In this paper, we consider the density estimation problem associated with the stationary measure of ergodic It^o diffusions from a discrete-time series that approximate the solutions of the stochastic differential equations. To take an advantage of t
he characterization of density function through the stationary solution of a parabolic-type Fokker-Planck PDE, we proceed as follows. First, we employ deep neural networks to approximate the drift and diffusion terms of the SDE by solving appropriate supervised learning tasks. Subsequently, we solve a steady-state Fokker-Plank equation associated with the estimated drift and diffusion coefficients with a neural-network-based least-squares method. We establish the convergence of the proposed scheme under appropriate mathematical assumptions, accounting for the generalization errors induced by regressing the drift and diffusion coefficients, and the PDE solvers. This theoretical study relies on a recent perturbation theory of Markov chain result that shows a linear dependence of the density estimation to the error in estimating the drift term, and generalization error results of nonparametric regression and of PDE regression solution obtained with neural-network models. The effectiveness of this method is reflected by numerical simulations of a two-dimensional Students t distribution and a 20-dimensional Langevin dynamics.
In this note the weak type estimates for fractional integrals are studied. More precisely, we adapt the arguments of Domingo-Salazar, Lacey, and Rey to obtain improvements for the endpoint weak type estimates for regular fractional sparse operators.
It is shown analytically that the energy-conserving implicit nonsymplectic scheme of Bacchini, Ripperda, Chen and Sironi provides a first-order accuracy to numerical solutions of a six-dimensional conservative Hamiltonian system. Because of this, a n
ew second-order energy-conserving implicit scheme is proposed. Numerical simulations of Galactic model hosting a BL Lacertae object and magnetized rotating black hole background support these analytical results. The new method with appropriate time steps is used to explore the effects of varying the parameters on the presence of chaos in the two physical models. Chaos easily occurs in the Galactic model as the mass of the nucleus, the internal perturbation parameter, and the anisotropy of the potential of the elliptical galaxy increase. The dynamics of charged particles around the magnetized Kerr spacetime is easily chaotic for larger energies of the particles, smaller initial angular momenta of the particles, and stronger magnetic fields. The chaotic properties are not necessarily weakened when the black hole spin increases. The new method can be used for any six-dimensional Hamiltonian problems, including globally hyperbolic spacetimes with readily available (3+1) split coordinates.
Deep neural networks suffer from catastrophic forgetting when learning multiple knowledge sequentially, and a growing number of approaches have been proposed to mitigate this problem. Some of these methods achieved considerable performance by associa
ting the flat local minima with forgetting mitigation in continual learning. However, they inevitably need (1) tedious hyperparameters tuning, and (2) additional computational cost. To alleviate these problems, in this paper, we propose a simple yet effective optimization method, called AlterSGD, to search for a flat minima in the loss landscape. In AlterSGD, we conduct gradient descent and ascent alternatively when the network tends to converge at each session of learning new knowledge. Moreover, we theoretically prove that such a strategy can encourage the optimization to converge to a flat minima. We verify AlterSGD on continual learning benchmark for semantic segmentation and the empirical results show that we can significantly mitigate the forgetting and outperform the state-of-the-art methods with a large margin under challenging continual learning protocols.
The study of superconductivity in compressed hydrides is of great interest due to measurements of high critical temperatures (Tc) in the vicinity of room temperature, beginning with the observations of LaH10 at 170-190 GPa. However, the pressures req
uired for synthesis of these high Tc superconducting hydrides currently remain extremely high. Here we show the investigation of crystal structures and superconductivity in the La-B-H system under pressure with particle-swarm intelligence structure searches methods in combination with first-principles calculations. Structures with six stoichiometries, LaBH, LaBH3, LaBH4, LaBH6, LaBH7 and LaBH8, were predicted to become stable under pressure. Remarkably, the hydrogen atoms in LaBH8 were found to bond with B atoms in a manner that is similar to that in H3S. Lattice dynamics calculations indicate that LaBH7 and LaBH8 become dynamically stable at pressures as low as 109.2 and 48.3 GPa, respectively. Moreover, the two phases were predicted to be superconducting with a critical temperature (Tc) of 93 K and 156 K at 110 GPa and 55 GPa, respectively. Our results provide guidance for future experiments targeting new hydride superconductors with both low synthesis pressures and high Tc.
This paper proposes a mesh-free computational framework and machine learning theory for solving elliptic PDEs on unknown manifolds, identified with point clouds, based on diffusion maps (DM) and deep learning. The PDE solver is formulated as a superv
ised learning task to solve a least-squares regression problem that imposes an algebraic equation approximating a PDE (and boundary conditions if applicable). This algebraic equation involves a graph-Laplacian type matrix obtained via DM asymptotic expansion, which is a consistent estimator of second-order elliptic differential operators. The resulting numerical method is to solve a highly non-convex empirical risk minimization problem subjected to a solution from a hypothesis space of neural-network type functions. In a well-posed elliptic PDE setting, when the hypothesis space consists of feedforward neural networks with either infinite width or depth, we show that the global minimizer of the empirical loss function is a consistent solution in the limit of large training data. When the hypothesis space is a two-layer neural network, we show that for a sufficiently large width, the gradient descent method can identify a global minimizer of the empirical loss function. Supporting numerical examples demonstrate the convergence of the solutions and the effectiveness of the proposed solver in avoiding numerical issues that hampers the traditional approach when a large data set becomes available, e.g., large matrix inversion.
