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Dynamic Tensor Inversion (DTI) is an emerging issue in recent research, prevalent in artificial intelligence development frameworks such as TensorFlow and PyTorch. Traditional numerical methods suffer significant lagging error when addressing this issue. To address this, Zeroing-type Neural Dynamics (ZND) and Gradient-type Neural Dynamics (GND) are employed to tackle the DTI. However, these two methods exhibit inherent limitations in the resolution process, i.e. high computational complexity and low solution accuracy, respectively. Motivated by this technology gap, this paper proposes an Adaptive Coefficient Gradient Neural Dynamics (ACGND) for dynamically solving the DTI with an efficient and precise manner. Through a series of simulation experiments and validations in engineering applications, the ACGND demonstrates advantages in resolving DTI. The ACGND enhances computational efficiency by circumventing matrix inversion, thereby reducing computational complexity. Moreover, its incorporation of adaptive coefficients and activation functions enables real-time adjustments of the computational solution, facilitating rapid convergence to theoretical solutions and adaptation to non-statinary scenarios. Code is available at https://github.com/Maia2333/ACGND-Code-Implementation .

Online solution of time-varying nonlinear optimization problems is considered an important issue in the fields of scientific and engineering research. In this study, the continuous-time derivative (CTD) model and two gradient dynamics (GD) models are developed for real-time varying nonlinear optimization (RTVNO). A continuous-time Zhang dynamics (CTZD) model is then generalized and investigated for RTVNO to remedy the weaknesses of CTD and GD models. For possible digital hardware realization, a discrete-time Zhang dynamics (DTZD) model, which can be further reduced to Newton-Raphson iteration (NRI), is also proposed and developed. Theoretical analyses indicate that the residual error of the CTZD model has an exponential convergence, and that the maximum steady-state residual error (MSSRE) of the DTZD model has an O ( τ 2 ) pattern with τ denoting the sampling gap. Simulation and numerical results further illustrate the efficacy and advantages of the proposed CTZD and DTZD models for RTVNO.

Turbulence small-scale behavior has been commonly investigated in literature by decomposing the velocity-gradient tensor (Aij) into the symmetric strain-rate (Sij) and anti-symmetric rotation-rate (Wij) tensors. To develop further insight, we revisit some of the key studies using a triple decomposition of the velocity-gradient tensor. The additive triple decomposition formally segregates the contributions of normal-strain-rate (Nij), pure-shear (Hij) and rigid-body-rotation-rate (Rij). The decomposition not only highlights the key role of shear, but it also provides a more accurate account of the influence of normal-strain and pure rotation on important small-scale features. First, the local streamline topology and geometry are described in terms of the three constituent tensors in velocity-gradient invariants' space. Using direct numerical simulation (DNS) data sets of forced isotropic turbulence, the velocity-gradient and pressure field fluctuations are examined at different Reynolds numbers. At all Reynolds numbers, shear contributes the most and rigid-body-rotation the least toward the velocity-gradient magnitude (A2 ≡ AijAij). Especially, shear contribution is dominant in regions of high values of A2 (intermittency). It is shown that the high-degree of enstrophy intermittency reported in literature is due to the shear contribution toward vorticity rather than that of rigid-body-rotation. The study also provides an explanation for the non-intermittent nature of pressure-Laplacian, despite the strong intermittency of enstrophy and dissipation fields. The study further investigates the alignment of the rotation axis with normal strain-rate and pressure Hessian eigenvectors. Overall, it is demonstrated that triple decomposition offers unique and deeper understanding of velocity-gradient behavior in turbulence.

In this paper, a gradient dynamics-based control method is proposed to directly tackle the singularity problem in the backstepping control design of the TORA system. This method is founded upon the construction of an energy-like positive function, which includes an auxiliary variable in terms of the intermediate virtual control law. On this basis, a gradient dynamics is created to obtain a new virtual control command, which is capable of making the auxiliary variable gradually approach zero, thereby mitigating the issue of division by zero. The core innovation is the integration of the gradient dynamics into the recursive backstepping design to overcome the singularity problem and stabilize the system at the equilibrium quickly. In addition, it rigorously proves that all the signals in the closed-loop control system are uniformly ultimately bounded, and the tracking errors converge to a small neighborhood around zero through a Lyapunov-based stability analysis. Comparative simulations demonstrate that the proposed approach not only avoids the singularity issue, but also achieves a better transient performance over other methods.

