Enhanced Differentiable Architecture Search Based on Asymptotic Regularization

In differentiable search architecture search methods,a more efficient search space design can significantly improve the performance of the searched architecture,thus requiring people to carefully define the search space with different complexity according to various operations.Meanwhile rationalizing the search strategies to explore the well-defined search space will further improve the speed and efficiency of architecture search.With this in mind,we propose a faster and more efficient differentiable architecture search method,AllegroNAS.Firstly,we introduce a more efficient search space enriched by the introduction of two redefined convolution modules.Secondly,we utilize a more efficient architectural parameter regularization method,mitigating the overfitting problem during the search process and reducing the error brought about by gradient approximation.Meanwhile,we introduce a natural exponential cosine annealing method to make the learning rate of the neural network training process more suitable for the search procedure.Moreover,group convolution and data augmentation are employed to reduce the computational cost.Finally,through extensive experiments on several public datasets,we demonstrate that our method can more swiftly search for better-performing neural network architectures in a more efficient search space,thus validating the effectiveness of our approach.

An inverse analysis of fluid flow through granular media using differentiable lattice Boltzmann method

This study presents a method for the inverse analysis of fluid flow problems.The focus is put on accurately determining boundary conditions and characterizing the physical properties of granular media,such as permeability,and fluid components,like viscosity.The primary aim is to deduce either constant pressure head or pressure profiles,given the known velocity field at a steady-state flow through a conduit containing obstacles,including walls,spheres,and grains.The lattice Boltzmann method(LBM)combined with automatic differentiation(AD)(AD-LBM)is employed,with the help of the GPU-capable Taichi programming language.A lightweight tape is used to generate gradients for the entire LBM simulation,enabling end-to-end backpropagation.Our AD-LBM approach accurately estimates the boundary conditions for complex flow paths in porous media,leading to observed steady-state velocity fields and deriving macro-scale permeability and fluid viscosity.The method demonstrates significant advantages in terms of prediction accuracy and computational efficiency,making it a powerful tool for solving inverse fluid flow problems in various applications.

CONSTRAINT QUALIFICATIONS AND DUAL PROBLEMS FOR QUASI-DIFFERENTIABLE PROGRAMMING

In classical nonlinear programming, it is a general method of developing optimality conditions that a nonlinear programming problem is linearized as a linear programming problem by using first order approximations of the functions at a given feasible point. The linearized procedure for differentiable nonlinear programming problems can be naturally generalized to the quasi differential case. As in classical case so called constraint qualifications have to be imposed on the constraint functions to guarantee that for a given local minimizer of the original problem the nullvector is an optimal solution of the corresponding 'quasilinearized' problem. In this paper, constraint qualifications for inequality constrained quasi differentiable programming problems of type min {f(x)|g(x)≤0} are considered, where f and g are qusidifferentiable functions in the sense of Demyanov. Various constraint qualifications for this problem are presented and a new one is proposed. The relations among these conditions are investigated. Moreover, a Wolf dual problem for this problem is introduced, and the corresponding dual theorems are given.

A continuous differentiable wavelet threshold function for speech enhancement

Enhanced speech based on the traditional wavelet threshold function had auditory oscillation distortion and the low signal-to-noise ratio (SNR). In order to solve these problems, a new continuous differentiable threshold function for speech enhancement was presented. Firstly, the function adopted narrow threshold areas, preserved the smaller signal speech, and improved the speech quality; secondly, based on the properties of the continuous differentiable and non-fixed deviation, each area function was attained gradually by using the method of mathematical derivation. It ensured that enhanced speech was continuous and smooth; it removed the auditory oscillation distortion; finally, combined with the Bark wavelet packets, it further improved human auditory perception. Experimental results show that the segmental SNR and PESQ (perceptual evaluation of speech quality) of the enhanced speech using this method increase effectively, compared with the existing speech enhancement algorithms based on wavelet threshold.

