We describe a recursive algorithm that decomposes an algebraic set into locally closed equidimensional sets, i.e. sets which each have irreducible components of the same dimension. At the core of this algorithm, we combine ideas from the theory of triangular sets, a.k.a. regular chains, with Gr\"obner bases to encode and work with locally closed algebraic sets. Equipped with this, our algorithm avoids projections of the algebraic sets that are decomposed and certain genericity assumptions frequently made when decomposing polynomial systems, such as assumptions about Noether position. This makes it produce fine decompositions on more structured systems where ensuring genericity assumptions often destroys the structure of the system at hand. Practical experiments demonstrate its efficiency compared to state-of-the-art implementations.
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High-dimensional transport equations frequently occur in science and engineering. Computing their numerical solution, however, is challenging due to its high dimensionality. In this work we develop an algorithm to efficiently solve the transport equation in moderately complex geometrical domains using a Galerkin method stabilized by streamline diffusion. The ansatz spaces are a tensor product of a sparse grid in space and discontinuous piecewise polynomials in time. Here, the sparse grid is constructed upon nested multilevel finite element spaces to provide geometric flexibility. This results in an implicit time-stepping scheme which we prove to be stable and convergent. If the solution has additional mixed regularity, the convergence of a $2d$-dimensional problem equals that of a $d$-dimensional one up to logarithmic factors. For the implementation, we rely on the representation of sparse grids as a sum of anisotropic full grid spaces. This enables us to store the functions and to carry out the computations on a sequence regular full grids exploiting the tensor product structure of the ansatz spaces. In this way existing finite element libraries and GPU acceleration can be used. The combination technique is used as a preconditioner for an iterative scheme to solve the transport equation on the sequence of time strips. Numerical tests show that the method works well for problems in up to six dimensions. Finally, the method is also used as a building block to solve nonlinear Vlasov-Poisson equations.
Natural language processing models are vulnerable to adversarial examples. Previous textual adversarial attacks adopt gradients or confidence scores to calculate word importance ranking and generate adversarial examples. However, this information is unavailable in the real world. Therefore, we focus on a more realistic and challenging setting, named hard-label attack, in which the attacker can only query the model and obtain a discrete prediction label. Existing hard-label attack algorithms tend to initialize adversarial examples by random substitution and then utilize complex heuristic algorithms to optimize the adversarial perturbation. These methods require a lot of model queries and the attack success rate is restricted by adversary initialization. In this paper, we propose a novel hard-label attack algorithm named LimeAttack, which leverages a local explainable method to approximate word importance ranking, and then adopts beam search to find the optimal solution. Extensive experiments show that LimeAttack achieves the better attacking performance compared with existing hard-label attack under the same query budget. In addition, we evaluate the effectiveness of LimeAttack on large language models, and results indicate that adversarial examples remain a significant threat to large language models. The adversarial examples crafted by LimeAttack are highly transferable and effectively improve model robustness in adversarial training.
We study the problem of uncertainty quantification for time series prediction, with the goal of providing easy-to-use algorithms with formal guarantees. The algorithms we present build upon ideas from conformal prediction and control theory, are able to prospectively model conformal scores in an online setting, and adapt to the presence of systematic errors due to seasonality, trends, and general distribution shifts. Our theory both simplifies and strengthens existing analyses in online conformal prediction. Experiments on 4-week-ahead forecasting of statewide COVID-19 death counts in the U.S. show an improvement in coverage over the ensemble forecaster used in official CDC communications. We also run experiments on predicting electricity demand, market returns, and temperature using autoregressive, Theta, Prophet, and Transformer models. We provide an extendable codebase for testing our methods and for the integration of new algorithms, data sets, and forecasting rules.
In recent years, there has been a growing interest in understanding complex microstructures and their effect on macroscopic properties. In general, it is difficult to derive an effective constitutive law for such microstructures with reasonable accuracy and meaningful parameters. One numerical approach to bridge the scales is computational homogenization, in which a microscopic problem is solved at every macroscopic point, essentially replacing the effective constitutive model. Such approaches are, however, computationally expensive and typically infeasible in multi-query contexts such as optimization and material design. To render these analyses tractable, surrogate models that can accurately approximate and accelerate the microscopic problem over a large design space of shapes, material and loading parameters are required. In previous works, such models were constructed in a data-driven manner using methods such as Neural Networks (NN) or Gaussian Process Regression (GPR). However, these approaches currently suffer from issues, such as need for large amounts of training data, lack of physics, and considerable extrapolation errors. In this work, we develop a reduced order model based on Proper Orthogonal Decomposition (POD), Empirical Cubature Method (ECM) and a geometrical transformation method with the following key features: (i) large shape variations of the microstructure are captured, (ii) only relatively small amounts of training data are necessary, and (iii) highly non-linear history-dependent behaviors are treated. The proposed framework is tested and examined in two numerical examples, involving two scales and large geometrical variations. In both cases, high speed-ups and accuracies are achieved while observing good extrapolation behavior.