The lack-of-fit statistical reduction, developed and formulated first by Bruce Turkington, is a general method taking Liouville equation for probability density (detailed level) and transforming it to reduced dynamics of projected quantities (less detailed level). In this paper the method is generalized. The Hamiltonian Liouville equation is replaced by an arbitrary Hamiltonian evolution combined with gradient dynamics (GENERIC), the Boltzmann entropy is replaced by an arbitrary entropy, and the kinetic energy by an arbitrary energy. The gradient part is a generalized gradient dynamics generated by a dissipation potential. The reduced evolution of the projected state variables is shown to preserve the GENERIC structure of the original (detailed level) evolution. The dissipation potential is obtained by solving a Hamilton–Jacobi equation. In summary, the lack-of-fit reduction can start with GENERIC and obtain GENERIC for the reduced state variables.

In contrast to the conventional entanglement-separability paradigm in quantum information theory, we embark on a different path by introducing a classical dichotomy. We aim to quantify a measurement’s disturbance by minimizing the difference between input and post-measurement states to distinguish either classical or quantum correlations. Theoretically, we apply a complex-valued gradient flow over Stiefel manifolds for minimization. Our focus extends beyond the practical application to encompass the well-known Łojasiewicz gradient inequality. This inequality is a fundamental tool that guarantees the global convergence of the flow from any initial starting point to the optimal solution. Numerically, we validate the effectiveness and robustness of our proposed method by performing a series of experiments in different scenarios. Experimental results suggest the capability of our approach to accurately and reliably characterize correlations as classical or quantum.

This paper is concerned with the distributed Nash equilibrium (NE) computation problem for non-cooperative games subject to partial-decision information. For the purpose of congestion mitigation, coding-decoding-based schemes are constructed on the basis of logarithmic and uniform quantizers, respectively. To be specific, the data (decision variable) are first mapped to codewords by an encoder scheme, and then sent to the neighboring agents through a directed communication network (with non-doubly stochastic weighted matrix). By using a decoder scheme, a new distributed algorithm is established for seeking the NE. In order to eliminate the convergence error caused by quantization, a dynamic variable is introduced and a modified coding-decoding-based algorithm is constructed under the uniform quantization scheme, which ensures the asymptotic convergence to the NE. The proposed algorithm only requires that the weighted adjacency matrix is row stochastic instead of double stochastic. Finally, one numerical example is provided to validate the effectiveness of our algorithms.

Wetting and dewetting dynamics of simple and complex liquids is described by kinetic equations in gradient dynamics form that incorporates the various coupled dissipative processes in a fully thermodynamically consistent manner. After briefly reviewing this, we also review how chemical reactions can be captured by a related gradient dynamics description, assuming detailed balanced mass action type kinetics. Then, we bring both aspects together and discuss mesoscopic reactive thin-film hydrodynamics illustrated by two examples, namely, models for reactive wetting and reactive surfactants. These models can describe the approach to equilibrium but may also be employed to study out-of-equilibrium chemo-mechanical dynamics. In the latter case, one breaks the gradient dynamics form by chemostatting to obtain active systems. In this way, for reactive wetting we recover running drops that are driven by chemically sustained wettability gradients and for drops covered by autocatalytic reactive surfactants we find complex forms of self-propulsion and self-excited oscillations.

Standard MCMC methods can scale poorly to big data settings due to the need to evaluate the likelihood at each iteration. There have been a number of approximate MCMC algorithms that use sub-sampling ideas to reduce this computational burden, but with the drawback that these algorithms no longer target the true posterior distribution. We introduce a new family of Monte Carlo methods based upon a multidimensional version of the Zig-Zag process of [Ann. Appl. Probab. 27 (2017) 846–882], a continuous-time piecewise deterministic Markov process. While traditional MCMC methods are reversible by construction (a property which is known to inhibit rapid convergence) the Zig-Zag process offers a flexible nonreversible alternative which we observe to often have favourable convergence properties. We show how the Zig-Zag process can be simulated without discretisation error, and give conditions for the process to be ergodic. Most importantly, we introduce a sub-sampling version of the Zig-Zag process that is an example of an exact approximate scheme, that is, the resulting approximate process still has the posterior as its stationary distribution. Furthermore, if we use a control-variate idea to reduce the variance of our unbiased estimator, then the Zig-Zag process can be super-efficient: after an initial preprocessing step, essentially independent samples from the posterior distribution are obtained at a computational cost which does not depend on the size of the data.

Reformulating constitutive relation in terms of gradient dynamics (being derivative of a dissipation potential) brings additional information on stability, metastability and instability of the dynamics with respect to perturbations of the constitutive relation, called CR-stability. CR-instability is connected to the loss of convexity of the dissipation potential, which makes the Legendre-conjugate dissipation potential multivalued and causes dissipative phase transitions that are not induced by non-convexity of free energy, but by non-convexity of the dissipation potential. CR-stability of the constitutive relation with respect to perturbations is then manifested by constructing evolution equations for the perturbations in a thermodynamically sound way (CR-extension). As a result, interesting experimental observations of behavior of complex fluids under shear flow and supercritical boiling curve can be explained.