Low-dimensional Model Research Articles

Uneven extraction in coffee brewing

A recent experiment showed that, contrary to theoretical predictions, beyond a cutoff point, grinding coffee more finely results in lower extraction. One potential explanation for this is that fine grinding promotes non-uniform extraction in the coffee bed. We investigate the possibility that this could occur due the interaction between dissolution and flow promoting uneven extraction. A low dimensional model in which there are two possible pathways for flow is derived and analyzed. This model shows that, below a critical grind size, there is a decreasing extraction with decreasing grind size as is seen experimentally. This is due to a complicated interplay between an initial imbalance in the porosities and permeabilities of the two pathways in the model, which is increased by flow and extraction, leading to the complete extraction of all soluble coffee from one pathway.

Dynamical Bifurcation for a Class of Large-Scale Fractional Delayed Neural Networks With Complex Ring-Hub Structure and Hybrid Coupling.

Real neural networks are characterized by large-scale and complex topology. However, the current dynamical analysis is limited to low-dimensional models with simplified topology. Therefore, there is still a huge gap between neural network theory and its application. This article proposes a class of large-scale neural networks with a ring-hub structure, where a hub node is connected to n peripheral nodes and these peripheral nodes are linked by a ring. In particular, there exists a hybrid coupling mode in the network topology. The mathematical model of such systems is described by fractional-order delayed differential equations. The aim of this article is to investigate the local stability and Hopf bifurcation of this high-dimensional neural network. First, the Coates flow graph is employed to obtain the characteristic equation of the linearized high-dimensional neural network model, which is a transcendental equation including multiple exponential items. Then, the sufficient conditions ensuring the stability of equilibrium and the existence of Hopf bifurcation are achieved by taking time delay as a bifurcation parameter. Finally, some numerical examples are given to support the theoretical results. It is revealed that the increasing time delay can effectively induce the occurrence of periodic oscillation. Moreover, the fractional order, the self-feedback coefficient, and the number of neurons also have effects on the onset of Hopf bifurcation.

Data-driven reduced order modeling for mechanical oscillators using Koopman approaches

Data-driven reduced order modeling methods that aim at extracting physically meaningful governing equations directly from measurement data are facing a growing interest in recent years. The HAVOK-algorithm is a Koopman-based method that distills a forced, low-dimensional state-space model for a given dynamical system from a univariate measurement time series. This article studies the potential of HAVOK for application to mechanical oscillators by investigating which information of the underlying system can be extracted from the state-space model generated by HAVOK. Extensive parameter studies are performed to point out the strengths and pitfalls of the algorithm and ultimately yield recommendations for choosing tuning parameters. The application of the algorithm to real-world friction brake system measurements concludes this study.

Machine learned compact kinetic models for methane combustion

Chemical kinetic models are an essential component in the development and optimisation of combustion devices through their coupling to multi-dimensional simulations such as computational fluid dynamics (CFD). Due to the significant level of detail contained within, detailed chemical kinetic models are computationally prohibitive for use in CFD. Therefore, low-dimensional kinetic models which retain good fidelity to the reality are needed, the production of which requires considerable human-time cost and expert knowledge.Here, we present a novel automated compute intensification methodology to produce overly-reduced and optimised (“compact”) chemical kinetic models. The Machine Learned Optimisation of Chemical Kinetics (MLOCK) coded algorithm systematically perturbs each of the four chemical kinetic model components to discover what combinations of terms results in a model with high fidelity calculations. A virtual reaction network comprised of n species is first obtained using conventional mechanism reduction procedures. Once n is lower than a threshold value, the model performance is typically of low fidelity.To adjust for this, the weights (virtual reaction rate constants) of important connections (virtual reactions) between each node (species) of the virtual reaction network are numerically optimised across four sequential phases to replicate select calculations.The first version of MLOCK (MLOCK1.0), simultaneously perturbs all three virtual Arrhenius reaction rate constant parameters for important connections and assesses the suitability of the new parameters through objective error functions, which quantify the error in the calculations of each model candidate to a set of optimisation targets, comprised of detailed model calculations.In this study, the MLOCK algorithm is demonstrated by automatic creation of compact models for the archetypal case of methane/air combustion. It is shown that the NUGMECH1.0 detailed model comprised of 2789 species is reliably compacted to 15 species (nodes), whilst retaining an overall fidelity of 79–90% to the detailed model at Industry-defined performance target calculations, outperforming the prior state-of-art.

