亚洲男人的天堂2018av,欧美草比,久久久久久免费视频精选,国色天香在线看免费,久久久久亚洲av成人片仓井空

We develop an \textit{a posteriori} error analysis for a numerical estimate of the time at which a functional of the solution to a partial differential equation (PDE) first achieves a threshold value on a given time interval. This quantity of interest (QoI) differs from classical QoIs which are modeled as bounded linear (or nonlinear) functionals {of the solution}. Taylor's theorem and an adjoint-based \textit{a posteriori} analysis is used to derive computable and accurate error estimates in the case of semi-linear parabolic and hyperbolic PDEs. The accuracy of the error estimates is demonstrated through numerical solutions of the one-dimensional heat equation and linearized shallow water equations (SWE), representing parabolic and hyperbolic cases, respectively.

Many future technologies rely on neural networks, but verifying the correctness of their behavior remains a major challenge. It is known that neural networks can be fragile in the presence of even small input perturbations, yielding unpredictable outputs. The verification of neural networks is therefore vital to their adoption, and a number of approaches have been proposed in recent years. In this paper we focus on semidefinite programming (SDP) based techniques for neural network verification, which are particularly attractive because they can encode expressive behaviors while ensuring a polynomial time decision. Our starting point is the DeepSDP framework proposed by Fazlyab et al, which uses quadratic constraints to abstract the verification problem into a large-scale SDP. When the size of the neural network grows, however, solving this SDP quickly becomes intractable. Our key observation is that by leveraging chordal sparsity and specific parametrizations of DeepSDP, we can decompose the primary computational bottleneck of DeepSDP -- a large linear matrix inequality (LMI) -- into an equivalent collection of smaller LMIs. Our parametrization admits a tunable parameter, allowing us to trade-off efficiency and accuracy in the verification procedure. We call our formulation Chordal-DeepSDP, and provide experimental evaluation to show that it can: (1) effectively increase accuracy with the tunable parameter and (2) outperform DeepSDP on deeper networks.

Most machine learning methods are used as a black box for modelling. We may try to extract some knowledge from physics-based training methods, such as neural ODE (ordinary differential equation). Neural ODE has advantages like a possibly higher class of represented functions, the extended interpretability compared to black-box machine learning models, ability to describe both trend and local behaviour. Such advantages are especially critical for time series with complicated trends. However, the known drawback is the high training time compared to the autoregressive models and long-short term memory (LSTM) networks widely used for data-driven time series modelling. Therefore, we should be able to balance interpretability and training time to apply neural ODE in practice. The paper shows that modern neural ODE cannot be reduced to simpler models for time-series modelling applications. The complexity of neural ODE is compared to or exceeds the conventional time-series modelling tools. The only interpretation that could be extracted is the eigenspace of the operator, which is an ill-posed problem for a large system. Spectra could be extracted using different classical analysis methods that do not have the drawback of extended time. Consequently, we reduce the neural ODE to a simpler linear form and propose a new view on time-series modelling using combined neural networks and an ODE system approach.

In a Federated Learning (FL) setup, a number of devices contribute to the training of a common model. We present a method for selecting the devices that provide updates in order to achieve improved generalization, fast convergence, and better device-level performance. We formulate a min-max optimization problem and decompose it into a primal-dual setup, where the duality gap is used to quantify the device-level performance. Our strategy combines \emph{exploration} of data freshness through a random device selection with \emph{exploitation} through simplified estimates of device contributions. This improves the performance of the trained model both in terms of generalization and personalization. A modified Truncated Monte-Carlo (TMC) method is applied during the exploitation phase to estimate the device's contribution and lower the communication overhead. The experimental results show that the proposed approach has a competitive performance, with lower communication overhead and competitive personalization performance against the baseline schemes.

We develop in this paper two classes of length preserving schemes for the Landau-Lifshitz equation based on two different Lagrange multiplier approaches. In the first approach, the Lagrange multiplier $\lambda(\bx,t)$ equals to $|\nabla m(\bx,t)|^2$ at the continuous level, while in the second approach, the Lagrange multiplier $\lambda(\bx,t)$ is introduced to enforce the length constraint at the discrete level and is identically zero at the continuous level. By using a predictor-corrector approach, we construct efficient and robust length preserving higher-order schemes for the Landau-Lifshitz equation, with the computational cost dominated by the predictor step which is simply a semi-implicit scheme. Furthermore, by introducing another space-independent Lagrange multiplier, we construct energy dissipative, in addition to length preserving, schemes for the Landau-Lifshitz equation, at the expense of solving one nonlinear algebraic equation. We present ample numerical experiments to validate the stability and accuracy for the proposed schemes, and also provide a performance comparison with some existing schemes.

