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The heterogeneous micromechanical properties of biological tissues have profound implications across diverse medical and engineering domains. However, identifying the full-field heterogeneous elastic properties of soft materials using traditional computational and engineering approaches is fundamentally challenging due to difficulties in estimating local stress fields. Recently, there has been a growing interest in using data-driven models to learn full-field mechanical responses such as displacement and strain from experimental or synthetic data. However, research studies on inferring the full-field elastic properties of materials, a more challenging problem, are scarce, particularly for large deformation, hyperelastic materials. Here, we propose a novel approach to identify the elastic modulus distribution in nonlinear, large deformation hyperelastic materials utilizing physics-informed neural networks (PINNs). We evaluate the prediction accuracies and computational efficiency of PINNs, informed by mechanic features and principles, across three synthetic materials with structural complexity that closely resemble real tissue patterns, such as brain tissue and tricuspid valve tissue. Our improved PINN architecture accurately estimates the full-field elastic properties, with relative errors of less than 5% across all examples. This research has significant potential for advancing our understanding of micromechanical behaviors in biological materials, impacting future innovations in engineering and medicine.

Continual learning algorithms strive to acquire new knowledge while preserving prior information. Often, these algorithms emphasise stability and restrict network updates upon learning new tasks. In many cases, such restrictions come at a cost to the model's plasticity, i.e. the model's ability to adapt to the requirements of a new task. But is all change detrimental? Here, we approach this question by proposing that activation spaces in neural networks can be decomposed into two subspaces: a readout range in which change affects prior tasks and a null space in which change does not alter prior performance. Based on experiments with this novel technique, we show that, indeed, not all activation change is associated with forgetting. Instead, the only change in the subspace visible to the readout of a task can lead to decreased stability, while restricting change outside of this subspace is associated only with a loss of plasticity. Analysing various commonly used algorithms, we show that regularisation-based techniques do not fully disentangle the two spaces and, as a result, restrict plasticity more than need be. We expand our results by investigating a linear model in which we can manipulate learning in the two subspaces directly and thus causally link activation changes to stability and plasticity. For hierarchical, nonlinear cases, we present an approximation that enables us to estimate functionally relevant subspaces at every layer of a deep nonlinear network, corroborating our previous insights. Together, this work provides novel means to derive insights into the mechanisms behind stability and plasticity in continual learning and may serve as a diagnostic tool to guide developments of future continual learning algorithms that stabilise inference while allowing maximal space for learning.

We consider the numerical behavior of the fixed-stress splitting method for coupled poromechanics as undrained regimes are approached. We explain that pressure stability is related to the splitting error of the scheme, not the fact that the discrete saddle point matrix never appears in the fixed-stress approach. This observation reconciles previous results regarding the pressure stability of the splitting method. Using examples of compositional poromechanics with application to geological CO$_2$ sequestration, we see that solutions obtained using the fixed-stress scheme with a low order finite element-finite volume discretization which is not inherently inf-sup stable can exhibit the same pressure oscillations obtained with the corresponding fully implicit scheme. Moreover, pressure jump stabilization can effectively remove these spurious oscillations in the fixed-stress setting, while also improving the efficiency of the scheme in terms of the number of iterations required at every time step to reach convergence.

For a model convection-diffusion problem, we obtain new error estimates for a general upwinding finite element discretization based on bubble modification of the test space. The key analysis tool is based on finding representations of the optimal norms on the trial spaces at the continuous and discrete levels. We analyze and compare the standard linear discretization, the saddle point least square and upwinding Petrov-Galerkin methods. We conclude that the bubble upwinding Petrov-Galerkin method is the most performant discretization for the one dimensional model. Our results for the model convection-diffusion problem can be extended for creating new and efficient discretizations for the multidimensional cases.

Random features models play a distinguished role in the theory of deep learning, describing the behavior of neural networks close to their infinite-width limit. In this work, we present a thorough analysis of the generalization performance of random features models for generic supervised learning problems with Gaussian data. Our approach, built with tools from the statistical mechanics of disordered systems, maps the random features model to an equivalent polynomial model, and allows us to plot average generalization curves as functions of the two main control parameters of the problem: the number of random features $N$ and the size $P$ of the training set, both assumed to scale as powers in the input dimension $D$. Our results extend the case of proportional scaling between $N$, $P$ and $D$. They are in accordance with rigorous bounds known for certain particular learning tasks and are in quantitative agreement with numerical experiments performed over many order of magnitudes of $N$ and $P$. We find good agreement also far from the asymptotic limits where $D\to \infty$ and at least one between $P/D^K$, $N/D^L$ remains finite.

