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In this article, we derive optimal relaxation rates for the Cahn-Hilliard equation on the one-dimensional torus and the line. We consider initial conditions with a finite (but not small) L1-distanc...

We consider a discrete-continuum model of a biomembrane with embedded particles. While the membrane is represented by a continuous surface, embedded particles are described by rigid discrete objects which are free to move and rotate in lateral direction. For the membrane we consider a linearized Canham-Helfrich energy functional and height and slope boundary conditions imposed on the particle boundaries resulting in a coupled minimization problem for the membrane shape and particle positions. When considering the energetically optimal membrane shape for each particle position we obtain a reduced energy functional that models the implicitly given interaction potential for the membrane-mediated mechanical particle-particle interactions. We show that this interaction potential is differentiable with respect to the particle positions and orientations. Furthermore we derive a fully practical representation of the derivative only in terms of well defined derivatives of the membrane. This opens the door for the application of minimization algorithms for the computation of minimizers of the coupled system and for further investigation of the interaction potential of membrane-mediated mechanical particle-particle interaction. The results are illustrated with numerical examples comparing the explicit derivative formula with difference quotient approximations. We furthermore demonstrate the application of the derived formula to implement a gradient flow for the approximation of optimal particle configurations.

In this study, we develop high-order time-accurate, efficient, and energy stable schemes for solving the conservative Swift–Hohenberg equation that can be used to describe the L 2 -gradient flow based phase-field crystal dynamics. By adopting a modified exponential scalar auxiliary variable approach, we first transform the original equations into an expanded system. Based on the expanded system, the first-, second-, and third-order time-accurate schemes are constructed using the backward Euler formula, second-order backward difference formula (BDF2), and third-order backward difference formula (BDF3), respectively. The energy dissipation law can be easily proved with respect to a modified energy. In each time step, the local variable is updated by solving one elliptic type equation and the non-local variables are explicitly computed. The whole algorithm is totally decoupled and easy to implement. Extensive numerical experiments in two- and three-dimensional spaces are performed to show the accuracy, energy stability, and practicability of the proposed schemes.

We prove that a Morse-Smale gradient-like flow on a closed manifold has a "system of compatible invariant stable foliations" that is analogous to the object introduced by Palis and Smale in their proof of the structural stability of Morse-Smale diffeomorphisms and flows, but with finer regularity and geometric properties. We show how these invariant foliations can be used in order to give a self-contained proof of the well-known but quite delicate theorem stating that the unstable manifolds of a Morse-Smale gradient-like flow on a closed manifold $M$ are the open cells of a $CW$-decomposition of $M$., 57 pages, 1 figure. The list of references has been expanded and the discussion about the history of the problem and future perspectives has been improved thanks to the suggestions of some readers

Bilayer plates are compound materials that exhibit large bending deformations when exposed to environmental changes that lead to different mechanical responses in the involved materials. In this article a new numerical method that is suitable for simulating the isometric deformation induced by a given material mismatch in a bilayer plate is discussed. A dimensionally reduced formulation of the bending energy is discretized generically in an abstract setting and specified for discrete Kirchhoff triangles; convergence towards the continuous formulation is proved. A practical semi-implicit discrete gradient flow employing a linearization of the isometry constraint is proposed as an iterative method for the minimization of the bending energy; stability and a bound on the violation of the isometry constraint are proved. The incorporation of obstacles is discussed and the practical performance of the method is illustrated with numerical experiments involving the simulation of large bending deformations and investigation of contact phenomena.

In this paper, we define a family of functionals generalizing the Yang–Mills–Higgs functionals on a closed Riemannian manifold. Then we prove the short-time existence of the corresponding gradient flow by a gauge-fixing technique. The lack of a maximum principle for the higher order operator brings us a lot of inconvenience during the estimates for the Higgs field. We observe that the$L^2$-bound of the Higgs field is enough for energy estimates in four dimensions and we show that, provided the order of derivatives appearing in the higher order Yang–Mills–Higgs functionals is strictly greater than one, solutions to the gradient flow do not hit any finite-time singularities. As for the Yang–Mills–Higgsk-functional with Higgs self-interaction, we show that, provided$\dim (M), for every smooth initial data the associated gradient flow admits long-time existence. The proof depends on local$L^2$-derivative estimates, energy estimates and blow-up analysis.

