In this paper we compute the spherical Fourier expansions coefficients for the restriction of the generalised Wendland functions from $d-$dimensional Euclidean space to the (d-1)-dimensional unit sphere. The development required to derive these coefficients relies heavily upon known asymptotic results for hypergeometric functions and the final result shows that they can be expressed in closed form as a multiple of a certain $_{3}F_{2}$ hypergeometric function. Using the closed form expressions we are able to provide the precise asymptotic rates of decay for the spherical Fourier coefficients which we observe have a close connection to the asymptotic decay rate of the corresponding Euclidean Fourier transform.

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This paper is concerned with convergence estimates for fully discrete tree tensor network approximations of high-dimensional functions from several model classes. For functions having standard or mixed Sobolev regularity, new estimates generalizing and refining known results are obtained, based on notions of linear widths of multivariate functions. In the main results of this paper, such techniques are applied to classes of functions with compositional structure, which are known to be particularly suitable for approximation by deep neural networks. As shown here, such functions can also be approximated by tree tensor networks without a curse of dimensionality -- however, subject to certain conditions, in particular on the depth of the underlying tree. In addition, a constructive encoding of compositional functions in tree tensor networks is given.

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We consider an optimal control problem for the steady-state Kirchhoff equation, a prototype for nonlocal partial differential equations, different from fractional powers of closed operators. Existence and uniqueness of solutions of the state equation, existence of global optimal solutions, differentiability of the control-to-state map and first-order necessary optimality conditions are established. The aforementioned results require the controls to be functions in $H^1$ and subject to pointwise upper and lower bounds. In order to obtain the Newton differentiability of the optimality conditions, we employ a Moreau-Yosida-type penalty approach to treat the control constraints and study its convergence. The first-order optimality conditions of the regularized problems are shown to be Newton diffentiable, and a generalized Newton method is detailed. A discretization of the optimal control problem by piecewise linear finite elements is proposed and numerical results are presented.

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In this work we propose a batch version of the Greenkhorn algorithm for multimarginal regularized optimal transport problems. Our framework is general enough to cover, as particular cases, some existing algorithms like Sinkhorn and Greenkhorn algorithm for the bi-marginal setting, and (greedy) MultiSinkhorn for multimarginal optimal transport. We provide a complete converge analysis, which is based on the properties of the iterative Bregman projections (IBP) method with greedy control. Global linear rate of convergence and explicit bound on the iteration complexity are obtained. When specialized to above mentioned algorithms, our results give new insights and/or improve existing ones.

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In this paper, we consider sampling from a class of distributions with thin tails supported on $\mathbb{R}^d$ and make two primary contributions. First, we propose a new Metropolized Algorithm With Optimization Step (MAO), which is well suited for such targets. Our algorithm is capable of sampling from distributions where the Metropolis-adjusted Langevin algorithm (MALA) is not converging or lacking in theoretical guarantees. Second, we derive upper bounds on the mixing time of MAO. Our results are supported by simulations on multiple target distributions.

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Deep neural networks are notorious for defying theoretical treatment. However, when the number of parameters in each layer tends to infinity the network function is a Gaussian process (GP) and quantitatively predictive description is possible. Gaussian approximation allows to formulate criteria for selecting hyperparameters, such as variances of weights and biases, as well as the learning rate. These criteria rely on the notion of criticality defined for deep neural networks. In this work we describe a new way to diagnose (both theoretically and empirically) this criticality. To that end, we introduce partial Jacobians of a network, defined as derivatives of preactivations in layer $l$ with respect to preactivations in layer $l_0<l$. These quantities are particularly useful when the network architecture involves many different layers. We discuss various properties of the partial Jacobians such as their scaling with depth and relation to the neural tangent kernel (NTK). We derive the recurrence relations for the partial Jacobians and utilize them to analyze criticality of deep MLP networks with (and without) LayerNorm. We find that the normalization layer changes the optimal values of hyperparameters and critical exponents. We argue that LayerNorm is more stable when applied to preactivations, rather than activations due to larger correlation depth.

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Gradient flows are a powerful tool for optimizing functionals in general metric spaces, including the space of probabilities endowed with the Wasserstein metric. A typical approach to solving this optimization problem relies on its connection to the dynamic formulation of optimal transport and the celebrated Jordan-Kinderlehrer-Otto (JKO) scheme. However, this formulation involves optimization over convex functions, which is challenging, especially in high dimensions. In this work, we propose an approach that relies on the recently introduced input-convex neural networks (ICNN) to parametrize the space of convex functions in order to approximate the JKO scheme, as well as in designing functionals over measures that enjoy convergence guarantees. We derive a computationally efficient implementation of this JKO-ICNN framework and experimentally demonstrate its feasibility and validity in approximating solutions of low-dimensional partial differential equations with known solutions. We also demonstrate its viability in high-dimensional applications through an experiment in controlled generation for molecular discovery.

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Machine learning-based data-driven modeling can be computationally efficient solutions of time-dependent subsurface geophysical systems. In this work, our previous approach of conditional generative adversarial networks (cGAN) developed for steady-state problems with heterogeneous materials is extended to time-dependent problems by adopting the concept of continuous cGAN (CcGAN). The CcGAN that can condition continuous variables in the cGAN framework is developed to incorporate the time domain through either element-wise addition or conditional batch normalization. As a demonstration case, the transient response of the coupled poroelastic process is studied in two different permeability fields: Zinn \& Harvey transformation and a bimodal transformation. The proposed CcGAN uses heterogeneous permeability fields as input parameters while pressure and displacement fields over time are model output. Our results show that the model provides sufficient accuracy with computational speed-up. This robust framework will enable us to perform real-time reservoir management and robust uncertainty quantification in realistic problems.

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The quality of mesh generation has long been considered a vital aspect in providing engineers with reliable simulation results throughout the history of the Finite Element Method (FEM). The element extraction method, which is currently the most robust method, is used in business software. However, in order to speed up extraction, the approach is done by finding the next element that optimizes a target function, which can result in local mesh of bad quality after many time steps. We provide TreeMesh, a method that uses this method in conjunction with reinforcement learning (also possible with supervised learning) and a novel Monte-Carlo tree search (MCTS) (Coulom(2006), Kocsis and Szepesv\'ari(2006), Browne et~al.(2012)). The algorithm is based on a previously proposed approach (Pan et~al.(2021)). After making many improvements on DRL (algorithm, state-action-reward setting) and adding a MCTS, it outperforms the former work on the same boundary. Furthermore, using tree search, our program reveals much preponderance on seed-density-changing boundaries, which is common on thin-film materials.

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In this paper, we consider a boundary value problem (BVP) for a fourth order nonlinear functional integro-differential equation. We establish the existence and uniqueness of solution and construct a numerical method for solving it. We prove that the method is of second order accuracy and obtain an estimate for the total error. Some examples demonstrate the validity of the obtained theoretical results and the efficiency of the numerical method.

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In this paper, we study the optimal convergence rate for distributed convex optimization problems in networks. We model the communication restrictions imposed by the network as a set of affine constraints and provide optimal complexity bounds for four different setups, namely: the function $F(\xb) \triangleq \sum_{i=1}^{m}f_i(\xb)$ is strongly convex and smooth, either strongly convex or smooth or just convex. Our results show that Nesterov's accelerated gradient descent on the dual problem can be executed in a distributed manner and obtains the same optimal rates as in the centralized version of the problem (up to constant or logarithmic factors) with an additional cost related to the spectral gap of the interaction matrix. Finally, we discuss some extensions to the proposed setup such as proximal friendly functions, time-varying graphs, improvement of the condition numbers.

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