Peridynamic (PD) theory is significant and promising in engineering and materials science; however, it imposes challenges owing to the enormous computational cost caused by its nonlocality. Our main contribution, which overcomes the restrictions of the existing fast method, is a general computational framework for the linear bond-based peridynamic models based on the meshfree method, called the matrix-structure-based fast method (MSBFM), which is suitable for the general case, including 2D/3D problems, and static/dynamic issues, as well as problems with general boundary conditions, in particular, problems with crack propagation. Consequently, we provide a general calculation flow chart. The proposed computational framework is practical and easily embedded into the existing computational algorithm. With this framework, the computational cost is reduced from $O(N^2)$ to $O(N\log N)$, and the storage request is reduced from $O(N^2)$ to $O(N)$, where N is the degree of freedom. Finally, the vast reduction of the computational and memory requirement is verified by numerical examples.
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The multi-armed bandit(MAB) problem is a simple yet powerful framework that has been extensively studied in the context of decision-making under uncertainty. In many real-world applications, such as robotic applications, selecting an arm corresponds to a physical action that constrains the choices of the next available arms (actions). Motivated by this, we study an extension of MAB called the graph bandit, where an agent travels over a graph to maximize the reward collected from different nodes. The graph defines the agent's freedom in selecting the next available nodes at each step. We assume the graph structure is fully available, but the reward distributions are unknown. Built upon an offline graph-based planning algorithm and the principle of optimism, we design a learning algorithm, G-UCB, that balances long-term exploration-exploitation using the principle of optimism. We show that our proposed algorithm achieves $O(\sqrt{|S|T\log(T)}+D|S|\log T)$ learning regret, where $|S|$ is the number of nodes and $D$ is the diameter of the graph, which matches the theoretical lower bound $\Omega(\sqrt{|S|T})$ up to logarithmic factors. To our knowledge, this result is among the first tight regret bounds in non-episodic, un-discounted learning problems with known deterministic transitions. Numerical experiments confirm that our algorithm outperforms several benchmarks.
During the past decades, evolutionary computation (EC) has demonstrated promising potential in solving various complex optimization problems of relatively small scales. Nowadays, however, ongoing developments in modern science and engineering are bringing increasingly grave challenges to the conventional EC paradigm in terms of scalability. As problem scales increase, on the one hand, the encoding spaces (i.e., dimensions of the decision vectors) are intrinsically larger; on the other hand, EC algorithms often require growing numbers of function evaluations (and probably larger population sizes as well) to work properly. To meet such emerging challenges, not only does it require delicate algorithm designs, but more importantly, a high-performance computing framework is indispensable. Hence, we develop a distributed GPU-accelerated algorithm library -- EvoX. First, we propose a generalized workflow for implementing general EC algorithms. Second, we design a scalable computing framework for running EC algorithms on distributed GPU devices. Third, we provide user-friendly interfaces to both researchers and practitioners for benchmark studies as well as extended real-world applications. To comprehensively assess the performance of EvoX, we conduct a series of experiments, including: (i) scalability test via numerical optimization benchmarks with problem dimensions/population sizes up to millions; (ii) acceleration test via a neuroevolution task with multiple GPU nodes; (iii) extensibility demonstration via the application to reinforcement learning tasks on the OpenAI Gym. The code of EvoX is available at //github.com/EMI-Group/EvoX.
Pre-trained wav2vec2.0 model has been proved its effectiveness for speaker recognition. However, current feature processing methods are focusing on classical pooling on the output features of the pre-trained wav2vec2.0 model, such as mean pooling, max pooling etc. That methods take the features as the independent and irrelevant units, ignoring the inter-relationship among all the features, and do not take the features as an overall representation of a speaker. Gated Recurrent Unit (GRU), as a feature fusion method, can also be considered as a complicated pooling technique, mainly focuses on the temporal information, which may show poor performance in some situations that the main information is not on the temporal dimension. In this paper, we investigate the graph neural network (GNN) as a backend processing module based on wav2vec2.0 framework to provide a solution for the mentioned matters. The GNN takes all the output features as the graph signal data and extracts the related graph structure information of features for speaker recognition. Specifically, we first give a simple proof that the GNN feature fusion method can outperform than the mean, max, random pooling methods and so on theoretically. Then, we model the output features of wav2vec2.0 as the vertices of a graph, and construct the graph adjacency matrix by graph attention network (GAT). Finally, we follow the message passing neural network (MPNN) to design our message function, vertex update function and readout function to transform the speaker features into the graph features. The experiments show our performance can provide a relative improvement compared to the baseline methods. Code is available at xxx.
Learning precise surrogate models of complex computer simulations and physical machines often require long-lasting or expensive experiments. Furthermore, the modeled physical dependencies exhibit nonlinear and nonstationary behavior. Machine learning methods that are used to produce the surrogate model should therefore address these problems by providing a scheme to keep the number of queries small, e.g. by using active learning and be able to capture the nonlinear and nonstationary properties of the system. One way of modeling the nonstationarity is to induce input-partitioning, a principle that has proven to be advantageous in active learning for Gaussian processes. However, these methods either assume a known partitioning, need to introduce complex sampling schemes or rely on very simple geometries. In this work, we present a simple, yet powerful kernel family that incorporates a partitioning that: i) is learnable via gradient-based methods, ii) uses a geometry that is more flexible than previous ones, while still being applicable in the low data regime. Thus, it provides a good prior for active learning procedures. We empirically demonstrate excellent performance on various active learning tasks.