We develop a first- and second-order finite-volume scheme to solve gradient flow equations with non-homogeneous properties, obtained in the framework of dynamical-density functional theory. The scheme takes advantage of an upwind approach for the space discretization to ensure positivity of the density under a CFL condition and decay of the discrete free energy. Our computational approach is used to study several one- and two-dimensional systems, with a general free-energy functional accounting for external fields and inter-particle potentials, and placed in non-homogeneous thermal baths characterized by anisotropic, space-dependent and time-dependent properties.

We study the dynamics of a droplet moving on an inclined rough surface in the absence of inertial and viscous stress effects. In this case, the dynamics of the droplet is a purely geometric motion in terms of the wetting domain and the capillary surface. Using a single graph representation, we interpret this geometric motion as a gradient flow on a Hilbert manifold. We propose unconditionally stable first/second order numerical schemes to simulate this geometric motion of the droplet, which is described using motion by mean curvature coupled with moving contact lines. The schemes are based on (i) explicit moving boundaries, which decouple the dynamic updates of the contact lines and the capillary surface, (ii) a semi-Lagrangian method on moving grids and (iii) a predictor-corrector method with a nonlinear elliptic solver upto second order accuracy. For the case of quasi-static dynamics with continuous spatial variable in the numerical schemes, we prove the stability and convergence of the first/second order numerical schemes. To demonstrate the accuracy and long-time validation of the proposed schemes, several challenging computational examples - including breathing droplets, droplets on inhomogeneous rough surfaces and quasi-static Kelvin pendant droplets - are constructed and compared with exact solutions to quasi-static dynamics obtained by desingularized differential-algebraic system of equations (DAEs)., 8 figures, 48 pages

We recast a network generating partial differential equation system into a singular limit of a dissipative gradient flow model, which not only identifies the consistent physical boundary conditions but also generates networks. We then develop a set of structure-preserving numerical algorithms for the gradient flow model. Using the energy quadratization (EQ) method, we reformulate the gradient flow system into an equivalent one with a quadratic energy density by introducing auxiliary variables. Subsequently, we devise a series of fully discrete, linear, second order, energy-production-rate preserving, finite difference algorithms to solve the EQ-reformulated PDE system subject to various compatible boundary conditions. We show that the numerical schemes are energy-production-rate preserving for any time steps. Numerical convergence tests are given to validate the accuracy of the fully discrete schemes. Several 2D numerical examples are given to demonstrate the capability of the schemes in predicting network generating phenomena with the gradient flow PDE system, especially, the original network generating PDE model.

In this paper, we devise and analyse three highly efficient second-order accurate (in time) schemes for solving the Functionalized Cahn–Hilliard (FCH) gradient flow equation where an asymmetric double-well potential function is considered. Based on the Scalar Auxiliary Variable (SAV) approach, we construct these schemes by splitting the FCH free energy in a novel and ingenious way. Utilizing the Crank–Nicolson formula, we firstly construct two semi-discrete second-order numerical schemes, which we denote by CN-SAV and CN-SAV-A, respectively. To be more specific, the CN-SAV scheme is constructed based on the fixed time step, while the CN-SAV-A scheme is a variable time step scheme. The BDF2-SAV scheme is another second-order scheme in which the fixed time step should be used. It is designed by applying the second-order backward difference (BDF2) formula. All the constructed schemes are proved to be unconditionally energy stable and uniquely solvable in theory. To the best of our knowledge, the CN-SAV-A scheme is the first unconditionally energy stable, second-order scheme with variable time steps for the FCH gradient flow equation. In addition, an effective adaptive time selection strategy introduced in Christlieb et al., (2014) is slightly modified and then adopted to select the time step for the CN-SAV-A scheme. Finally, several numerical experiments based on the Fourier pseudo-spectral method are carried out in two and three dimensions, respectively, to confirm the numerical accuracy and efficiency of the constructed schemes.