Turbulence control in plane Couette flow using low-dimensional neural ODE-based models and deep reinforcement learning

The high dimensionality and complex dynamics of turbulent flows remain an obstacle to the discovery and implementation of control strategies. Deep reinforcement learning (RL) is a promising avenue for overcoming these obstacles, but requires a training phase in which the RL agent iteratively interacts with the flow environment to learn a control policy, which can be prohibitively expensive when the environment involves slow experiments or large-scale simulations. We overcome this challenge using a framework we call “DManD-RL” (data-driven manifold dynamics-RL), which generates a data-driven low-dimensional model of our system that we use for RL training. With this approach, we seek to minimize drag in a direct numerical simulation (DNS) of a turbulent minimal flow unit of plane Couette flow at Re=400 using two slot jets on one wall. We obtain, from DNS data with O(105) degrees of freedom, a 25-dimensional DManD model of the dynamics by combining an autoencoder and neural ordinary differential equation. Using this model as the environment, we train an RL control agent, yielding a 440-fold speedup over training on the DNS, with equivalent control performance. The agent learns a policy that laminarizes 84% of unseen DNS test trajectories within 900 time units, significantly outperforming classical opposition control (58%), despite the actuation authority being much more restricted. The agent often achieves laminarization through a counterintuitive strategy that drives the formation of two low-speed streaks, with a spanwise wavelength that is too small to be self-sustaining. The agent demonstrates the same performance when we limit observations to wall shear rate.

Low-dimensional models of single neurons: a review.

The classical Hodgkin-Huxley (HH) point-neuron model of action potential generation is four-dimensional. It consists of four ordinary differential equations describing the dynamics of the membrane potential and three gating variables associated to a transient sodium and a delayed-rectifier potassium ionic currents. Conductance-based models of HH type are higher-dimensional extensions of the classical HH model. They include a number of supplementary state variables associated with other ionic current types, and are able to describe additional phenomena such as subthreshold oscillations, mixed-mode oscillations (subthreshold oscillations interspersed with spikes), clustering and bursting. In this manuscript we discuss biophysically plausible and phenomenological reduced models that preserve the biophysical and/or dynamic description of models of HH type and the ability to produce complex phenomena, but the number of effective dimensions (state variables) is lower. We describe several representative models. We also describe systematic and heuristic methods of deriving reduced models from models of HH type.

Data-driven low-dimensional dynamic model of Kolmogorov flow

Minimal dimensional models are desirable for reduced computational costs in simulations as well as for applications such as model-based control. Long-time dynamics of flows often evolve on a low-dimensional manifold M in the full state space. We use neural networks to estimate M and the dynamics on it for two-dimensional Kolmogorov flow in a chaotic bursting regime. Outcomes include: a minimal dimension estimate, good short-time tracking and long-time statistics, as well as accurate predictions of bursting events.

Dynamic Modeling and Attitude–Vibration Cooperative Control for a Large-Scale Flexible Spacecraft

Modern spacecraft usually have larger and more flexible appendages whose vibration becomes more and more prominent, and it has a great influence on the precision of spacecraft attitude. Therefore, the cooperative control of attitude maneuvering and structural vibration of the system has become a significant issue in the spacecraft design process. We developed a low-dimensional and high-precision mathematical model for a large-scale flexible spacecraft (LSFS) equipped with a pair of hinged solar arrays in this paper. The analytic global modes are used to obtain the rigid–flexible coupling discrete dynamic model, and the governing equations with multiple DOFs for the system are derived by using the Hamiltonian principle. The rigid–flexible coupled oscillating responses of LSFS under the three-axis attitude-driving torque pulse during the in-orbit attitude maneuvering process are investigated. A study on the flexibility of the hinge was also conducted. Based on the simplified and accurate dynamic model of the system, we can obtain a state-space model for LSFS conveniently, and the cooperative control schemes for rigid motion and flexible oscillation control are designed by using the LQR, PD, and PD + IS algorithms. The simulation results show that three cooperative controllers can realize spacecraft attitude adjustment and synchronously eliminate flexible oscillation successfully. By comparison, the PD + IS controller is simpler so that it is suitable for the real-time attitude–vibration cooperative control of spacecraft.