Weakly-Supervised Single-view Dense 3D Point Cloud Reconstruction via Differentiable Renderer

In recent years,addressing ill-posed problems by leveraging prior knowledge contained in databases on learning techniques has gained much attention.In this paper,we focus on complete three-dimensional(3D)point cloud reconstruction based on a single red-green-blue(RGB)image,a task that cannot be approached using classical reconstruction techniques.For this purpose,we used an encoder-decoder framework to encode the RGB information in latent space,and to predict the 3D structure of the considered object from different viewpoints.The individual predictions are combined to yield a common representation that is used in a module combining camera pose estimation and rendering,thereby achieving differentiability with respect to imaging process and the camera pose,and optimization of the two-dimensional prediction error of novel viewpoints.Thus,our method allows end-to-end training and does not require supervision based on additional ground-truth(GT)mask annotations or ground-truth camera pose annotations.Our evaluation of synthetic and real-world data demonstrates the robustness of our approach to appearance changes and self-occlusions,through outperformance of current state-of-the-art methods in terms of accuracy,density,and model completeness.

SIMULTANEOUSE APPROXIMATION TO A DIFFERENTIABLE FUNCTION AND ITS DERIVATIVES BY LAGRANGE INTERPOLATING POLYNOMIALS

This paper establishes the following pointwise result for simultancous Lagrange imterpolating approxima- tion:,then |f^(k)(x)-P_n^(k)(f,x)|=O(1)△_n^(q-k)(x)ω where P_n(f,x)is the Lagrange interpolating potynomial of deereeon the nodes X_nUY_n(see the definition of the next).

A DIFFERENTIABLE SPHERE THEOREM WITH PINCHING INTEGRAL RICCI CURVATURE

In this article, we introduce the Hausdorff convergence to derive a differentiable sphere theorem which shows an interesting rigidity phenomenon on some kind of manifolds.

Star-shaped Differentiable Functions and Star-shaped Differentials

Based on the isomorphism between the space of star-shaped sets and the space of continuous positively homogeneous real-valued functions, the star-shaped differential of a directionally differentiable function is defined. Formulas for star-shaped differential of a pointwise maximum and a pointwise minimum of a finite number of directionally differentiable functions, and a composite of two directionaUy differentiable functions are derived. Furthermore, the mean-value theorem for a directionaUy differentiable function is demonstrated.

Language Model Using Differentiable Neural Computer Based on Forget Gate-Based Memory Deallocation

A differentiable neural computer(DNC)is analogous to the Von Neumann machine with a neural network controller that interacts with an external memory through an attention mechanism.Such DNC’s offer a generalized method for task-specific deep learning models and have demonstrated reliability with reasoning problems.In this study,we apply a DNC to a language model(LM)task.The LM task is one of the reasoning problems,because it can predict the next word using the previous word sequence.However,memory deallocation is a problem in DNCs as some information unrelated to the input sequence is not allocated and remains in the external memory,which degrades performance.Therefore,we propose a forget gatebased memory deallocation(FMD)method,which searches for the minimum value of elements in a forget gate-based retention vector.The forget gatebased retention vector indicates the retention degree of information stored in each external memory address.In experiments,we applied our proposed NTM architecture to LM tasks as a task-specific example and to rescoring for speech recognition as a general-purpose example.For LM tasks,we evaluated DNC using the Penn Treebank and enwik8 LM tasks.Although it does not yield SOTA results in LM tasks,the FMD method exhibits relatively improved performance compared with DNC in terms of bits-per-character.For the speech recognition rescoring tasks,FMD again showed a relative improvement using the LibriSpeech data in terms of word error rate.

ON SIMULTANEOUS APPROXIMATION TO A DIFFERENTIABLE FUNCTION AND ITS DERIVATIVE BY INVERSE PAL-TYPE INTERPOLATION POLYNOMIALS

Let ξn-1<ξn-2 <ξn-2 <… < ξ1 be the zeros of the the (n -1)-th Legendre polynomial Pn-1(x) and - 1 = xn < xn-1 <… < x1 = 1 the zeros of the polynomial W n(x) =- n(n - 1) Pn-1(t)dt = (1 -x2)P'n-1(x). By the theory of the inverse Pal-Type interpolation, for a function f(x) ∈ C[-1 1], there exists a unique polynomial Rn(x) of degree 2n - 2 (if n is even) satisfying conditions Rn(f,ξk) = f(∈ek)(1≤ k≤ n - 1) ;R'n(f,xk) = f'(xk)(1≤ k≤ n). This paper discusses the simultaneous approximation to a differentiable function f by inverse Pal-Type interpolation polynomial {Rn(f,x)} (n is even) and the main result of this paper is that if f ∈ C'[1,1], r≥2, n≥ + 2> and n is even thenholds uniformly for all x ∈ [- 1,1], where h(x) = 1 +