We explore equivariant dynamics under the symmetric group $S_N$ of all permutations of $N$ elements. Specifically we study one-parameter vector fields, up to cubic order, which commute with the standard real $(N-1)$-dimensional irreducible representation of $S_N$. The parameter is the linearization at the trivial 1-cluster equilibrium of total synchrony. All equilibria are cluster solutions involving up to three clusters. The resulting global dynamics is of gradient type: all bounded solutions are cluster equilibria and heteroclinic orbits between them. In the limit of large $N$, we present a detailed analysis of the web of heteroclinic orbits among the plethora of 2-cluster equilibria. Our focus is on the global dynamics of 3-cluster solutions with one rebel cluster of small size. These solutions describe slow relative growth and decay of 2-cluster states. For $N\rightarrow\infty$, the limiting heteroclinic web defines an integrable \emph{rebel flow} in the space of 2-cluster equilibrium configurations. We identify and study the seven qualitatively distinct global rebel flows which arise in this setting. Applications include oscillators with all-to-all coupling, and electrochemistry. For illustration we consider synchronization clusters among $N$ complex Stuart-Landau oscillators with complex linear global coupling., 46 pages, 21 figures

We consider a multi-agent system whose dynamics is governed by a gradient flow on the unit sphere associated with the interaction potential between positions of all agents measured by a weighted distance | x i − x j | p + 2 for any p ≠ 0 . In this paper, we employ both attractive and repulsive couplings to study the asymptotic behavior of the system accompanied by both p > 0 (positive range) and p 0 (negative range), and this enables to yield richer dynamical phenomena. Firstly in an attractive regime, we focus on the emergence of the complete aggregation; however, the relaxation dynamics towards the aggregated state for the positive range differs from the one for the negative range. More precisely for p > 0 , the complete aggregation occurs with an algebraic rate O ( t − 1 / p ) . On the other hand for p 0 , the issue of global existence arises due to the singular interaction and is crucially related to the aggregation estimate. To this end, we show that the complete aggregation emerges in finite time and thus a solution exists until such a time. Lastly in a repulsive regime, we mainly consider the splay state for both positive and negative ranges, and several case studies are presented to obtain the qualitative insight. Finally, we compare our results with the case of p = 0 .

The Cahn–Hilliard equation is one of the most common models to describe phase separation processes of a mixture of two materials. For a better description of short-range interactions between the material and the boundary, various dynamic boundary conditions for the Cahn–Hilliard equation have been proposed and investigated in recent times. Of particular interests are the model by Goldstein et al. [Phys. D 240 (2011) 754–766] and the model by Liu and Wu [Arch. Ration. Mech. Anal. 233 (2019) 167–247]. Both of these models satisfy similar physical properties but differ greatly in their mass conservation behaviour. In this paper we introduce a new model which interpolates between these previous models, and investigate analytical properties such as the existence of unique solutions and convergence to the previous models mentioned above in both the weak and the strong sense. For the strong convergences we also establish rates in terms of the interpolation parameter, which are supported by numerical simulations obtained from a fully discrete, unconditionally stable and convergent finite element scheme for the new interpolation model.

International audience; The existence of weak solutions to the obstacle problem for a nonlocal semilinear fourth-order parabolic equation is shown, using its underlying gradient flow structure. The model governs the dynamics of a microelectromechanical system with heterogeneous dielectric properties.

In this note, we establish certain regularity estimates for the spinor flow introduced and initially studied in \cite{AWW2016}. Consequently, we obtain that the norm of the second order covariant derivative of the spinor field becoming unbounded is the only obstruction for long-time existence of the spinor flow. This generalizes the blow up criteria obtained in \cite{Sc2018} for surfaces to general dimensions. As another application of the estimates, we also obtain a lower bound for the existence time in terms of the initial data. Our estimates are based on an observation that, up to pulling back by a one-parameter family of diffeomorphisms, the metric part of the spinor flow is equivalent to a modified Ricci flow., Comment: 32 pages

We construct short retractions of a CAT(1) space to its small convex subsets. This construction provides an alternative geometric description of an analytic tool introduced by Wilfrid Kendall. Our construction uses a tractrix flow which can be defined as a gradient flow for a family of functions of certain type. In an appendix we prove a general existence result for gradient flows of time-dependent locally Lipschitz semiconcave functions, which is of independent interest., Comment: 11 pages, 2 figures. arXiv admin note: text overlap with arXiv:1903.08539

In the Stiefel manifold X n , k , we replace Frankel linear height function by a quadratic one. We prove this is still a Morse–Bott function, whose structure of critical levels presents a dichotomy according to the sign of n − 2 k . The critical submanifolds are no longer Grassmannians but total spaces of fibrations of basis a product of two Grassmannians. We explicitly integrate the gradient flow.

