Abstract Math Word Problems (MWP) solving involves language comprehension and mathematical reasoning. Most of the existing models are primarily based on deep learning methods. However, deep learning models often exhibit large model scales (e.g., large number of parameters), high feature dimensions, and an imbalance between accuracy and computational efficiency. To address these issues, We propose a novel dual-decoder model for solving Mathematical Word Problems (MWP), which is constructed using a Lie Group intrinsic mean feature matrix and named the Dual-Decoder Neural Symbolic Machine (DDNSM). This model projects data samples onto a Lie Group manifold to effectively reduce the output feature dimensions of the encoder through a feature matrix built around the Lie Group intrinsic mean, thereby reducing the computational load in the decoding stage and improving model inference efficiency. Additionally, we designed a dual-decoder system, comprising a decoder based on global and local attention mechanisms and another structured around a tree built from the Lie Group intrinsic mean. Extensive experiments on multiple challenging public datasets demonstrate that our model achieves competitive results in MWP tasks, achieving competitive results in MWP tasks with improved accuracy and computational efficiency.

Traditional ionospheric models were mostly constructed based on a single layer assumption from Global Navigation Satellite System (GNSS) observations, while it cannot capture vertical information of the ionosphere. This study proposes a new method to construct a double-layer ionospheric model based on constraints from a three-dimensional ionospheric model, whereby the bottom and topside ionospheric TEC can be represented by two spherical harmonic (SH) functions. The new improved model allows two SH functions to capture the spatiotemporal TEC variations across the vertical range of the ionosphere. The determination of the two thin layer heights (TLHs) in the double-layer model is achieved through minimum mapping function error. Moreover, the performance of the new model is validated using GPS, BDS, and Galileo data from the International GNSS Server (IGS) Network, and compared with the global ionospheric map (GIM). During the experiment period, the results indicate that (1) the TLHs of the bottom and topside ionosphere exhibit distinct spatiotemporal trends with the optimal global heights as 350 km and 650 km, respectively; (2) the average relative accuracies of the bottom and topside ionospheric models are up to 86.80 % and 85.33 %, respectively; (3) the new model demonstrates an improvement of approximately 20–27 % in terms of TEC when compared to the GIM model, with the RMS better than 4.64 TECU, 2.99 TECU, and 3.61 TECU in the low, middle, and high latitudes, respectively; and (4) with the increase of geomagnetic activity, the performance of the double-layer model shows a slight decline, but its relative accuracy can still reach over 84.8%.

Current fully supervised facial landmark detection methods have progressed rapidly and achieved remarkable performance. However, they still suffer when coping with faces under large poses and heavy occlusions for inaccurate facial shape constraints and insufficient labeled training samples. In this article, we propose a semisupervised framework, that is, a self-calibrated pose attention network (SCPAN) to achieve more robust and precise facial landmark detection in challenging scenarios. To be specific, a boundary-aware landmark intensity (BALI) field is proposed to model more effective facial shape constraints by fusing boundary and landmark intensity field information. Moreover, a self-calibrated pose attention (SCPA) model is designed to provide a self-learned objective function that enforces intermediate supervision without label information by introducing a self-calibrated mechanism and a pose attention mask. We show that by integrating the BALI fields and SCPA model into a novel SCPAN, more facial prior knowledge can be learned and the detection accuracy and robustness of our method for faces with large poses and heavy occlusions have been improved. The experimental results obtained for challenging benchmark datasets demonstrate that our approach outperforms state-of-the-art methods in the literature.

In real-world project scheduling applications, activity durations are often uncertain. Proactive scheduling can effectively cope with the duration uncertainties, by generating robust baseline solutions according to a priori stochastic knowledge. However, most of the existing proactive approaches assume that the duration uncertainty of an activity is not related to its scheduled start time, which may not hold in many real-world scenarios. In this paper, we relax this assumption by allowing the duration uncertainty to be time-dependent, which is caused by the uncertainty of whether the activity can be executed on each time slot. We propose a stochastic optimization model to find an optimal Partial-order Schedule (POS) that minimizes the expected makespan. This model can cover both the time-dependent uncertainty studied in this paper and the traditional time-independent duration uncertainty. To circumvent the underlying complexity in evaluating a given solution, we approximate the stochastic optimization model based on Sample Average Approximation (SAA). Finally, we design two efficient branch-and-bound algorithms to solve the NP-hard SAA problem. Empirical evaluation confirms that our approach can generate high-quality proactive solutions for a variety of uncertainty distributions.

In this paper, we consider the challenging problem of riskaware proactive scheduling with the objective of minimizing robust makespan. State-of-the-art approaches based on probabilistic constrained optimization lead to Mixed Integer Linear Programs that must be heuristically approximated. We optimize the robust makespan via a coherent risk measure, Conditional Value-at-Risk (CVaR). Since traditional CVaR optimization approaches assuming linear spaces does not suit our problem, we propose a general branch-and-bound framework for combinatorial CVaR minimization. We then design an approximate complete algorithm, and employ resource reasoning to enable constraint propagation for multiple samples. Empirical results show that our algorithm outperforms state-of-the-art approaches with higher solution quality.

Most of the existing proactive scheduling approaches assume the durations of activities can be described by independent random variables that have no relation with time. We deal with the more challenging problem where the duration uncertainty is related to the scheduled time period. We propose a sampling based approach by extending the Consensus method from stochastic optimization. Experimental results show the effectiveness of our approach in solution quality and stability.

In agent-based simulation, emergent equilibrium describes the macroscopic steady states of agents' interactions. While the state of individual agents might be changing, the collective behavior pattern remains the same in macroscopic equilibrium states. Traditionally, these emergent equilibriums are calculated using Monte Carlo methods. However, these methods require thousands of repeated simulation runs, which are extremely time-consuming. In this paper, we propose a novel three-layer framework to efficiently compute emergent equilibriums. The framework consists of a macro-level pseudo-arclength equilibrium solver (PAES), a micro-level simulator (MLS) and a macro-micro bridge (MMB). It can adaptively explore parameter space and recursively compute equilibrium states using the predictor-corrector scheme. We apply the framework to the popular opinion dynamics and labour market models. The experimental results show that our framework outperformed Monte Carlo experiments in terms of computation efficiency while maintaining the accuracy.