Oscillating Behavior of a Compartmental Model with Retarded Noisy Dynamic Infection Rate

Our study is based on an epidemiological compartmental model, the SIRS model. In the SIRS model, each individual is in one of the states susceptible (S), infected (I) or recovered (R), depending on its state of health. In compartment R, an individual is assumed to stay immune within a finite time interval only and then transfers back to the S compartment. We extend the model and allow for a feedback control of the infection rate by mitigation measures which are related to the number of infections. A finite response time of the feedback mechanism is supposed that changes the low-dimensional SIRS model into an infinite-dimensional set of integro-differential (delay-differential) equations. It turns out that the retarded feedback renders the originally stable endemic equilibrium of SIRS (stable focus) to an unstable focus if the delay exceeds a certain critical value. Nonlinear solutions show persistent regular oscillations of the number of infected and susceptible individuals. In the last part we include noise effects from the environment and allow for a fluctuating infection rate. This results in multiplicative noise terms and our model turns into a set of stochastic nonlinear integro-differential equations. Numerical solutions reveal an irregular behavior of repeated disease outbreaks in the form of infection waves with a variety of frequencies and amplitudes.

Reduction of High Dimensional Noninteger Commensurate Systems Based on Differential Evolution

This work relates to the reduction of a noninteger commensurate high dimensional system. The essential objective of this article is to come up with an approximating technique to replace the original high dimensional system with a low dimensional model preserving the properties of the original system in its shortened model. Superiority of the proposed technique is exhibited by correlating the reduced model with the models of other current methods. The simulation results are cited to approve that the recommended technique has high efficiency with a closing value of time-domain specifications. Conclusively, more logical comparisons were made between other existing methods. The performance indices are calculated for both original system and reduced model system and presented in the manuscript.

Grid-based state space exploration for molecular binding

Binding processes are difficult to sample with molecular dynamics (MD) simulations. In particular, the state space exploration is often incomplete. Evaluating the molecular interaction energy on a grid circumvents this problem but is heavily limited by state space dimensionality. Here, we make the first steps towards a low-dimensional grid-based model of molecular binding. We discretize the state space of relative positions and orientations of the two molecules under the rigid body assumption. The corresponding program is published as the Python package molgri. For the rotational component of the grids, we test algorithms based on Euler angles, polyhedra and quaternions, of which the polyhedra-based are the most uniform. The program outputs a sequence of molecular structures that can be easily processed by standard MD programs to calculate grid point energies. We demonstrate the grid-based approach on two molecular systems: a water dimer and a coiled-coil protein interacting with a chloride anion. For the second system we relax the rigid-body assumption and improve the accuracy of the grid point energies by an energy minimization. In both cases, oriented bonding patterns and energies confirm expectations from chemical intuition and MD simulations. We also demonstrate how analysis of energy contributions on a grid can be performed and demonstrate that electrostatically driven association is sufficiently resolved by point-energy calculations. Overall, grid-based models of molecular binding are potentially a powerful complement to molecular sampling approaches, and we see the potential to expand the method to quantum chemistry and flexible docking applications.

First blood: An efficient, hybrid one- and zero-dimensional, modular hemodynamic solver.

Low-dimensional (1D or 0D) models can describe the whole human blood circulation, for example, 1D distributed parameter model for the arterial network and 0D concentrated models for the heart or other organs. This paper presents a combined 1D-0D solver, called first_blood, that solves the governing equations of fluid dynamics to model low-dimensional hemodynamic effects. An extended method of characteristics is applied here to solve the momentum, and mass conservation equations and the viscoelastic wall model equation, mimicking the material properties of arterial walls. The heart and the peripheral lumped models are solved with a general zero-dimensional (0D) nonlinear solver. The model topology can be modular, that is, first_blood can solve any 1D-0D hemodynamic model. To demonstrate the applicability of first_blood, the human arterial system, the heart and the peripherals are modelled using the solver. The simulation time of a heartbeat takes around 2 s, that is, first_blood requires only twice the real-time for the simulation using an average PC, which highlights the computational efficiency. The source code is available on GitHub, that is, it is open source. The model parameters are based on the literature suggestions and on the validation of output data to obtain physiologically relevant results.

Intelligent operation, maintenance, and control system for public building: Towards infection risk mitigation and energy efficiency

During the post-COVID-19 era, it is important but challenging to synchronously mitigate the infection risk and optimize the energy savings in public buildings. While, ineffective control of ventilation and purification systems can result in increased energy consumption and cross-contamination. This paper is to develop intelligent operation, maintenance, and control systems by coupling intelligent ventilation and air purification systems (negative ion generators). Optimal deployment of sensors is determined by Fuzzy C-mean (FCM), based on which CO2 concentration fields are rapidly predicted by combing the artificial neural network (ANN) and self-adaptive low-dimensional linear model (LLM). Negative oxygen ion and particle concentrations are simulated with different numbers of negative ion generators. Optimal ventilation rates and number of negative ion generators are decided. A visualization platform is established to display the effects of ventilation control, epidemic prevention, and pollutant removal. The rapid prediction error of LLM-based ANN for CO2 concentration was below 10% compared with the simulation. Fast decision reduced CO2 concentration below 1000 ppm, infection risk below 1.5%, and energy consumption by 27.4%. The largest removal efficiency was 81% when number of negative ion generators was 10. This work can promote intelligent operation, maintenance, and control systems considering infection prevention and energy sustainability.