The harmonic map energy of a map from a closed, constant-curvature surface to a closed target manifold can be seen as a functional on the space of maps and domain metrics. We consider the gradient flow for this energy. In the absence of singularities, previous theory established that the flow converges to a branched minimal immersion, but only at a sequence of times converging to infinity, and only after pulling back by a sequence of diffeomorphisms. In this paper, we investigate whether it is necessary to pull back by these diffeomorphisms, and whether the convergence is uniform as t → ∞ {t\to\infty} .

Assignment flows denote a class of dynamical models for contextual data labelling (classification) on graphs. We derive a novel parametrisation of assignment flows that reveals how the underlying information geometry induces two processes for assignment regularisation and for gradually enforcing unambiguous decisions, respectively, that seamlessly interact when solving for the flow. Our result enables to characterise the dominant part of the assignment flow as a Riemannian gradient flow with respect to the underlying information geometry. We consider a continuous-domain formulation of the corresponding potential and develop a novel algorithm in terms of solving a sequence of linear elliptic partial differential equations (PDEs) subject to a simple convex constraint. Our result provides a basis for addressing learning problems by controlling such PDEs in future work.

We consider the one-constant Landau-de Gennes model for nematic liquid crystals. The order parameter is a traceless tensor field $$\mathbf{Q}$$ , which is constrained to be uniaxial: $$\mathbf{Q}= s (\mathbf{n}\otimes \mathbf{n}- d^{-1}\mathbf{I})$$ where $$\mathbf{n}$$ is a director field, $$s\in \mathbb {R}$$ is the degree of orientation, and $$d\ge 2$$ is the dimension. Building on similarities with the one-constant Ericksen energy, we propose a structure-preserving finite element method for the computation of equilibrium configurations. We prove stability and consistency of the method without regularization, and $$\Gamma $$ -convergence of the discrete energies towards the continuous one as the mesh size goes to zero. We design an alternating direction gradient flow algorithm for the solution of the discrete problems, and we show that such a scheme decreases the energy monotonically. Finally, we illustrate the method’s capabilities by presenting some numerical simulations in two and three dimensions including non-orientable line fields.

While deep learning is successful in a number of applications, it is not yet well understood theoretically. A theoretical characterization of deep learning should answer questions about their approximation power, the dynamics of optimization, and good out-of-sample performance, despite overparameterization and the absence of explicit regularization. We review our recent results toward this goal. In approximation theory both shallow and deep networks are known to approximate any continuous functions at an exponential cost. However, we proved that for certain types of compositional functions, deep networks of the convolutional type (even without weight sharing) can avoid the curse of dimensionality. In characterizing minimization of the empirical exponential loss we consider the gradient flow of the weight directions rather than the weights themselves, since the relevant function underlying classification corresponds to normalized networks. The dynamics of normalized weights turn out to be equivalent to those of the constrained problem of minimizing the loss subject to a unit norm constraint. In particular, the dynamics of typical gradient descent have the same critical points as the constrained problem. Thus there is implicit regularization in training deep networks under exponential-type loss functions during gradient flow. As a consequence, the critical points correspond to minimum norm infima of the loss. This result is especially relevant because it has been recently shown that, for overparameterized models, selection of a minimum norm solution optimizes cross-validation leave-one-out stability and thereby the expected error. Thus our results imply that gradient descent in deep networks minimize the expected error.

Flow along isobaths of a sloping lower boundary generates an across-isobath Ekman transport in the bottom boundary layer. When this Ekman transport is down the slope it causes convective mixing—much like a downfront wind in the surface boundary layer—destroying stratification and potential vorticity. In this manuscript we show how this can lead to the development of a forced centrifugal or symmetric instability regime, where the potential vorticity flux generated by friction along the boundary is balanced by submesoscale instabilities that return the boundary layer potential vorticity to zero. This balance provides a strong constraint on the boundary layer evolution, which we use to develop a theory that explains the evolution of the boundary layer thickness, the rate at which the instabilities extract energy from the geostrophic flow field, and the magnitude and vertical structure of the dissipation. Finally, we show using theory and a high-resolution numerical model how the presence of centrifugal or symmetric instabilities alters the time-dependent Ekman adjustment of the boundary layer, delaying Ekman buoyancy arrest and enhancing the total energy removed from the balanced flow field. Submesoscale instabilities of the bottom boundary layer may therefore play an important, largely overlooked, role in the energetics of flow over topography in the ocean.

