In this series of works we establish homogenized lattice Boltzmann methods (HLBM) for the simulation of fluid flow through porous media. Our contributions in part I are twofold. First, we assemble the targeted partial differential equation system by formally unifying the governing equations for nonstationary fluid flow in porous media. To this end, a matrix of regularly arranged obstacles of equal size is placed into the domain to model fluid flow through structures of different porosities that is governed by the incompressible nonstationary Navier--Stokes equations. Depending on the ratio of geometric parameters in the matrix arrangement, several cases of homogenized equations are obtained. We review the existing methods to homogenize the nonstationary Navier--Stokes equations for specific porosities and interpret connections between the resulting model equations from the perspective of applicability. Consequently, the homogenized Navier--Stokes equations are formulated as targeted partial differential equations which jointly incorporate the derived aspects. Second, we propose a kinetic model, named homogenized Bhatnagar--Gross--Krook Boltzmann equation, which approximates the homogenized nonstationary Navier--Stokes equations. We formally prove that the zeroth and first order moments of the kinetic model provide solutions to the mass and momentum balance variables of the macrocopic model up to specific orders in the scaling parameter. Based on the present contributions, in the sequel (part II) the homogenized Navier--Stokes equations are consistently approximated by deriving a limit consistent HLBM discretization of the homogenized Bhatnagar--Gross--Krook Boltzmann equation.
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We present Neural Spectral Methods, a technique to solve parametric Partial Differential Equations (PDEs), grounded in classical spectral methods. Our method uses orthogonal bases to learn PDE solutions as mappings between spectral coefficients. In contrast to current machine learning approaches which enforce PDE constraints by minimizing the numerical quadrature of the residuals in the spatiotemporal domain, we leverage Parseval's identity and introduce a new training strategy through a \textit{spectral loss}. Our spectral loss enables more efficient differentiation through the neural network, and substantially reduces training complexity. At inference time, the computational cost of our method remains constant, regardless of the spatiotemporal resolution of the domain. Our experimental results demonstrate that our method significantly outperforms previous machine learning approaches in terms of speed and accuracy by one to two orders of magnitude on multiple different problems. When compared to numerical solvers of the same accuracy, our method demonstrates a $10\times$ increase in performance speed.
We give a recursive construction for projective Reed-Muller codes in terms of affine Reed-Muller codes and projective Reed-Muller codes in fewer variables. From this construction, we obtain the dimension of the subfield subcodes of projective Reed-Muller codes for some particular degrees that give codes with good parameters. Moreover, from this recursive construction we are able to derive a lower bound for the generalized Hamming weights of projective Reed-Muller codes which, together with the basic properties of the generalized Hamming weights, allows us to determine most of the weight hierarchy of projective Reed-Muller codes in many cases.
This paper considers the problem of robust iterative Bayesian smoothing in nonlinear state-space models with additive noise using Gaussian approximations. Iterative methods are known to improve smoothed estimates but are not guaranteed to converge, motivating the development of more robust versions of the algorithms. The aim of this article is to present Levenberg-Marquardt (LM) and line-search extensions of the classical iterated extended Kalman smoother (IEKS) as well as the iterated posterior linearisation smoother (IPLS). The IEKS has previously been shown to be equivalent to the Gauss-Newton (GN) method. We derive a similar GN interpretation for the IPLS. Furthermore, we show that an LM extension for both iterative methods can be achieved with a simple modification of the smoothing iterations, enabling algorithms with efficient implementations. Our numerical experiments show the importance of robust methods, in particular for the IEKS-based smoothers. The computationally expensive IPLS-based smoothers are naturally robust but can still benefit from further regularisation.
This work aims to investigate the problem of 3D modeling using single free-hand sketches, which is one of the most natural ways we humans express ideas. Although sketch-based 3D modeling can drastically make the 3D modeling process more accessible, the sparsity and ambiguity of sketches bring significant challenges for creating high-fidelity 3D models that reflect the creators' ideas. In this work, we propose a view- and structural-aware deep learning approach, \textit{Deep3DSketch}, which tackles the ambiguity and fully uses sparse information of sketches, emphasizing the structural information. Specifically, we introduced random pose sampling on both 3D shapes and 2D silhouettes, and an adversarial training scheme with an effective progressive discriminator to facilitate learning of the shape structures. Extensive experiments demonstrated the effectiveness of our approach, which outperforms existing methods -- with state-of-the-art (SOTA) performance on both synthetic and real datasets.
As a surrogate for computationally intensive meso-scale simulation of woven composites, this article presents Recurrent Neural Network (RNN) models. Leveraging the power of transfer learning, the initialization challenges and sparse data issues inherent in cyclic shear strain loads are addressed in the RNN models. A mean-field model generates a comprehensive data set representing elasto-plastic behavior. In simulations, arbitrary six-dimensional strain histories are used to predict stresses under random walking as the source task and cyclic loading conditions as the target task. Incorporating sub-scale properties enhances RNN versatility. In order to achieve accurate predictions, the model uses a grid search method to tune network architecture and hyper-parameter configurations. The results of this study demonstrate that transfer learning can be used to effectively adapt the RNN to varying strain conditions, which establishes its potential as a useful tool for modeling path-dependent responses in woven composites.
