Physical Information Neural Networks for Solving High-index Differential-algebraic Equation Systems Based on Radau Methods

As is well known, differential algebraic equations (DAEs), which are able to describe dynamic changes and underlying constraints, have been widely applied in engineering fields such as fluid dynamics, multi-body dynamics, mechanical systems and control theory. In practical physical modeling within these domains, the systems often generate high-index DAEs. Classical implicit numerical methods typically result in varying order reduction of numerical accuracy when solving high-index systems.~Recently, the physics-informed neural network (PINN) has gained attention for solving DAE systems. However, it faces challenges like the inability to directly solve high-index systems, lower predictive accuracy, and weaker generalization capabilities. In this paper, we propose a PINN computational framework, combined Radau IIA numerical method with a neural network structure via the attention mechanisms, to directly solve high-index DAEs. Furthermore, we employ a domain decomposition strategy to enhance solution accuracy. We conduct numerical experiments with two classical high-index systems as illustrative examples, investigating how different orders of the Radau IIA method affect the accuracy of neural network solutions. The experimental results demonstrate that the PINN based on a 5th-order Radau IIA method achieves the highest level of system accuracy. Specifically, the absolute errors for all differential variables remains as low as $10^{-6}$, and the absolute errors for algebraic variables is maintained at $10^{-5}$, surpassing the results found in existing literature. Therefore, our method exhibits excellent computational accuracy and strong generalization capabilities, providing a feasible approach for the high-precision solution of larger-scale DAEs with higher indices or challenging high-dimensional partial differential algebraic equation systems.

Reinforcement Learning Guided Multi-Objective Exam Paper Generation

To reduce the repetitive and complex work of instructors, exam paper generation (EPG) technique has become a salient topic in the intelligent education field, which targets at generating high-quality exam paper automatically according to instructor-specified assessment criteria. The current advances utilize the ability of heuristic algorithms to optimize several well-known objective constraints, such as difficulty degree, number of questions, etc., for producing optimal solutions. However, in real scenarios, considering other equally relevant objectives (e.g., distribution of exam scores, skill coverage) is extremely important. Besides, how to develop an automatic multi-objective solution that finds an optimal subset of questions from a huge search space of large-sized question datasets and thus composes a high-quality exam paper is urgent but non-trivial. To this end, we skillfully design a reinforcement learning guided Multi-Objective Exam Paper Generation framework, termed MOEPG, to simultaneously optimize three exam domain-specific objectives including difficulty degree, distribution of exam scores, and skill coverage. Specifically, to accurately measure the skill proficiency of the examinee group, we first employ deep knowledge tracing to model the interaction information between examinees and response logs. We then design the flexible Exam Q-Network, a function approximator, which automatically selects the appropriate question to update the exam paper composition process. Later, MOEPG divides the decision space into multiple subspaces to better guide the updated direction of the exam paper. Through extensive experiments on two real-world datasets, we demonstrate that MOEPG is feasible in addressing the multiple dilemmas of exam paper generation scenario.