A sequential quadratic optimization algorithm is proposed for solving smooth nonlinear equality constrained optimization problems in which the objective function is defined by an expectation of a stochastic function. The algorithmic structure of the proposed method is based on a step decomposition strategy that is known in the literature to be widely effective in practice, wherein each search direction is computed as the sum of a normal step (toward linearized feasibility) and a tangential step (toward objective decrease in the null space of the constraint Jacobian). However, the proposed method is unique from others in the literature in that it both allows the use of stochastic objective gradient estimates and possesses convergence guarantees even in the setting in which the constraint Jacobians may be rank deficient. The results of numerical experiments demonstrate that the algorithm offers superior performance when compared to popular alternatives.
We consider the problem of minimizing a non-convex function over a smooth manifold $\mathcal{M}$. We propose a novel algorithm, the Orthogonal Directions Constrained Gradient Method (ODCGM) which only requires computing a projection onto a vector space. ODCGM is infeasible but the iterates are constantly pulled towards the manifold, ensuring the convergence of ODCGM towards $\mathcal{M}$. ODCGM is much simpler to implement than the classical methods which require the computation of a retraction. Moreover, we show that ODCGM exhibits the near-optimal oracle complexities $\mathcal{O}(1/\varepsilon^2)$ and $\mathcal{O}(1/\varepsilon^4)$ in the deterministic and stochastic cases, respectively. Furthermore, we establish that, under an appropriate choice of the projection metric, our method recovers the landing algorithm of Ablin and Peyr\'e (2022), a recently introduced algorithm for optimization over the Stiefel manifold. As a result, we significantly extend the analysis of Ablin and Peyr\'e (2022), establishing near-optimal rates both in deterministic and stochastic frameworks. Finally, we perform numerical experiments which shows the efficiency of ODCGM in a high-dimensional setting.
Prescriptive Process Monitoring is a prominent problem in Process Mining, which consists in identifying a set of actions to be recommended with the goal of optimising a target measure of interest or Key Performance Indicator (KPI). One challenge that makes this problem difficult is the need to provide Prescriptive Process Monitoring techniques only based on temporally annotated (process) execution data, stored in, so-called execution logs, due to the lack of well crafted and human validated explicit models. In this paper we aim at proposing an AI based approach that learns, by means of Reinforcement Learning (RL), an optimal policy (almost) only from the observation of past executions and recommends the best activities to carry on for optimizing a KPI of interest. This is achieved first by learning a Markov Decision Process for the specific KPIs from data, and then by using RL training to learn the optimal policy. The approach is validated on real and synthetic datasets and compared with off-policy Deep RL approaches. The ability of our approach to compare with, and often overcome, Deep RL approaches provides a contribution towards the exploitation of white box RL techniques in scenarios where only temporal execution data are available.
This paper is concerned with low-rank matrix optimization, which has found a wide range of applications in machine learning. This problem in the special case of matrix sensing has been studied extensively through the notion of Restricted Isometry Property (RIP), leading to a wealth of results on the geometric landscape of the problem and the convergence rate of common algorithms. However, the existing results can handle the problem in the case with a general objective function subject to noisy data only when the RIP constant is close to 0. In this paper, we develop a new mathematical framework to solve the above-mentioned problem with a far less restrictive RIP constant. We prove that as long as the RIP constant of the noiseless objective is less than $1/3$, any spurious local solution of the noisy optimization problem must be close to the ground truth solution. By working through the strict saddle property, we also show that an approximate solution can be found in polynomial time. We characterize the geometry of the spurious local minima of the problem in a local region around the ground truth in the case when the RIP constant is greater than $1/3$. Compared to the existing results in the literature, this paper offers the strongest RIP bound and provides a complete theoretical analysis on the global and local optimization landscapes of general low-rank optimization problems under random corruptions from any finite-variance family.
We present DiffXPBD, a novel and efficient analytical formulation for the differentiable position-based simulation of compliant constrained dynamics (XPBD). Our proposed method allows computation of gradients of numerous parameters with respect to a goal function simultaneously leveraging a performant simulation model. The method is efficient, thus enabling differentiable simulations of high resolution geometries and degrees of freedom (DoFs). Collisions are naturally included in the framework. Our differentiable model allows a user to easily add additional optimization variables. Every control variable gradient requires the computation of only a few partial derivatives which can be computed using automatic differentiation code. We demonstrate the efficacy of the method with examples such as elastic material parameter estimation, initial value optimization, optimizing for underlying body shape and pose by only observing the clothing, and optimizing a time-varying external force sequence to match sparse keyframe shapes at specific times. Our approach demonstrates excellent efficiency and we demonstrate this on high resolution meshes with optimizations involving over 26 million degrees of freedom. Making an existing solver differentiable requires only a few modifications and the model is compatible with both modern CPU and GPU multi-core hardware.
Recent years have witnessed the enormous success of low-dimensional vector space representations of knowledge graphs to predict missing facts or find erroneous ones. Currently, however, it is not yet well-understood how ontological knowledge, e.g. given as a set of (existential) rules, can be embedded in a principled way. To address this shortcoming, in this paper we introduce a framework based on convex regions, which can faithfully incorporate ontological knowledge into the vector space embedding. Our technical contribution is two-fold. First, we show that some of the most popular existing embedding approaches are not capable of modelling even very simple types of rules. Second, we show that our framework can represent ontologies that are expressed using so-called quasi-chained existential rules in an exact way, such that any set of facts which is induced using that vector space embedding is logically consistent and deductively closed with respect to the input ontology.
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