Robust Markov Decision Processes (RMDPs) are a widely used framework for sequential decision-making under parameter uncertainty. RMDPs have been extensively studied when the objective is to maximize the discounted return, but little is known for average optimality (optimizing the long-run average of the rewards obtained over time) and Blackwell optimality (remaining discount optimal for all discount factors sufficiently close to 1). In this paper, we prove several foundational results for RMDPs beyond the discounted return. We show that average optimal policies can be chosen stationary and deterministic for sa-rectangular RMDPs but, perhaps surprisingly, that history-dependent (Markovian) policies strictly outperform stationary policies for average optimality in s-rectangular RMDPs. We also study Blackwell optimality for sa-rectangular RMDPs, where we show that {\em approximate} Blackwell optimal policies always exist, although Blackwell optimal policies may not exist. We also provide a sufficient condition for their existence, which encompasses virtually any examples from the literature. We then discuss the connection between average and Blackwell optimality, and we describe several algorithms to compute the optimal average return. Interestingly, our approach leverages the connections between RMDPs and stochastic games.
To understand high precision observations of exoplanets and brown dwarfs, we need detailed and complex general circulation models (GCMs) that incorporate hydrodynamics, chemistry, and radiation. For this study, we specifically examined the coupling between chemistry and radiation in GCMs and compared different methods for the mixing of opacities of different chemical species in the correlated-k assumption, when equilibrium chemistry cannot be assumed. We propose a fast machine learning method based on DeepSets (DS), which effectively combines individual correlated-k opacities (k-tables). We evaluated the DS method alongside other published methods such as adaptive equivalent extinction (AEE) and random overlap with rebinning and resorting (RORR). We integrated these mixing methods into our GCM (expeRT/MITgcm) and assessed their accuracy and performance for the example of the hot Jupiter HD~209458 b. Our findings indicate that the DS method is both accurate and efficient for GCM usage, whereas RORR is too slow. Additionally, we observed that the accuracy of AEE depends on its specific implementation and may introduce numerical issues in achieving radiative transfer solution convergence. We then applied the DS mixing method in a simplified chemical disequilibrium situation, where we modeled the rainout of TiO and VO, and confirmed that the rainout of TiO and VO would hinder the formation of a stratosphere. To further expedite the development of consistent disequilibrium chemistry calculations in GCMs, we provide documentation and code for coupling the DS mixing method with correlated-k radiative transfer solvers. The DS method has been extensively tested to be accurate enough for GCMs; however, other methods might be needed for accelerating atmospheric retrievals.
The Conformer has become the most popular encoder model for automatic speech recognition (ASR). It adds convolution modules to a transformer to learn both local and global dependencies. In this work we describe a faster, more memory-efficient, and better-performing transformer, called Zipformer. Modeling changes include: 1) a U-Net-like encoder structure where middle stacks operate at lower frame rates; 2) reorganized block structure with more modules, within which we re-use attention weights for efficiency; 3) a modified form of LayerNorm called BiasNorm allows us to retain some length information; 4) new activation functions SwooshR and SwooshL work better than Swish. We also propose a new optimizer, called ScaledAdam, which scales the update by each tensor's current scale to keep the relative change about the same, and also explictly learns the parameter scale. It achieves faster convergence and better performance than Adam. Extensive experiments on LibriSpeech, Aishell-1, and WenetSpeech datasets demonstrate the effectiveness of our proposed Zipformer over other state-of-the-art ASR models. Our code is publicly available at //github.com/k2-fsa/icefall.
Among the commonly used non-destructive techniques, the Ground Penetrating Radar (GPR) is one of the most widely adopted today for assessing pavement conditions in France. However, conventional radar systems and their forward processing methods have shown their limitations for the physical and geometrical characterization of very thin layers such as tack coats. However, the use of Machine Learning methods applied to GPR with an inverse approach showed that it was numerically possible to identify the tack coat characteristics despite masking effects due to low timefrequency resolution noted in the raw B-scans. Thus, we propose in this paper to apply the inverse approach based on Machine Learning, already validated in previous works on numerical data, on two experimental cases with different pavement structures. The first case corresponds to a validation on known pavement structures on the Gustave Eiffel University (Nantes, France) with its pavement fatigue carousel and the second case focuses on a new real road in Vend{\'e}e department (France). In both case studies, the performances of SVM/SVR methods showed the efficiency of supervised learning methods to classify and estimate the emulsion proportioning in the tack coats.
A semi-implicit in time, entropy stable finite volume scheme for the compressible barotropic Euler system is designed and analyzed and its weak convergence to a dissipative measure-valued (DMV) solution [E. Feireisl et al., Dissipative measure-valued solutions to the compressible Navier-Stokes system, Calc. Var. Partial Differential Equations, 2016] of the Euler system is shown. The entropy stability is achieved by introducing a shifted velocity in the convective fluxes of the mass and momentum balances, provided some CFL-like condition is satisfied to ensure stability. A consistency analysis is performed in the spirit of the Lax's equivalence theorem under some physically reasonable boundedness assumptions. The concept of K-convergence [E. Feireisl et al., K-convergence as a new tool in numerical analysis, IMA J. Numer. Anal., 2020] is used in order to obtain some strong convergence results, which are then illustrated via rigorous numerical case studies. The convergence of the scheme to a DMV solution, a weak solution and a strong solution of the Euler system using the weak-strong uniqueness principle and relative entropy are presented.
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