Emergence of time persistence in a data-driven neural network model.

Establishing accurate as well as interpretable models of network activity is an open challenge in systems neuroscience. Here, we infer an energy-based model of the anterior rhombencephalic turning region (ARTR), a circuit that controls zebrafish swimming statistics, using functional recordings of the spontaneous activity of hundreds of neurons. Although our model is trained to reproduce the low-order statistics of the network activity at short time scales, its simulated dynamics quantitatively captures the slowly alternating activity of the ARTR. It further reproduces the modulation of this persistent dynamics by the water temperature and visual stimulation. Mathematical analysis of the model unveils a low-dimensional landscape-based representation of the ARTR activity, where the slow network dynamics reflects Arrhenius-like barriers crossings between metastable states. Our work thus shows how data-driven models built from large neural populations recordings can be reduced to low-dimensional functional models in order to reveal the fundamental mechanisms controlling the collective neuronal dynamics.

An optimal regulation of fluxes dictates microbial growth in and out of steady-state.

Effective coordination of cellular processes is critical to ensure the competitive growth of microbial organisms. Pivotal to this coordination is the appropriate partitioning of cellular resources between protein synthesis via translation and the metabolism needed to sustain it. Here, we extend a low-dimensional allocation model to describe the dynamic regulation of this resource partitioning. At the core of this regulation is the optimal coordination of metabolic and translational fluxes, mechanistically achieved via the perception of charged- and uncharged-tRNA turnover. An extensive comparison with ≈ 60 data sets from Escherichia coli establishes this regulatory mechanism's biological veracity and demonstrates that a remarkably wide range of growth phenomena in and out of steady state can be predicted with quantitative accuracy. This predictive power, achieved with only a few biological parameters, cements the preeminent importance of optimal flux regulation across conditions and establishes low-dimensional allocation models as an ideal physiological framework to interrogate the dynamics of growth, competition, and adaptation in complex and ever-changing environments.

Self-propulsion via slipping: Frictional swimming in multilegged locomotors

Locomotion is typically studied either in continuous media where bodies and legs experience forces generated by the flowing medium or on solid substrates dominated by friction. In the former, centralized whole-body coordination is believed to facilitate appropriate slipping through the medium for propulsion. In the latter, slip is often assumed minimal and thus avoided via decentralized control schemes. We find in laboratory experiments that terrestrial locomotion of a meter-scale multisegmented/legged robophysical model resembles undulatory fluid swimming. Experiments varying waves of leg stepping and body bending reveal how these parameters result in effective terrestrial locomotion despite seemingly ineffective isotropic frictional contacts. Dissipation dominates over inertial effects in this macroscopic-scaled regime, resulting in essentially geometric locomotion on land akin to microscopic-scale swimming in fluids. Theoretical analysis demonstrates that the high-dimensional multisegmented/legged dynamics can be simplified to a centralized low-dimensional model, which reveals an effective resistive force theory with an acquired viscous drag anisotropy. We extend our low-dimensional, geometric analysis to illustrate how body undulation can aid performance in non-flat obstacle-rich terrains and also use the scheme to quantitatively model how body undulation affects performance of biological centipede locomotion (the desert centipede Scolopendra polymorpha) moving at relatively high speeds (∼0.5 body lengths/sec). Our results could facilitate control of multilegged robots in complex terradynamic scenarios.

The nonparametric Box–Cox model for high-dimensional regression analysis

The mainstream theory for high-dimensional regression assumes that the underlying true model is a low-dimensional linear regression model. On the other hand, a standard technique in regression analysis, even in the traditional low-dimensional setting, is to employ the Box–Cox transformation for reducing anomalies such as non-additivity and heteroscedasticity in linear regression. In this paper, we propose a new high-dimensional regression method based on a nonparametric Box–Cox model with an unspecified monotone transformation function. Model fitting and computation become much more challenging than the usual penalized regression method, and a two-step method is proposed for the estimation of this model in high-dimensional settings. First, we propose a novel technique called composite probit regression (CPR) and use the folded concave penalized CPR for estimating the regression parameters. The strong oracle property of the estimator is established without knowing the nonparametric transformation function. Next, the nonparametric function is estimated by conducting univariate monotone regression. The computation is done efficiently by using a coordinate-majorization-descent algorithm. Extensive simulation studies show that the proposed method performs well in various settings. Our analysis of the supermarket data demonstrates the superior performance of the proposed method over the standard high-dimensional regression method.