Channel prediction is critical to address the channel aging issue in mobile scenarios. Existing channel prediction techniques are mainly designed for discrete channel prediction, which can only predict the future channel in a fixed time slot per frame, while the other intra-frame channels are usually recovered by interpolation. However, these approaches suffer from a serious interpolation loss, especially for mobile millimeter wave communications. To solve this challenging problem, we propose a tensor neural ordinary differential equation (TN-ODE) based continuous-time channel prediction scheme to realize the direct prediction of intra-frame channels. Specifically, inspired by the recently developed continuous mapping model named neural ODE in the field of machine learning, we first utilize the neural ODE model to predict future continuous-time channels. To improve the channel prediction accuracy and reduce computational complexity, we then propose the TN-ODE scheme to learn the structural characteristics of the high-dimensional channel by low dimensional learnable transform. Simulation results show that the proposed scheme is able to achieve higher intra-frame channel prediction accuracy than existing schemes.
We study self-regulating processes modeling biological transportation networks as presented in \cite{portaro2023}. In particular, we focus on the 1D setting for Dirichlet and Neumann boundary conditions. We prove an existence and uniqueness result under the assumption of positivity of the diffusivity $D$. We explore systematically various scenarios and gain insights into the behavior of $D$ and its impact on the studied system. This involves analyzing the system with a signed measure distribution of sources and sinks. Finally, we perform several numerical tests in which the solution $D$ touches zero, confirming the previous hints of local existence in particular cases.
A locked $t$-omino tiling is a tiling of a finite grid or torus by $t$-ominoes such that, if you remove any pair of tiles, the only way to fill in the remaining space with $t$-ominoes is to use the same two tiles in the exact same configuration as before. We exclude degenerate cases where there is only one tiling due to the grid/torus dimensions. Locked $t$-omino tilings arise as obstructions to popular political redistricting algorithms. It is a classic (and straightforward) result that grids do not admit locked 2-omino tilings. In this paper, we construct explicit locked 3-, 4-, and 5-omino tilings of grids of various sizes. While 3-omino tilings are plentiful, we find that 4- and 5-omino tilings are remarkably elusive. Using an exhaustive computational search, we find that, up to symmetries, the $10 \times 10$ grid admits a locked 4-omino tiling, the $20 \times 20$ grid admits a locked 5-omino tiling, and there are no others for any other grid size attempted. Finally, we construct an infinite family of locked $t$-omino tilings on tori with unbounded $t$.
Yes if the context, the list of variables defining the motion control problem, is dimensionally similar. Here we show that by modifying the problem formulation using dimensionless variables, we can re-use the optimal control law generated numerically for a specific system to a sub-space of dimensionally similar systems. This is demonstrated, with numerically generated optimal controllers, for the classic motion control problem of swinging-up a torque-limited inverted pendulum. We also discuss the concept of regime, a region in the space of context variables, that can help relax the condition on dimensional similarity. Futhermore, we discuss how applying dimensionnal scaling of the input and output of a context-specific policy is equivalent to substituing the new systems parameters in an analytical equation for dimentionnaly similar systems. It remains to be seen if this approach can also help generalizing policies for more complex high-dimensional problems.
We are interested in creating statistical methods to provide informative summaries of random fields through the geometry of their excursion sets. To this end, we introduce an estimator for the length of the perimeter of excursion sets of random fields on $\mathbb{R}^2$ observed over regular square tilings. The proposed estimator acts on the empirically accessible binary digital images of the excursion regions and computes the length of a piecewise linear approximation of the excursion boundary. The estimator is shown to be consistent as the pixel size decreases, without the need of any normalization constant, and with neither assumption of Gaussianity nor isotropy imposed on the underlying random field. In this general framework, even when the domain grows to cover $\mathbb{R}^2$, the estimation error is shown to be of smaller order than the side length of the domain. For affine, strongly mixing random fields, this translates to a multivariate Central Limit Theorem for our estimator when multiple levels are considered simultaneously. Finally, we conduct several numerical studies to investigate statistical properties of the proposed estimator in the finite-sample data setting.
The scattering of electromagnetic waves by three--dimensional periodic structures is important for many problems of crucial scientific and engineering interest. Due to the complexity and three-dimensional nature of these waves, the fast, accurate, and reliable numerical simulations of these are indispensable for engineers and scientists alike. For this, High Order Spectral methods are frequently employed and here we describe an algorithm in this class. Our approach is perturbative in nature where we view the deviation of the permittivity from a constant value as the deformation and we pursue regular perturbation theory. This work extends our previous contribution regarding the Helmholtz equation to the full vector Maxwell equations, by providing a rigorous analyticity theory, both in deformation size and spatial variable (provided that the permittivity is, itself, analytic).
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