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We present a machine learning framework capable of consistently inferring mathematical expressions of the hyperelastic energy functionals for incompressible materials from sparse experimental data and physical laws. To achieve this goal, we propose a polyconvex neural additive model (PNAM) that enables us to express the hyperelasticity model in a learnable feature space while enforcing polyconvexity. An upshot of this feature space obtained via PNAM is that (1) it is spanned by a set univariate basis that can be re-parametrized with a more complex mathematical form, and (2) the resultant elasticity model is guaranteed to fulfill the polyconvexity, which ensures that the acoustic tensor remains elliptic for any deformation. To further improve the interpretability, we use genetic programming to convert each univariate basis into a compact mathematical expression. The resultant multi-variable mathematical models obtained from this proposed framework are not only more interpretable but are also proven to fulfill physical laws. By controlling the compactness of the learned symbolic form, the machine learning-generated mathematical model also requires fewer arithmetic operations than the deep neural network counterparts during deployment. This latter attribute is crucial for scaling large-scale simulations where the constitutive responses of every integration point must be updated within each incremental time step. We compare our proposed model discovery framework against other state-of-the-art alternatives to assess the robustness and efficiency of the training algorithms and examine the trade-off between interpretability, accuracy, and precision of the learned symbolic hyperelasticity models obtained from different approaches. Our numerical results suggest that our approach extrapolates well outside the training data regime due to the precise incorporation of physics-based knowledge.

Application of deep learning methods to physical simulations such as CFD (Computational Fluid Dynamics), have been so far of limited industrial relevance. This paper demonstrates the development and application of a deep learning framework for real-time predictions of the impact of tip clearance variations on the aerodynamic performance of multi-stage axial compressors in gas turbines. The proposed C(NN)FD architecture is proven to be scalable to industrial applications, and achieves in real-time accuracy comparable to the CFD benchmark. The deployed model, is readily integrated within the manufacturing and build process of gas turbines, thus providing the opportunity to analytically assess the impact on performance and potentially reduce requirements for expensive physical tests.

We propose a finite element discretization for the steady, generalized Navier-Stokes equations for fluids with shear-dependent viscosity, completed with inhomogeneous Dirichlet boundary conditions and an inhomogeneous divergence constraint. We establish (weak) convergence of discrete solutions as well as a priori error estimates for the velocity vector field and the scalar kinematic pressure. Numerical experiments complement the theoretical findings.

Markov chain Monte Carlo (MCMC) algorithms are based on the construction of a Markov Chain with transition probabilities $P_\mu(x,\cdot)$, where $\mu$ indicates an invariant distribution of interest. In this work, we look at these transition probabilities as functions of their invariant distributions, and we develop a notion of derivative in the invariant distribution of a MCMC kernel. We build around this concept a set of tools that we refer to as Markov Chain Monte Carlo Calculus. This allows us to compare Markov chains with different invariant distributions within a suitable class via what we refer to as mean value inequalities. We explain how MCMC Calculus provides a natural framework to study algorithms using an approximation of an invariant distribution, also illustrating how it suggests practical guidelines for MCMC algorithms efficiency. We conclude this work by showing how the tools developed can be applied to prove convergence of interacting and sequential MCMC algorithms, which arise in the context of particle filtering.

A conjecture attributed to Smith states that every pair of longest cycles in a $k$-connected graph intersect each other in at least $k$ vertices. In this paper, we show that every pair of longest cycles in a~$k$-connected graph on $n$ vertices intersect each other in at least~$\min\{n,8k-n-16\}$ vertices, which confirms Smith's conjecture when $k\geq (n+16)/7$. An analog conjecture for paths instead of cycles was stated by Hippchen. By a simple reduction, we relate both conjectures, showing that Hippchen's conjecture is valid when either $k \leq 6$ or $k \geq (n+9)/7$.

