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We describe and analyze a quasi-Trefftz DG method for solving boundary value problems for the homogeneous diffusion-advection-reaction equation with piecewise-smooth coefficients. Trefftz schemes are high-order Galerkin methods whose discrete functions are elementwise exact solutions of the underlying PDE. Trefftz basis functions can be computed for many PDEs that are linear, homogeneous and with piecewise-constant coefficients. However, if the equation has varying coefficients, in general, exact solutions are unavailable, hence the construction of discrete Trefftz spaces is impossible. Quasi-Trefftz methods have been introduced to overcome this limitation, relying on discrete spaces of functions that are elementwise "approximate solutions" of the PDE. A space-time quasi-Trefftz DG method for the acoustic wave equation with smoothly varying coefficients has recently been studied; since it has shown excellent results, we propose a related method that can be applied to second-order elliptic equations. The DG weak formulation is derived using an interior penalty parameter and the upwind numerical fluxes. We choose polynomial quasi-Trefftz basis functions, whose coefficients can be computed with a simple algorithm based on the Taylor expansion of the PDE's coefficients. The main advantage of Trefftz and quasi-Trefftz schemes over more classical ones is the higher accuracy for comparable numbers of degrees of freedom. We prove that the dimension of the quasi-Trefftz space is smaller than the dimension of the full polynomial space of the same degree and that yields the same optimal convergence rates. The quasi-Trefftz DG method is well-posed, consistent and stable and we prove its high-order convergence. We present some numerical experiments in two dimensions that show excellent properties in terms of approximation and convergence rate.

Sentence embeddings induced with various transformer architectures encode much semantic and syntactic information in a distributed manner in a one-dimensional array. We investigate whether specific grammatical information can be accessed in these distributed representations. Using data from a task developed to test rule-like generalizations, our experiments on detecting subject-verb agreement yield several promising results. First, we show that while the usual sentence representations encoded as one-dimensional arrays do not easily support extraction of rule-like regularities, a two-dimensional reshaping of these vectors allows various learning architectures to access such information. Next, we show that various architectures can detect patterns in these two-dimensional reshaped sentence embeddings and successfully learn a model based on smaller amounts of simpler training data, which performs well on more complex test data. This indicates that current sentence embeddings contain information that is regularly distributed, and which can be captured when the embeddings are reshaped into higher dimensional arrays. Our results cast light on representations produced by language models and help move towards developing few-shot learning approaches.

This paper develops and analyses semi-discrete numerical method for two dimensional Vlasov-Stokes' system with periodic boundary condition. The method is based on coupling of semi-discrete discontinuous Galerkin method for the Vlasov equation with discontinuous Galerkin scheme for the stationary incompressible Stokes' equation. The proposed method is both mass and momentum conservative. Since it is difficult to establish non-negativity of the discrete local density, the generalized discrete Stokes' operator become non-coercive and indefinite and under smallness condition on the discretization parameter, optimal error estimates are established with help of a modified the Stokes' projection to deal with Stokes' part and with the help of a special projection to tackle the Vlasov part. Finally, numerical experiments based on the dG method combined with a splitting algorithm are performed.

In this paper, we consider variational autoencoders (VAE) for general state space models. We consider a backward factorization of the variational distributions to analyze the excess risk associated with VAE. Such backward factorizations were recently proposed to perform online variational learning and to obtain upper bounds on the variational estimation error. When independent trajectories of sequences are observed and under strong mixing assumptions on the state space model and on the variational distribution, we provide an oracle inequality explicit in the number of samples and in the length of the observation sequences. We then derive consequences of this theoretical result. In particular, when the data distribution is given by a state space model, we provide an upper bound for the Kullback-Leibler divergence between the data distribution and its estimator and between the variational posterior and the estimated state space posterior distributions.Under classical assumptions, we prove that our results can be applied to Gaussian backward kernels built with dense and recurrent neural networks.

Dynamical systems across the sciences, from electrical circuits to ecological networks, undergo qualitative and often catastrophic changes in behavior, called bifurcations, when their underlying parameters cross a threshold. Existing methods predict oncoming catastrophes in individual systems but are primarily time-series-based and struggle both to categorize qualitative dynamical regimes across diverse systems and to generalize to real data. To address this challenge, we propose a data-driven, physically-informed deep-learning framework for classifying dynamical regimes and characterizing bifurcation boundaries based on the extraction of topologically invariant features. We focus on the paradigmatic case of the supercritical Hopf bifurcation, which is used to model periodic dynamics across a wide range of applications. Our convolutional attention method is trained with data augmentations that encourage the learning of topological invariants which can be used to detect bifurcation boundaries in unseen systems and to design models of biological systems like oscillatory gene regulatory networks. We further demonstrate our method's use in analyzing real data by recovering distinct proliferation and differentiation dynamics along pancreatic endocrinogenesis trajectory in gene expression space based on single-cell data. Our method provides valuable insights into the qualitative, long-term behavior of a wide range of dynamical systems, and can detect bifurcations or catastrophic transitions in large-scale physical and biological systems.