The defocus deblurring raised from the finite aperture size and exposure time is an essential problem in the computational photography. It is very challenging because the blur kernel is spatially varying and difficult to estimate by traditional metho
ds. Due to its great breakthrough in low-level tasks, convolutional neural networks (CNNs) have been introduced to the defocus deblurring problem and achieved significant progress. However, they apply the same kernel for different regions of the defocus blurred images, thus it is difficult to handle these nonuniform blurred images. To this end, this study designs a novel blur-aware multi-branch network (BaMBNet), in which different regions (with different blur amounts) should be treated differentially. In particular, we estimate the blur amounts of different regions by the internal geometric constraint of the DP data, which measures the defocus disparity between the left and right views. Based on the assumption that different image regions with different blur amounts have different deblurring difficulties, we leverage different networks with different capacities (emph{i.e.} parameters) to process different image regions. Moreover, we introduce a meta-learning defocus mask generation algorithm to assign each pixel to a proper branch. In this way, we can expect to well maintain the information of the clear regions while recovering the missing details of the blurred regions. Both quantitative and qualitative experiments demonstrate that our BaMBNet outperforms the state-of-the-art methods. Source code will be available at https://github.com/junjun-jiang/BaMBNet.
The transition-metal-based kagome metals provide a versatile platform for correlated topological phases hosting various electronic instabilities. While superconductivity is rare in layered kagome compounds, its interplay with nontrivial topology coul
d offer an engaging space to realize exotic excitations of quasiparticles. Here, we use scanning tunneling microscopy (STM) to study a newly discovered Z$_2$ topological kagome metal CsV$_3$Sb$_5$ with a superconducting ground state. We observe charge modulation associated with the opening of an energy gap near the Fermi level. When across single-unit-cell surface step edges, the intensity of this charge modulation exhibits a {pi}-phase shift, suggesting a three-dimensional 2$times$2$times$2 charge density wave ordering. Interestingly, a robust zero-bias conductance peak is observed inside the superconducting vortex core on the Cs 2$times$2 surfaces that does not split in a large distance when moving away from the vortex center, resembling the Majorana bound states arising from the superconducting Dirac surface states in Bi$_2$Te$_3$/NbSe$_2$ heterostructures. Our findings establish CsV$_3$Sb$_5$ as a promising candidate for realizing exotic excitations at the confluence of nontrivial lattice geometry, topology and multiple electronic orders.
Recently, numerous algorithms have been developed to tackle the problem of vision-language navigation (VLN), i.e., entailing an agent to navigate 3D environments through following linguistic instructions. However, current VLN agents simply store thei
r past experiences/observations as latent states in recurrent networks, failing to capture environment layouts and make long-term planning. To address these limitations, we propose a crucial architecture, called Structured Scene Memory (SSM). It is compartmentalized enough to accurately memorize the percepts during navigation. It also serves as a structured scene representation, which captures and disentangles visual and geometric cues in the environment. SSM has a collect-read controller that adaptively collects information for supporting current decision making and mimics iterative algorithms for long-range reasoning. As SSM provides a complete action space, i.e., all the navigable places on the map, a frontier-exploration based navigation decision making strategy is introduced to enable efficient and global planning. Experiment results on two VLN datasets (i.e., R2R and R4R) show that our method achieves state-of-the-art performance on several metrics.
In this paper, an implicit nonsymplectic exact energy-preserving integrator is specifically designed for a ten-dimensional phase-space conservative Hamiltonian system with five degrees of freedom. It is based on a suitable discretization-averaging of
the Hamiltonian gradient, with a second-order accuracy to numerical solutions. A one-dimensional disordered discrete nonlinear Schr{o}dinger equation and a post-Newtonian Hamiltonian system of spinning compact binaries are taken as our two examples. We demonstrate numerically that the proposed algorithm exhibits good long-term performance in the preservation of energy, if roundoff errors are neglected. This result is independent of time steps, initial orbital eccentricities, and regular and chaotic orbital dynamical behavior. In particular, the application of appropriately large time steps to the new algorithm is helpful in reducing time-consuming and roundoff errors. This new method, combined with fast Lyapunov indicators, is well suited related to chaos in the two example problems. It is found that chaos in the former system is mainly responsible for one of the parameters. In the latter problem, a combination of small initial separations and high initial eccentricities can easily induce chaos.