A new (n+1)-dimensional generalized Kadomtsev–Petviashvili equation: integrability characteristics and localized solutions

Searching for higher-dimensional integrable models is one of the most significant and challenging issues in nonlinear mathematical physics. This paper aims to extend the classic lower-dimensional integrable models to arbitrary spatial dimension. We investigate the celebrated Kadomtsev–Petviashvili (KP) equation and propose its (n+1)-dimensional integrable extension. Based on the singularity manifold analysis and binary Bell polynomial method, it is found that the (n+1)-dimensional generalized KP equation has N-soliton solutions, and it also possesses the Painlevé property, Lax pair, Bäcklund transformation as well as infinite conservation laws, and thus the (n+1)-dimensional generalized KP equation is proven to be completely integrable. Moreover, various types of localized solutions can be constructed starting from the N-soliton solutions. The abundant interactions including overtaking solitons, head-on solitons, one-order lump, two-order lump, breather, breather-soliton mixed solutions are analyzed by some graphs.

Deformation conjecture: deforming lower dimensional integrable systems to higher dimensional ones by using conservation laws

Utilizing some conservation laws of (1+1)-dimensional integrable local evolution systems, it is conjectured that higher dimensional integrable equations may be regularly constructed by a deformation algorithm. The algorithm can be applied to Lax pairs and higher order flows. In other words, if the original lower dimensional model is Lax integrable (possesses Lax pairs) and symmetry integrable (possesses infinitely many higher order symmetries and/or infinitely many conservation laws), then the deformed higher order systems are also Lax integrable and symmetry integrable. For concreteness, the deformation algorithm is applied to the usual (1 + 1)-dimensional Korteweg-de Vries (KdV) equation and the (1 + 1)-dimensional Ablowitz-Kaup-Newell-Segur (AKNS) system (including nonlinear Schrödinger (NLS) equation as a special example). It is interesting that the deformed (3+1)-dimensional KdV equation is also an extension of the (1 + 1)-dimensional Harry-Dym (HD) type equations which are reciprocal links of the (1+1)-dimensional KdV equation. The Lax pairs of the (3 + 1)-dimensional KdV-HD system and the (2 + 1)-dimensional AKNS system are explicitly given. The higher order symmetries, i.e., the whole (3 + 1)-dimensional KdV-HD hierarchy, are also explicitly obtained via the deformation algorithm. The single soliton solution of the (3 + 1)-dimensional KdV-HD equation is implicitly given. Because of the effects of the deformation, the symmetric soliton shape of the usual KdV equation is no longer conserved and deformed to be asymmetric and/or multi-valued. The deformation conjecture holds for all the known (1 +1)-dimensional integrable local evolution systems that have been checked, and we have not yet found any counter-example so far. The introduction of a large number of (D + 1)-dimensional integrable systems of this paper explores a serious challenge to all mathematicians and theoretical physicists because the traditional methods are no longer directly valid to solve these integrable equations.

Where are the coexisting parallel climates? Large ensemble climate projections from the point of view of chaos theory.

We review the recent results of large ensemble climate projections considering them to be the simulations of chaotic systems. The quick spread of an initially localized ensemble in the first weeks after initialization is an appearance of the butterfly effect, illustrating the unpredictability of the dynamics. We show that the growth rate of uncertainty (an analog of the Lyapunov exponent) can be determined right after initialization. The next phase corresponds to a convergence of the no longer localized ensemble to the time-dependent climate attractor and requires a much longer time. After convergence takes place, the ensemble faithfully represents the climate dynamics. Concerning a credible simulation, the observed signal should then wander within the spread of the converged ensemble all the time, i.e., to behave just as any of the ensemble members. As a manifestation of the chaotic-like climate dynamics, one can imagine that beyond the single, observed time-dependent climate, a plethora of parallel climate realizations exists. Converged climate ensembles also define the probability distribution by which the physical quantities of the different climate realizations occur. Large ensemble simulations were shown earlier to be credible in the sense formulated. Here, in addition, an extended credibility condition is given, which requires the ensemble to be a converged ensemble, valid also for low-dimensional models. Interestingly, to the best of our knowledge, no low-order physical or engineering systems subjected to time-dependent forcings are known for which a comparison between simulation and experiment would be available. As illustrative examples, the CESM1-LE climate model and a chaotic pendulum are taken.