We study the gradient flow of the Mobius energy introduced by O’Hara (Topology 30:2 (1991), 241–247). We will show a fundamental 𝜀-regularity result that allows us to bound the infinity norm of all derivatives for some time if the energy is small on a certain scale. This result enables us to characterize the formation of a singularity in terms of concentrations of energy and allows us to construct a blow-up profile at a possible singularity. This solves one of the open problems listed by Zheng-Xu He (

Inspired by recent work on minimizers and gradient flows of constrained interaction energies, we prove that these energies arise as the slow diffusion limit of well-known aggregation-diffusion energies. We show that minimizers of aggregation-diffusion energies converge to a minimizer of the constrained interaction energy and gradient flows converge to a gradient flow. Our results apply to a range of interaction potentials, including singular attractive and repulsive-attractive power-law potentials. In the process of obtaining the slow diffusion limit, we also extend the well-posedness theory for aggregation-diffusion equations and Wasserstein gradient flows to admit a wide range of nonconvex interaction potentials. We conclude by applying our results to develop a numerical method for constrained interaction energies, which we use to investigate open questions on set valued minimizers.

We study a Lagrangian numerical scheme for solving a nonlinear drift diffusion equations of the form \begin{document}$ \partial_t u = \partial_x(u \cdot ({\sf c}^*)^\prime[\partial_x \mathit{h}^\prime(u)+ \mathit{v}^\prime]) $\end{document} , like Fokker-Plank and \begin{document}$ q $\end{document} -Laplace equations, on an interval. This scheme will consist of a spatio-temporal discretization founded on the formulation of the equation in terms of inverse distribution functions. It is based on the gradient flow structure of the equation with respect to optimal transport distances for a family of costs that are in some sense \begin{document}$ p $\end{document} -Wasserstein like. Additionally we will show that, under a regularity assumption on the initial data, this also includes a family of discontinuous, flux-limiting cost inducing equations like Rosenau's relativistic heat equation. We show that this discretization inherits various properties from the continuous flow, like entropy monotonicity, mass preservation, a minimum/maximum principle and flux-limitation in the case of the corresponding cost. Convergence in the limit of vanishing mesh size will be proven as the main result. Finally we will present numerical experiments including a numerical convergence analysis.

We provide an estimation of the dissipation of the Wasserstein 2 distance between the law of some interacting \begin{document}$ N $\end{document} -particle system, and the \begin{document}$ N $\end{document} times tensorized product of the solution to the corresponding limit nonlinear conservation law. It then enables to recover classical propagation of chaos results [ 20 ] in the case of Lipschitz coefficients, uniform in time propagation of chaos in [ 17 ] in the case of strictly convex coefficients. And some recent results [ 7 ] as the case of particle in a double well potential.

We consider a second order gradient flow of the p-elastic energy for a planar theta-network of three curves with fixed lengths. We construct a weak solution of the flow by means of an implicit variational scheme. We show long-time existence of the evolution and convergence to a critical point of the energy., 27 pages

Path planning is an important and classical problem in theoretical research and practical applications. In the complex and hierarchical space scenarios, path planning faces more difficulties and challenges due to the structural particularity. Considering the directivity of paths in hierarchical spaces, it is more important to guarantee the fluency and efficiency of the paths in hierarchical spaces. In this paper, we introduce a path network planning problem from multi-source to multi-destination in hierarchical spaces, namely Balanced-Flow Path Network Planning (BF-PNP) problem, and prove its NP-completeness. To balance the flow rate among the layers in the space, we propose a batch scheduling algorithm with the objective of minimizing the scheduling time consumption. To evaluate the performance on efficiency and time complexity, we perform a series of simulations and the results indicate the advantages of the proposed algorithm.

