Search

This paper introduces a tensor neural network (TNN) to address nonparametric regression problems. Characterized by its distinct sub-network structure, the TNN effectively facilitates variable separation, thereby enhancing the approximation of complex, unknown functions. Our comparative analysis reveals that the TNN outperforms conventional Feed-Forward Networks (FFN) and Radial Basis Function Networks (RBN) in terms of both approximation accuracy and generalization potential, despite a similar scale of parameters. A key innovation of our approach is the integration of statistical regression and numerical integration within the TNN framework. This integration allows for the efficient computation of high-dimensional integrals associated with the regression function. The implications of this advancement extend to a broader range of applications, particularly in scenarios demanding precise high-dimensional data analysis and prediction.

Based on tensor neural network, we propose an interpolation method for high dimensional non-tensor-product-type functions. This interpolation scheme is designed by using the tensor neural network based machine learning method. This means that we use a tensor neural network to approximate high dimensional functions which has no tensor product structure. In some sense, the non-tenor-product-type high dimensional function is transformed to the tensor neural network which has tensor product structure. It is well known that the tensor product structure can bring the possibility to design highly accurate and efficient numerical methods for dealing with high dimensional functions. In this paper, we will concentrate on computing the high dimensional integrations and solving high dimensional partial differential equations. The corresponding numerical methods and numerical examples will be provided to validate the proposed tensor neural network interpolation., Comment: 14 pages, 2 figures. arXiv admin note: text overlap with arXiv:2402.00040, arXiv:2311.02732

This work starts from definition of randomness, the results of algorithmic randomness are analyzed from the perspective of application. Then, the source and nature of randomness is explored, and the relationship between infinity and randomness is found. The properties of randomness are summarized from the perspective of interaction between systems, that is, the set composed of sequences generated by randomness has the property of asymptotic completeness. Finally, the importance of randomness in AI research is emphasized.

Modern statistical analyses often encounter datasets with massive sizes and heavy-tailed distributions. For datasets with massive sizes, traditional estimation methods can hardly be used to estimate the extreme value index directly. To address the issue, we propose here a subsampling-based method. Specifically, multiple subsamples are drawn from the whole dataset by using the technique of simple random subsampling with replacement. Based on each subsample, an approximate maximum likelihood estimator can be computed. The resulting estimators are then averaged to form a more accurate one. Under appropriate regularity conditions, we show theoretically that the proposed estimator is consistent and asymptotically normal. With the help of the estimated extreme value index, we can estimate high-level quantiles and tail probabilities of a heavy-tailed random variable consistently. Extensive simulation experiments are provided to demonstrate the promising performance of our method. A real data analysis is also presented for illustration purpose.

Materials that can produce large controllable strains are widely used in shape memory devices, actuators and sensors. Great efforts have been made to improve the strain outputs of various material systems. Among them, ferroelastic transitions underpin giant reversible strains in electrically-driven ferro/piezoelectrics and thermally- or magneticallydriven shape memory alloys. However, large-strain ferroelastic switching in conventional ferroelectrics is very challenging while magnetic and thermal controls are not desirable for applications. Here, we demonstrate an unprecedentedly large shear strain up to 21.5 % in a hybrid ferroelectric, C6H5N(CH3)3CdCl3. The strain response is about two orders of magnitude higher than those of top-performing conventional ferroelectric polymers and oxides. It is achieved via inorganic bond switching and facilitated by the structural confinement of the large organic moieties, which prevents the undesired 180-degree polarization switching. Furthermore, Br substitution can effectively soften the bonds and result in giant shear piezoelectric coefficient (d35 ~ 4800 pm/V) in Br-rich end of the solid solution, C6H5N(CH3)3CdBr3xCl3(1-x). The superior electromechanical properties of the compounds promise their potential in lightweight and high energy density devices, and the strategy described here should inspire the development of next-generation piezoelectrics and electroactive materials based on hybrid ferroelectrics., Comment: 32 pages, 14 figures, 5 tables