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We present the library \texttt{lymph} for the finite element numerical discretization of coupled multi-physics problems. \texttt{lymph} is a Matlab library for the discretization of partial differential equations based on high-order discontinuous Galerkin methods on polytopal grids (PolyDG) for spatial discretization coupled with suitable finite-difference time marching schemes. The objective of the paper is to introduce the library by describing it in terms of installation, input/output data, and code structure, highlighting -- when necessary -- key implementation aspects related to the method. A user guide, proceeding step-by-step in the implementation and solution of a Poisson problem, is also provided. In the last part of the paper, we show the results obtained for several differential problems, namely the Poisson problem, the heat equation, and the elastodynamics system. Through these examples, we show the convergence properties and highlight some of the main features of the proposed method, i.e. geometric flexibility, high-order accuracy, and robustness with respect to heterogeneous physical parameters.

This paper describes $\pi2\text{vec}$, a method for representing behaviors of black box policies as feature vectors. The policy representations capture how the statistics of foundation model features change in response to the policy behavior in a task agnostic way, and can be trained from offline data, allowing them to be used in offline policy selection. This work provides a key piece of a recipe for fusing together three modern lines of research: Offline policy evaluation as a counterpart to offline RL, foundation models as generic and powerful state representations, and efficient policy selection in resource constrained environments.

Cross-validation is a widely used technique for assessing the performance of predictive models on unseen data. Many predictive models, such as Kernel-Based Partial Least-Squares (PLS) models, require the computation of $\mathbf{X}^{\mathbf{T}}\mathbf{X}$ and $\mathbf{X}^{\mathbf{T}}\mathbf{Y}$ using only training set samples from the input and output matrices, $\mathbf{X}$ and $\mathbf{Y}$, respectively. In this work, we present three algorithms that efficiently compute these matrices. The first one allows no column-wise preprocessing. The second one allows column-wise centering around the training set means. The third one allows column-wise centering and column-wise scaling around the training set means and standard deviations. Demonstrating correctness and superior computational complexity, they offer significant cross-validation speedup compared with straight-forward cross-validation and previous work on fast cross-validation - all without data leakage. Their suitability for parallelization is highlighted with an open-source Python implementation combining our algorithms with Improved Kernel PLS.

This paper presents a {\delta}-PI algorithm which is based on damped Newton method for the H{\infty} tracking control problem of unknown continuous-time nonlinear system. A discounted performance function and an augmented system are used to get the tracking Hamilton-Jacobi-Isaac (HJI) equation. Tracking HJI equation is a nonlinear partial differential equation, traditional reinforcement learning methods for solving the tracking HJI equation are mostly based on the Newton method, which usually only satisfies local convergence and needs a good initial guess. Based upon the damped Newton iteration operator equation, a generalized tracking Bellman equation is derived firstly. The {\delta}-PI algorithm can seek the optimal solution of the tracking HJI equation by iteratively solving the generalized tracking Bellman equation. On-policy learning and off-policy learning {\delta}-PI reinforcement learning methods are provided, respectively. Off-policy version {\delta}-PI algorithm is a model-free algorithm which can be performed without making use of a priori knowledge of the system dynamics. NN-based implementation scheme for the off-policy {\delta}-PI algorithms is shown. The suitability of the model-free {\delta}-PI algorithm is illustrated with a nonlinear system simulation.

This paper proposes to develop a new variant of the two-time-scale stochastic approximation to find the roots of two coupled nonlinear operators, assuming only noisy samples of these operators can be observed. Our key idea is to leverage the classic Ruppert-Polyak averaging technique to dynamically estimate the operators through their samples. The estimated values of these averaging steps will then be used in the two-time-scale stochastic approximation updates to find the desired solution. Our main theoretical result is to show that under the strongly monotone condition of the underlying nonlinear operators the mean-squared errors of the iterates generated by the proposed method converge to zero at an optimal rate $\mathcal{O}(1/k)$, where $k$ is the number of iterations. Our result significantly improves the existing result of two-time-scale stochastic approximation, where the best known finite-time convergence rate is $\mathcal{O}(1/k^{2/3})$.