Discovering underlying structures of causal relations from observational studies poses a great challenge in scientific research where randomized trials or intervention-based studies are infeasible. This challenge pertains to the lack of knowledge on pre-specified roles of cause and effect in observations studies. Leveraging Shannon's seminal work on information theory, we propose a new conceptual framework of asymmetry where any causal link between putative cause and effect is captured by unequal information flows from one variable to another. We present an entropy-based asymmetry coefficient that not only enables us to assess for whether one variable is a stronger predictor of the other, but also detects an imprint of the underlying causal relation in observational studies. Our causal discovery analytics can accommodate low-dimensional confounders naturally. The proposed methodology relies on scalable non-parametric density estimation using fast Fourier transformation, making the resulting estimation method manyfold faster than the classical bandwidth-based density estimation while maintaining comparable mean integrated squared error rates. We investigate key asymptotic properties of our methodology and utilize a data-splitting and cross-fitting technique to facilitate inference for the direction of causal relations. We illustrate the performance of our methodology through simulation studies and real data examples.

We propose a new method called the Metropolis-adjusted Mirror Langevin algorithm for approximate sampling from distributions whose support is a compact and convex set. This algorithm adds an accept-reject filter to the Markov chain induced by a single step of the mirror Langevin algorithm (Zhang et al., 2020), which is a basic discretisation of the mirror Langevin dynamics. Due to the inclusion of this filter, our method is unbiased relative to the target, while known discretisations of the mirror Langevin dynamics including the mirror Langevin algorithm have an asymptotic bias. We give upper bounds for the mixing time of the proposed algorithm when the potential is relatively smooth, convex, and Lipschitz with respect to a self-concordant mirror function. As a consequence of the reversibility of the Markov chain induced by the algorithm, we obtain an exponentially better dependence on the error tolerance for approximate sampling. We also present numerical experiments that corroborate our theoretical findings.

We develop a fourth-order Magnus expansion based quantum algorithm for the simulation of many-body problems involving two-level quantum systems with time-dependent Hamiltonians, $\mathcal{H}(t)$. A major hurdle in the utilization of the Magnus expansion is the appearance of a commutator term which leads to prohibitively long circuits. We present a technique for eliminating this commutator and find that a single time-step of the resulting algorithm is only marginally costlier than that required for time-stepping with a time-independent Hamiltonian, requiring only three additional single-qubit layers. For a large class of Hamiltonians appearing in liquid-state nuclear magnetic resonance (NMR) applications, we further exploit symmetries of the Hamiltonian and achieve a surprising reduction in the expansion, whereby a single time-step of our fourth-order method has a circuit structure and cost that is identical to that required for a fourth-order Trotterized time-stepping procedure for time-independent Hamiltonians. Moreover, our algorithms are able to take time-steps that are larger than the wavelength of oscillation of the time-dependent Hamiltonian, making them particularly suited for highly-oscillatory controls. The resulting quantum circuits have shorter depths for all levels of accuracy when compared to first and second-order Trotterized methods, as well as other fourth-order Trotterized methods, making the proposed algorithm a suitable candidate for simulation of time-dependent Hamiltonians on near-term quantum devices.

In this paper, we propose a general unified tracking-servoing approach for controlling the shape of elastic deformable objects using robotic arms. Our approach works by forming a lattice around the object, binding the object to the lattice, and tracking and servoing the lattice instead of the object. This makes our approach have full 3D control over deformable objects of any general form (linear, thin-shell, volumetric). Furthermore, it decouples the runtime complexity of the approach from the objects' geometric complexity. Our approach is based on the As-Rigid-As-Possible (ARAP) deformation model. It requires no mechanical parameter of the object to be known and can drive the object toward desired shapes through large deformations. The inputs to our approach are the point cloud of the object's surface in its rest shape and the point cloud captured by a 3D camera in each frame. Overall, our approach is more broadly applicable than existing approaches. We validate the efficiency of our approach through numerous experiments with deformable objects of various shapes and materials (paper, rubber, plastic, foam). Experiment videos are available on the project website: //sites.google.com/view/tracking-servoing-approach.

Enumeration problems are often encountered as key subroutines in the exact computation of graph parameters such as chromatic number, treewidth, or treedepth. In the case of treedepth computation, the enumeration of inclusion-wise minimal separators plays a crucial role. However and quite surprisingly, the complexity status of this problem has not been settled since it has been posed as an open direction by Kloks and Kratsch in 1998. Recently at the PACE 2020 competition dedicated to treedepth computation, solvers have been circumventing that by listing all minimal $a$-$b$ separators and filtering out those that are not inclusion-wise minimal, at the cost of efficiency. Naturally, having an efficient algorithm for listing inclusion-wise minimal separators would drastically improve such practical algorithms. In this note, however, we show that no efficient algorithm is to be expected from an output-sensitive perspective, namely, we prove that there is no output-polynomial time algorithm for inclusion-wise minimal separators enumeration unless P = NP.