The greedy and nearest-neighbor TSP heuristics can both have $\log n$ approximation factors from optimal in worst case, even just for $n$ points in Euclidean space. In this note, we show that this approximation factor is only realized when the optimal tour is unusually short. In particular, for points from any fixed $d$-Ahlfor's regular metric space (which includes any $d$-manifold like the $d$-cube $[0,1]^d$ in the case $d$ is an integer but also fractals of dimension $d$ when $d$ is real-valued), our results imply that the greedy and nearest-neighbor heuristics have \emph{additive} errors from optimal on the order of the \emph{optimal} tour length through \emph{random} points in the same space, for $d>1$.

We extend classical work by Janusz Czelakowski on the closure properties of the class of matrix models of entailment relations - nowadays more commonly called multiple-conclusion logics - to the setting of non-deterministic matrices (Nmatrices), characterizing the Nmatrix models of an arbitrary logic through a generalization of the standard class operators to the non-deterministic setting. We highlight the main differences that appear in this more general setting, in particular: the possibility to obtain Nmatrix quotients using any compatible equivalence relation (not necessarily a congruence); the problem of determining when strict homomorphisms preserve the logic of a given Nmatrix; the fact that the operations of taking images and preimages cannot be swapped, which determines the exact sequence of operators that generates, from any complete semantics, the class of all Nmatrix models of a logic. Many results, on the other hand, generalize smoothly to the non-deterministic setting: we show for instance that a logic is finitely based if and only if both the class of its Nmatrix models and its complement are closed under ultraproducts. We conclude by mentioning possible developments in adapting the Abstract Algebraic Logic approach to logics induced by Nmatrices and the associated equational reasoning over non-deterministic algebras.

We present a priori error estimates for a multirate time-stepping scheme for coupled differential equations. The discretization is based on Galerkin methods in time using two different time meshes for two parts of the problem. We aim at surface coupled multiphysics problems like two-phase flows. Special focus is on the handling of the interface coupling to guarantee a coercive formulation as key to optimal order error estimates. In a sequence of increasing complexity, we begin with the coupling of two ordinary differential equations, coupled heat conduction equation, and finally a coupled Stokes problem. For this we show optimal multi-rate estimates in velocity and a suboptimal result in pressure. The a priori estimates prove that the multirate method decouples the two subproblems exactly. This is the basis for adaptive methods which can choose optimal lattices for the respective subproblems.

Positive spanning sets span a given vector space by nonnegative linear combinations of their elements. These have attracted significant attention in recent years, owing to their extensive use in derivative-free optimization. In this setting, the quality of a positive spanning set is assessed through its cosine measure, a geometric quantity that expresses how well such a set covers the space of interest. In this paper, we investigate the construction of positive $k$-spanning sets with geometrical guarantees.Our results build on recently identified positive spanning sets, called orthogonally structured positive bases. We first describe how to identify such sets and compute their cosine measures efficiently. We then focus our study on positive $k$-spanning sets, for which we provide a complete description, as well as a new notion of cosine measure that accounts for the resilient nature of such sets. By combining our results, we are able to use orthogonally structured positive bases to create positive $k$-spanning sets with guarantees on the value of their cosine measures.

Classical forgery attacks against Offset Two-round (OTR) structures require some harsh conditions, such as some plaintext and ciphertext pairs need to be known, and the success probability is not too high. To solve these problems, a quantum forgery attack on OTR structure using Simon's algorithm is proposed. The attacker intercept the ciphertext-tag pair $(C,T)$ between the sender and receiver, while Simon's algorithm is used to find the period of the tag generation function in OTR, then we can successfully forge new ciphertext $C'$ ($C'\ne C$) for intercepted tag $T$. For a variant of OTR structure (Pr{/o}st-OTR-Even-Mansour structure), a universal forgery attack, in which it is easy to generate the correct tag of any given message if the attacker is allowed to change a single block in it, is proposed. It first obtains the secret parameter L using Simon's algorithm, then the secret parameter L is used to find the keys $k_1$ and $k_2$, so that an attacker can forge the changed messages. It only needs several plaintext blocks to help obtain the keys to forge any messages. Performance analysis shows that the query complexity of our attack is $O(n)$, and its success probability is very close to 1.