A key concern in integrating machine learning models in medicine is the ability to interpret their reasoning. Popular explainability methods have demonstrated satisfactory results in natural image recognition, yet in medical image analysis, many of these approaches provide partial and noisy explanations. Recently, attention mechanisms have shown compelling results both in their predictive performance and in their interpretable qualities. A fundamental trait of attention is that it leverages salient parts of the input which contribute to the model's prediction. To this end, our work focuses on the explanatory value of attention weight distributions. We propose a multi-layer attention mechanism that enforces consistent interpretations between attended convolutional layers using convex optimization. We apply duality to decompose the consistency constraints between the layers by reparameterizing their attention probability distributions. We further suggest learning the dual witness by optimizing with respect to our objective; thus, our implementation uses standard back-propagation, hence it is highly efficient. While preserving predictive performance, our proposed method leverages weakly annotated medical imaging data and provides complete and faithful explanations to the model's prediction.

Solving the time-dependent Schr\"odinger equation is an important application area for quantum algorithms. We consider Schr\"odinger's equation in the semi-classical regime. Here the solutions exhibit strong multiple-scale behavior due to a small parameter $\hbar$, in the sense that the dynamics of the quantum states and the induced observables can occur on different spatial and temporal scales. Such a Schr\"odinger equation finds many applications, including in Born-Oppenheimer molecular dynamics and Ehrenfest dynamics. This paper considers quantum analogues of pseudo-spectral (PS) methods on classical computers. Estimates on the gate counts in terms of $\hbar$ and the precision $\varepsilon$ are obtained. It is found that the number of required qubits, $m$, scales only logarithmically with respect to $\hbar$. When the solution has bounded derivatives up to order $\ell$, the symmetric Trotting method has gate complexity $\mathcal{O}\Big({ (\varepsilon \hbar)^{-\frac12} \mathrm{polylog}(\varepsilon^{-\frac{3}{2\ell}} \hbar^{-1-\frac{1}{2\ell}})}\Big),$ provided that the diagonal unitary operators in the pseudo-spectral methods can be implemented with $\mathrm{poly}(m)$ operations. When physical observables are the desired outcomes, however, the step size in the time integration can be chosen independently of $\hbar$. The gate complexity in this case is reduced to $\mathcal{O}\Big({\varepsilon^{-\frac12} \mathrm{polylog}( \varepsilon^{-\frac3{2\ell}} \hbar^{-1} )}\Big),$ with $\ell$ again indicating the smoothness of the solution.

Change-point analysis plays a significant role in various fields to reveal discrepancies in distribution in a sequence of observations. While a number of algorithms have been proposed for high-dimensional data, kernel-based methods have not been well explored due to difficulties in controlling false discoveries and mediocre performance. In this paper, we propose a new kernel-based framework that makes use of an important pattern of data in high dimensions to boost power. Analytic approximations to the significance of the new statistics are derived and fast tests based on the asymptotic results are proposed, offering easy off-the-shelf tools for large datasets. The new tests show superior performance for a wide range of alternatives when compared with other state-of-the-art methods. We illustrate these new approaches through an analysis of a phone-call network data.

An evolving surface finite element discretisation is analysed for the evolution of a closed two-dimensional surface governed by a system coupling a generalised forced mean curvature flow and a reaction--diffusion process on the surface, inspired by a gradient flow of a coupled energy. Two algorithms are proposed, both based on a system coupling the diffusion equation to evolution equations for geometric quantities in the velocity law for the surface. One of the numerical methods is proved to be convergent in the $H^1$ norm with optimal-order for finite elements of degree at least two. We present numerical experiments illustrating the convergence behaviour and demonstrating the qualitative properties of the flow: preservation of mean convexity, loss of convexity, weak maximum principles, and the occurrence of self-intersections.

A method for practical realization of the inverse scattering transform method for the Korteweg-de Vries equation is proposed. It is based on analytical representations for Jost solutions and for integral kernels of transformation operators obtained recently by the authors. The representations have the form of functional series in which the first coefficient plays a crucial role both in solving the direct scattering and the inverse scattering problems. The direct scattering problem reduces to computation of a number of the coefficients following a simple recurrent integration procedure with a posterior calculation of scattering data by well known formulas. The inverse scattering problem reduces to a system of linear algebraic equations from which the first component of the solution vector leads to the recovery of the potential. We prove the applicability of the finite section method to the system of linear algebraic equations and discuss numerical aspects of the proposed method. Numerical examples are given, which reveal the accuracy and speed of the method.