroclinic cycle for a single cycle, giving rise to either quasiperiodic motion with separate frequencies from the two cycles or periodic motion in which the two cycles are synchronized. The synchronization transition, which occurs via a Hopf bifurcation, is not induced by the commensurability of the two cycle frequencies but rather by the disappearance of the weaker frequency oscillation. For even larger coupling the motion changes via a resonant heteroclinic bifurcation to a slowing-down state corresponding to a single attracting heteroclinic orbit. Coexistence of multiple attractors can be found for some parameter regions. These results are of interest in ecological, sociological, neuronal, and other dynamical systems, which have the structure of coupled heteroclinic cycles.

General DFT++ method implemented with projector augmented waves: electronic structure of SrVO3 and the Mott transition in Ca(2-x)Sr(x)RuO4

The realistic description of correlated electron systems took an important step forward a few years ago as the combination of density functional methods and dynamical mean-field theory was conceived. This framework allows access to both high and low energy physics and is capable of the description of the specific physics of strongly correlated materials, like the Mott metal-insulator transition. A very important step in the procedure is the interface between the band structure method and the dynamical mean-field theory and its impurity solver. We present a general interface between a projector augmented-wave-based density functional code and many-body methods based on Wannier functions obtained from a projection on local orbitals. The implementation is very flexible and allows for various applications. Quantities like the momentum-resolved spectral function are accessible. We present applications to SrVO(3) and the metal-insulator transition in Ca(2-x)Sr(x)RuO(4).

Taming turbulence in the complex Ginzburg-Landau equation

Taming turbulence in the complex Ginzburg-Landau equation (CGLE) by using a global feedback control method and choosing traveling-wave solutions as our target state is investigated. The problem of optimal control for the smallest driving strength is studied by systematically comparing the stabilities of all traveling waves. Within the Benjamin-Feir-Newell unstable parameter region (c2<-c1 -1, a critical control curve is determined, which is located at c2=alphacbeta1 , with alpha approximately -4.0 and beta approximately -0.87. It characterizes the transition of chosen traveling-wave target state from long wavelength to short one. This finding is of great significance for taming turbulence in the CGLE and some other spatiotemporal systems as well.

Coordinate transformation and matrix measure approach for synchronization of complex networks

Global synchronization in complex networks has attracted considerable interest in various fields. There are mainly two analytical approaches for studying such time-varying networks. The first approach is Lyapunov function-based methods. For such an approach, the connected-graph-stability (CGS) method arguably gives the best results. Nevertheless, CGS is limited to the networks with cooperative couplings. The matrix measure approach (MMA) proposed by Chen, although having a wider range of applications in the network topologies than that of CGS, works for smaller numbers of nodes in most network topologies. The approach also has a limitation with networks having partial-state coupling. Other than giving yet another MMA, we introduce a new and, in some cases, optimal coordinate transformation to study such networks. Our approach fixes all the drawbacks of CGS and MMA. In addition, by merely checking the structure of the vector field of the individual oscillator, we shall be able to determine if the system is globally synchronized. In summary, our results can be applied to rather general time-varying networks with a large number of nodes.

Transition zone in controlling spatiotemporal chaos

The controllability of spatiotemporal chaos is investigated. Contrary to our common sense that there is only one transition point (i.e., a critical control strength) for successful control, we find that actually a transition zone exists, connecting two transition points for the local and global stabilities of the controlled state, respectively. Within the zone, the controllable probability increases from zero to one for random initial conditions. This behavior is found to be very generic and is expected to have a severe consequence in realistic applications in the control of spatiotemporal chaos.

Chaos synchronization in coupled systems by applying pinning control

Chaos synchronization in coupled chaotic oscillator systems with diffusive and gradient couplings forced by only one local feedback injection signal (boundary pinning control) is studied. By using eigenvalue analysis, we obtain controllable regions directly in control parameter space for different types of coupling links (including diagonal coupling and nondiagonal couplings). The effects of both diffusive and gradient couplings on chaos synchronization become clear. Some relevant factors on control efficiency such as coupled system size, transient process, and feedback signal intensity are also studied.

Dynamics of a neural system with a multiscale architecture

The architecture of the brain is characterized by a modular organization repeated across a hierarchy of spatial scales-neurons, minicolumns, cortical columns, functional brain regions, and so on. It is important to consider that the processes governing neural dynamics at any given scale are not only determined by the behaviour of other neural structures at that scale, but also by the emergent behaviour of smaller scales, and the constraining influence of activity at larger scales. In this paper, we introduce a theoretical framework for neural systems in which the dynamics are nested within a multiscale architecture. In essence, the dynamics at each scale are determined by a coupled ensemble of nonlinear oscillators, which embody the principle scale-specific neurobiological processes. The dynamics at larger scales are 'slaved' to the emergent behaviour of smaller scales through a coupling function that depends on a multiscale wavelet decomposition. The approach is first explicated mathematically. Numerical examples are then given to illustrate phenomena such as between-scale bifurcations, and how synchronization in small-scale structures influences the dynamics in larger structures in an intuitive manner that cannot be captured by existing modelling approaches. A framework for relating the dynamical behaviour of the system to measured observables is presented and further extensions to capture wave phenomena and mode coupling are suggested.

Network synchronization, diffusion, and the paradox of heterogeneity

Many complex networks display strong heterogeneity in the degree (connectivity) distribution. Heterogeneity in the degree distribution often reduces the average distance between nodes but, paradoxically, may suppress synchronization in networks of oscillators coupled symmetrically with uniform coupling strength. Here we offer a solution to this apparent paradox. Our analysis is partially based on the identification of a diffusive process underlying the communication between oscillators and reveals a striking relation between this process and the condition for the linear stability of the synchronized states. We show that, for a given degree distribution, the maximum synchronizability is achieved when the network of couplings is weighted and directed and the overall cost involved in the couplings is minimum. This enhanced synchronizability is solely determined by the mean degree and does not depend on the degree distribution and system size. Numerical verification of the main results is provided for representative classes of small-world and scale-free networks.

Controllability of flow turbulence

In this paper, we study the controllability of real-world flow turbulence governed by the two-dimensional Navier-Stokes equations, using strategies developed in chaos control. A case of control/synchronization of turbulent dynamics is observed when only one component of the velocity field vector is unidirectionally coupled to a target state, while the other component is uncoupled. Unlike previous results, it is shown that the dynamics of the whole velocity field cannot be completely controlled/synchronized to the target, even in the limit of long time and strong coupling strength. It is further revealed that the controlled component of the velocity field can be fully controlled/synchronized to the target, but the other component, which is not directly coupled to the target, can only be partially controlled/synchronized to the target. By extending an auxiliary method to distributed dynamic systems, the partial synchronization of two turbulent orbits in the present study can be categorized in the domain of generalized synchronization of spatiotemporal dynamics.

Complete synchronization and generalized synchronization of one-way coupled time-delay systems

The complete synchronization and generalized synchronization (GS) of one-way coupled time-delay systems are studied. We find that GS can be achieved by a single scalar signal, and its synchronization threshold for different delay times shows the parameter resonance effect, i.e., we can obtain stable synchronization at a smaller coupling if the delay time of the driven system is chosen such that it is in resonance with the driving system. Near chaos synchronization, the desynchronization dynamics displays periodic bursts with the period equal to the delay time of the driven system. These features can be easily applied to the recovery of time-delay systems.