In this paper, we design a new kind of high order inverse Lax-Wendroff (ILW) boundary treatment for solving hyperbolic conservation laws with finite difference method on a Cartesian mesh. This new ILW method decomposes the construction of ghost point values near inflow boundary into two steps: interpolation and extrapolation. At first, we impose values of some artificial auxiliary points through a polynomial interpolating the interior points near the boundary. Then, we will construct a Hermite extrapolation based on those auxiliary point values and the spatial derivatives at boundary obtained via the ILW procedure. This polynomial will give us the approximation to the ghost point value. By an appropriate selection of those artificial auxiliary points, high-order accuracy and stable results can be achieved. Moreover, theoretical analysis indicates that comparing with the original ILW method, especially for higher order accuracy, the new proposed one would require fewer terms using the relatively complicated ILW procedure and thus improve computational efficiency on the premise of maintaining accuracy and stability. We perform numerical experiments on several benchmarks, including one- and two-dimensional scalar equations and systems. The robustness and efficiency of the proposed scheme is numerically verified.

We consider the classical problems of interpolating a polynomial given a black box for evaluation, and of multiplying two polynomials, in the setting where the bit-lengths of the coefficients may vary widely, so-called unbalanced polynomials. Writing s for the total bit-length and D for the degree, our new algorithms have expected running time $\tilde{O}(s \log D)$, whereas previous methods for (resp.) dense or sparse arithmetic have at least $\tilde{O}(sD)$ or $\tilde{O}(s^2)$ bit complexity.

In this paper, we introduce a new simple approach to developing and establishing the convergence of splitting methods for a large class of stochastic differential equations (SDEs), including additive, diagonal and scalar noise types. The central idea is to view the splitting method as a replacement of the driving signal of an SDE, namely Brownian motion and time, with a piecewise linear path that yields a sequence of ODEs $-$ which can be discretized to produce a numerical scheme. This new way of understanding splitting methods is inspired by, but does not use, rough path theory. We show that when the driving piecewise linear path matches certain iterated stochastic integrals of Brownian motion, then a high order splitting method can be obtained. We propose a general proof methodology for establishing the strong convergence of these approximations that is akin to the general framework of Milstein and Tretyakov. That is, once local error estimates are obtained for the splitting method, then a global rate of convergence follows. This approach can then be readily applied in future research on SDE splitting methods. By incorporating recently developed approximations for iterated integrals of Brownian motion into these piecewise linear paths, we propose several high order splitting methods for SDEs satisfying a certain commutativity condition. In our experiments, which include the Cox-Ingersoll-Ross model and additive noise SDEs (noisy anharmonic oscillator, stochastic FitzHugh-Nagumo model, underdamped Langevin dynamics), the new splitting methods exhibit convergence rates of $O(h^{3/2})$ and outperform schemes previously proposed in the literature.

In the realm of cost-sharing mechanisms, the vulnerability to Sybil strategies - also known as false-name strategies, where agents create fake identities to manipulate outcomes - has not yet been studied. In this paper, we delve into the details of different cost-sharing mechanisms proposed in the literature, highlighting their non-Sybil-resistant nature. Furthermore, we prove that a Sybil-proof cost-sharing mechanism for public excludable goods under mild conditions is at least $(n+1)/2-$approximate. This finding reveals an exponential increase in the worst-case social cost in environments where agents are restricted from using Sybil strategies. To circumvent these negative results, we introduce the concept of \textit{Sybil Welfare Invariant} mechanisms, where a mechanism does not decrease its welfare under Sybil-strategies when agents choose weak dominant strategies and have subjective prior beliefs over other players' actions. Finally, we prove that the Shapley value mechanism for symmetric and submodular cost functions holds this property, and so deduce that the worst-case social cost of this mechanism is the $n$th harmonic number $\mathcal H_n$ under equilibrium with Sybil strategies, matching the worst-case social cost bound for cost-sharing mechanisms. This finding suggests that any group of agents, each with private valuations, can fund public excludable goods both permissionless and anonymously, achieving efficiency comparable to that of permissioned and non-anonymous domains, even when the total number of participants is unknown.

Quantization for a Borel probability measure refers to the idea of estimating a given probability by a discrete probability with support containing a finite number of elements. In this paper, we have considered a Borel probability measure $P$ on $\mathbb R^2$, which has support a nonuniform stretched Sierpi\'{n}ski triangle generated by a set of three contractive similarity mappings on $\mathbb R^2$. For this probability measure, we investigate the optimal sets of $n$-means and the $n$th quantization errors for all positive integers $n$.