Preactivation ResNets consistently outperforms the original postactivation ResNets on the CIFAR10/100 classification benchmark. However, these results surprisingly do not carry over to the standard ImageNet benchmark. First, we theoretically analyze this incongruity in terms of how the two variants differ in handling the propagation of gradients. Although identity shortcuts are critical in both variants for improving optimization and performance, we show that postactivation variants enable early layers to receive a diverse dynamic composition of gradients from effectively deeper paths in comparison to preactivation variants, enabling the network to make maximal use of its representational capacity. Second, we show that downsampling projections (while only a few in number) have a significantly detrimental effect on performance. We show that by simply replacing downsampling projections with identitylike dense-reshape shortcuts, the classification results of standard residual architectures such as ResNets, ResNeXts, and SE-Nets improve by up to 1.2% on ImageNet, without any increase in computational complexity (FLOPs).

In this work, we investigate a variational formulation for a time-fractional Fokker-Planck equation which arises in the study of complex physical systems involving anomalously slow diffusion. The model involves a fractional-order Caputo derivative in time, and thus inherently nonlocal. The study follows the Wasserstein gradient flow approach pioneered by [26]. We propose a JKO type scheme for discretizing the model, using the L1 scheme for the Caputo fractional derivative in time, and establish the convergence of the scheme as the time step size tends to zero. Illustrative numerical results in one- and two-dimensional problems are also presented to show the approach.

In this paper, we develop a new square phase-field crystal model using the L 2 -gradient flow approach, where the total mass of atoms is conserved through a nonlocal Lagrange multiplier. We construct a fast, provably unconditionally energy stable, second-order scheme by using the recently developed SAV approach with the stabilization technique, where an extra stabilization term is added to enhance the stability and keep the required accuracy while using large time steps. Through the comparisons with the classical Cahn–Hilliard type square phase-field crystal model and the non-stabilized SAV scheme for simulating some benchmark numerical examples, we demonstrate the robustness of the new model, as well as the stability and the accuracy of the developed scheme, numerically.

We study the steady states and dynamics of a thin-film-type equation with non-conserved mass in one dimension. The evolution equation is a non-linear fourth-order degenerate parabolic partial differential equation (PDE) motivated by a model of volatile viscous fluid films allowing for condensation or evaporation. We show that by changing the sign of the non-conserved flux and breaking from a gradient flow structure, the problem can exhibit novel behaviours including having two distinct classes of co-existing steady-state solutions. Detailed analysis of the bifurcation structure for these steady states and their stability reveals several possibilities for the dynamics. For some parameter regimes, solutions can lead to finite-time rupture singularities. Interestingly, we also show that a finite-amplitude limit cycle can occur as a singular perturbation in the nearly conserved limit.

We present a framework for Nesterov's accelerated gradient flows in probability space to design efficient mean-field Markov chain Monte Carlo (MCMC) algorithms for Bayesian inverse problems. Here four examples of information metrics are considered, including Fisher-Rao metric, Wasserstein-2 metric, Kalman-Wasserstein metric and Stein metric. For both Fisher-Rao and Wasserstein-2 metrics, we prove convergence properties of accelerated gradient flows. In implementations, we propose a sampling-efficient discrete-time algorithm for Wasserstein-2, Kalman-Wasserstein and Stein accelerated gradient flows with a restart technique. We also formulate a kernel bandwidth selection method, which learns the gradient of logarithm of density from Brownian-motion samples. Numerical experiments, including Bayesian logistic regression and Bayesian neural network, show the strength of the proposed methods compared with state-of-the-art algorithms.

The Functionalized Cahn–Hilliard equation has been proposed as a model for the interfacial energy of phase-separated mixtures of amphiphilic molecules. We study the existence of a nonnegative weak solutions of a gradient flow of the Functionalized Cahn–Hilliard equation subject to a degenerate mobility M ( u ) that is zero for u ≤ 0 . Assuming the initial data u 0 ( x ) is positive, we construct a weak solution as the limit of solutions corresponding to non-degenerate mobilities and verify that it satisfies an energy dissipation inequality. Our approach is a combination of Galerkin approximation, energy estimates, and weak convergence methods.