Differentiable programming and density matrix based Hartree–Fock method

Differentiable programming is an emerging programming paradigm that allows people to take derivative of an output of arbitrary code snippet with respect to its input. It is the workhorse behind several well known deep learning frameworks,and has attracted significant attention in scientific machine learning community. In this paper, we introduce and implement a density matrix based Hartree–Fock method that naturally fits into the demands of this paradigm, and demonstrate it by performing fully variational ground state calculation on several representative chemical molecules.

THE CONTINUAL DIFFERENTIABLE PEAK-UNIMODAL SOLUTIONS OF FEIGENBAUM’S FUNCTIONAL EQUATIONS

For the famous Feigenbaum's equations, in this paper, we established its constructive theorem of the peak-unimodal, then we found out other paths to explore the peak-unimodal solutions. For example, we proceed on the direction to try the non-symmetrical continuous peak-unimodal solutions and C1 solutions.

Umbilical cord:an unlimited source of cells differentiable towards dopaminergic neurons

Cell replacement therapy utilizing mesenchymal stem cells as its main resource holds great promise for ultimate treatment of human neurological disorders.Parkinson＇s disease（PD）is a common,chronic neurodegenerative disorder hallmarked by localized degeneration of a specific set of dopaminergic neurons within a midbrain sub-region.The specific cell type and confined location of degenerating neurons make cell replacement therapy ideal for PD treatment since it mainly requires replenishment of lost dopaminergic neurons with fresh and functional ones.Endogenous as well as exogenous cell sources have been identified as candidate targets for cell replacement therapy in PD.In this review,umbilical cord mesenchymal stem cells（UCMSCs）are discussed as they provide an inexpensive unlimited reservoir differentiable towards functional dopaminergic neurons that potentially lead to long-lasting behavioral recovery in PD patients.We also present mi RNAs-mediated neuronal differentiation of UCMSCs.The UCMSCs bear a number of outstanding characteristics including their non-tumorigenic,low-immunogenic properties that make them ideal for cell replacement therapy purposes.Nevertheless,more investigations as well as controlled clinical trials are required to thoroughly confirm the efficacy of UCMSCs for therapeutic medical-grade applications in PD.

MHD Maxwell Fluid with Heat Transfer Analysis under Ramp Velocity and Ramp Temperature Subject to Non-Integer Differentiable Operators

The main focus of this study is to investigate the impact of heat generation/absorption with ramp velocity and ramp temperature on magnetohydrodynamic(MHD)time-dependent Maxwell fluid over an unbounded plate embedded in a permeable medium.Non-dimensional parameters along with Laplace transformation and inversion algorithms are used to find the solution of shear stress,energy,and velocity profile.Recently,new fractional differential operators are used to define ramped temperature and ramped velocity.The obtained analytical solutions are plotted for different values of emerging parameters.Fractional time derivatives are used to analyze the impact of fractional parameters(memory effect)on the dynamics of the fluid.While making a comparison,it is observed that the fractional-order model is best to explain the memory effect as compared to classical models.Our results suggest that the velocity profile decrease by increasing the effective Prandtl number.The existence of an effective Prandtl number may reflect the control of the thickness of momentum and enlargement of thermal conductivity.The incremental value of the M is observed for a decrease in the velocity field,which reflects to control resistive force.Further,it is noted that the Atangana-Baleanu derivative in Caputo sense(ABC)is the best to highlight the dynamics of the fluid.The influence of pertinent parameters is analyzed graphically for velocity and energy profile.Expressions for skin friction and Nusselt number are also derived for fractional differential operators.