Adhesive and quasiadhesive categories provide a general framework for the study of algebraic graph rewriting systems. In a quasiadhesive category any two regular subobjects have a join which is again a regular subobject. Vice versa, if regular monos are adhesive, then the existence of a regular join for any pair of regular subobjects entails quasiadhesivity. It is also known (quasi)adhesive categories can be embedded in a Grothendieck topos via a functor preserving pullbacks and pushouts along (regular) monomorphisms. In this paper we extend these results to $\mathcal{M}, \mathcal{N}$-adhesive categories, a concept recently introduced to generalize the notion of (quasi)adhesivity. We introduce the notion of $\mathcal{N}$-adhesive morphism, which allows us to express $\mathcal{M}, \mathcal{N}$-adhesivity as a condition on the subobjects's posets. Moreover, $\mathcal{N}$-adhesive morphisms allows us to show how an $\mathcal{M},\mathcal{N}$-adhesive category can be embedded into a Grothendieck topos, preserving pullbacks and $\mathcal{M}, \mathcal{N}$-pushouts.

We design an algorithm for computing the $L$-series associated to an Anderson $t$-motives, exhibiting quasilinear complexity with respect to the target precision. Based on experiments, we conjecture that the order of vanishing at $T=1$ of the $v$-adic $L$-series of a given Anderson $t$-motive with good reduction does not depend on the finite place $v$.

Center-based clustering has attracted significant research interest from both theory and practice. In many practical applications, input data often contain background knowledge that can be used to improve clustering results. In this work, we build on widely adopted $k$-center clustering and model its input background knowledge as must-link (ML) and cannot-link (CL) constraint sets. However, most clustering problems including $k$-center are inherently $\mathcal{NP}$-hard, while the more complex constrained variants are known to suffer severer approximation and computation barriers that significantly limit their applicability. By employing a suite of techniques including reverse dominating sets, linear programming (LP) integral polyhedron, and LP duality, we arrive at the first efficient approximation algorithm for constrained $k$-center with the best possible ratio of 2. We also construct competitive baseline algorithms and empirically evaluate our approximation algorithm against them on a variety of real datasets. The results validate our theoretical findings and demonstrate the great advantages of our algorithm in terms of clustering cost, clustering quality, and running time.

Diffusion models have emerged as powerful generative tools, rivaling GANs in sample quality and mirroring the likelihood scores of autoregressive models. A subset of these models, exemplified by DDIMs, exhibit an inherent asymmetry: they are trained over $T$ steps but only sample from a subset of $T$ during generation. This selective sampling approach, though optimized for speed, inadvertently misses out on vital information from the unsampled steps, leading to potential compromises in sample quality. To address this issue, we present the S$^{2}$-DMs, which is a new training method by using an innovative $L_{skip}$, meticulously designed to reintegrate the information omitted during the selective sampling phase. The benefits of this approach are manifold: it notably enhances sample quality, is exceptionally simple to implement, requires minimal code modifications, and is flexible enough to be compatible with various sampling algorithms. On the CIFAR10 dataset, models trained using our algorithm showed an improvement of 3.27% to 14.06% over models trained with traditional methods across various sampling algorithms (DDIMs, PNDMs, DEIS) and different numbers of sampling steps (10, 20, ..., 1000). On the CELEBA dataset, the improvement ranged from 8.97% to 27.08%. Access to the code and additional resources is provided in the github.

Click-through rate (CTR) prediction plays a critical role in recommender systems and online advertising. The data used in these applications are multi-field categorical data, where each feature belongs to one field. Field information is proved to be important and there are several works considering fields in their models. In this paper, we proposed a novel approach to model the field information effectively and efficiently. The proposed approach is a direct improvement of FwFM, and is named as Field-matrixed Factorization Machines (FmFM, or $FM^2$). We also proposed a new explanation of FM and FwFM within the FmFM framework, and compared it with the FFM. Besides pruning the cross terms, our model supports field-specific variable dimensions of embedding vectors, which acts as soft pruning. We also proposed an efficient way to minimize the dimension while keeping the model performance. The FmFM model can also be optimized further by caching the intermediate vectors, and it only takes thousands of floating-point operations (FLOPs) to make a prediction. Our experiment results show that it can out-perform the FFM, which is more complex. The FmFM model's performance is also comparable to DNN models which require much more FLOPs in runtime.