Identifying internal waves in complex flow fields is a long‐standing problem in fluid dynamics, oceanography and atmospheric science, owing to the overlap of internal waves temporal and spatial scales with other flow regimes. Lagrangian filtering—that is, temporal filtering in a frame of reference moving with the flow—is one proposed methodology for performing this separation. Here we (a) describe an improved implementation of the Lagrangian filtering methodology and (b) introduce a new freely available, parallelized Python package that applies the method. We show that the package can be used to directly filter output from a variety of common ocean models including MITgcm, Regional Ocean Modeling System and MOM5 for both regional and global domains at high resolution. The Lagrangian filtering is shown to be a useful tool to both identify (and thereby quantify) internal waves, and to remove internal waves to isolate the non‐wave flow field.

There are currently two singularity-free universal expressions for the topological susceptibility in QCD, one based on the Yang-Mills gradient flow and the other on density-chain correlation functions. While the latter link the susceptibility to the anomalous chiral Ward identities, the gradient flow permits the emergence of the topological sectors in lattice QCD to be understood. Here the two expressions are shown to coincide in the continuum theory, for any number of quark flavours in the range where the theory is asymptotically free., Plain TeX source, 9 pages, no figures

We consider the gradient flow of a quadratic non-autonomous energy under monotonicity constraints. First, we provide a notion of weak solution, inspired by the theory of curves of maximal slope, and then we prove existence (employing time-discrete schemes with different implementations of the constraint), uniqueness, power and energy identity, comparison principle and continuous dependence. As a by-product, we show that the energy identity gives a selection criterion for the (non-unique) evolutions obtained by other notions of solutions. Finally, we show that for autonomous energies the evolution obtained with the monotonicity constraint actually coincides with the evolution obtained by replacing the constraint with a fixed obstacle, given by the initial datum.

Convolutional neural networks (CNNs) are widely used among the various deep learning techniques available because of their superior performance in the fields of computer vision and natural language processing. CNNs can effectively extract the locality and correlation of input data using structures in which convolutional layers are successively applied to the input data. In general, the performance of neural networks has improved as the depth of CNNs has increased. However, an increase in the depth of a CNN is not always accompanied by an increase in the accuracy of the neural network. This is because the gradient vanishing problem may arise, causing the weights of the weighted layers to fail to converge. Accordingly, the gradient flows of the VGGNet, ResNet, SENet, and DenseNet models were analyzed and compared in this study, and the reasons for the differences in the error rate performances of the models were derived.

We present a first-order aggregation model for a homogeneous Lohe matrix ensemble with higher order couplings via a gradient flow approach. For homogeneous free flow with the same Hamiltonian, it is well known that the Lohe matrix model with cubic couplings can be recast as a gradient system with a potential which is a squared Frobenius norm of of averaged state. In this paper, we further derive a generalized Lohe matrix model with higher-order couplings via gradient flow approach for a polynomial potential. For the proposed model, we also provide a sufficient framework in terms of coupling strengths and initial data leading to the emergent dynamics of a homogeneous ensemble.

The gradient flow exact renormalization group (GFERG) is an exact renormalization group motivated by the Yang--Mills gradient flow and its salient feature is a manifest gauge invariance. We generalize this GFERG, originally formulated for the pure Yang--Mills theory, to vector-like gauge theories containing fermion fields, keeping the manifest gauge invariance. For the chiral symmetry we have two options: one possible formulation preserves the conventional form of the chiral symmetry and the other simpler formulation realizes the chiral symmetry in a modified form \`a la Ginsparg--Wilson. We work out a gauge-invariant local Wilson action in quantum electrodynamics to the lowest nontrivial order of perturbation theory. This Wilson action reproduces the correct axial anomaly in~$D=2$., Comment: 23 pages, the final version to appear in PTEP

Journal of computational physics 436, 110165 (2021). doi:10.1016/j.jcp.2021.110165, Published by Elsevier, Amsterdam

We show how the sphaleron rate (the Minkowski rate for topological charge diffusion) can be determined by analytical continuation of the Euclidean topological-charge-density two-point function, which we investigate on the lattice, using gradient flow to reduce noise and provide improved operators which more accurately measure topology. We measure the correlators on large, fine lattices in the quenched approximation at 1.5Tc with high precision. Based on these data we first perform a continuum extrapolation at fixed physical flow time and then extrapolate the continuum estimates to zero flow time. The extrapolated correlators are then used to study the sphaleron rate by spectral reconstruction based on perturbatively motivated models.