Complements to the Theory of Higher-Order Types of Asymptotic Variation for Differentiable Functions

The purpose of this paper is to add some complements to the general theory of higher-order types of asymptotic variation developed in two previous papers so as to complete our elementary (but not too much!) theory in view of applications to the theory of finite asymptotic expansions in the real domain, the asymptotic study of ordinary differential equations and the like. The main results concern: 1) a detailed study of the types of asymptotic variation of an infinite series so extending the results known for the sole power series;2) the type of asymptotic variation of a Wronskian completing the many already-published results on the asymptotic behaviors of Wronskians;3) a comparison between the two main standard approaches to the concept of “type of asymptotic variation”: via an asymptotic differential equation or an asymptotic functional equation;4) a discussion about the simple concept of logarithmic variation making explicit and completing the results which, in the literature, are hidden in a quite-complicated general theory.

Efficient participating media rendering with differentiable regularization

Highly scattering media,such as milk,skin,and clouds,are common in the real world.Rendering participating media is challenging,especially for highorder scattering dominant media,because the light may undergo a large number of scattering events before leaving the surface.Monte Carlo-based methods typically require a long time to produce noise-free results.Based on the observation that low-albedo media contain less noise than high-albedo media,we propose reducing the variance of the rendered results using differentiable regularization.We first render an image with low-albedo participating media together with the gradient with respect to the albedo,and then predict the final rendered image with a low-albedo image and gradient image via a novel prediction function.To achieve high quality,we also consider the gradients of neighboring frames to provide a noise-free gradient image.Ultimately,our method can produce results with much less overall eror than equal-time path tracing methods.

Dynamic ocean inverse modeling based on differentiable rendering

Learning and inferring underlying motion patterns of captured 2D scenes and then re-creating dynamic evolution consistent with the real-world natural phenomena have high appeal for graphics and animation.To bridge the technical gap between virtual and real environments,we focus on the inverse modeling and reconstruction of visually consistent and property-verifiable oceans,taking advantage of deep learning and differentiable physics to learn geometry and constitute waves in a self-supervised manner.First,we infer hierarchical geometry using two networks,which are optimized via the differentiable renderer.We extract wave components from the sequence of inferred geometry through a network equipped with a differentiable ocean model.Then,ocean dynamics can be evolved using the reconstructed wave components.Through extensive experiments,we verify that our new method yields satisfactory results for both geometry reconstruction and wave estimation.Moreover,the new framework has the inverse modeling potential to facilitate a host of graphics applications,such as the rapid production of physically accurate scene animation and editing guided by real ocean scenes.

Learning accurate template matching with differentiable coarseto-fine correspondence refinement

Template matching is a fundamental task in computer vision and has been studied for decades.It plays an essential role in manufacturing industry for estimating the poses of different parts,facilitating downstream tasks such as robotic grasping.Existing methods fail when the template and source images have different modalities,cluttered backgrounds,or weak textures.They also rarely consider geometric transformations via homographies,which commonly exist even for planar industrial parts.To tackle the challenges,we propose an accurate template matching method based on differentiable coarse-tofine correspondence refinement.We use an edge-aware module to overcome the domain gap between the mask template and the grayscale image,allowing robust matching.An initial warp is estimated using coarse correspondences based on novel structure-aware information provided by transformers.This initial alignment is passed to a refinement network using references and aligned images to obtain sub-pixel level correspondences which are used to give the final geometric transformation.Extensive evaluation shows that our method to be significantly better than state-of-the-art methods and baselines,providing good generalization ability and visually plausible results even on unseen real data.

Weighted Integral of Infinitely Differentiable Multivariate Functions is Exponentially Convergent

We study the problem of a weighted integral of infinitely differentiable mul-tivariate functions defined on the unit cube with the L∞-norm of partial derivative of all orders bounded by 1.We consider the algorithms that use finitely many function values as information(called standard information).On the one hand,we obtained that the interpolatory quadratures based on the extended Chebyshev nodes of the second kind have almost the same quadrature weights.On the other hand,by using the Smolyak al-gorithm with the above interpolatory quadratures,we proved that the weighted integral problem is of exponential convergence in the worst case setting.

A differentiable sphere theorem with positive Ricci curvature and reverse volume pinching

Let Mn be a compact, simply connected n (≥3)-dimensional Riemannian manifold without bound-ary and Sn be the unit sphere Euclidean space Rn+1. We derive a differentiable sphere theorem whenever themanifold concerned satisfies that the sectional curvature KM is not larger than 1, while Ric(M)≥n+2 4 and the volume V (M) is not larger than (1 + η)V (Sn) for some positive number η depending only on n.