Rui Du

Yangkun Chen, Joseph Suarez, Junjie Zhang et al.

We present the results of the second Neural MMO challenge, hosted at IJCAI 2022, which received 1600+ submissions. This competition targets robustness and generalization in multi-agent systems: participants train teams of agents to complete a multi-task objective against opponents not seen during training. The competition combines relatively complex environment design with large numbers of agents in the environment. The top submissions demonstrate strong success on this task using mostly standard reinforcement learning (RL) methods combined with domain-specific engineering. We summarize the competition design and results and suggest that, as an academic community, competitions may be a powerful approach to solving hard problems and establishing a solid benchmark for algorithms. We will open-source our benchmark including the environment wrapper, baselines, a visualization tool, and selected policies for further research.

Zetao Ma, Jingrun Chen, Rui Du et al.

In this work, we develop a numerical homogenization approach for the fully nonlinear Landau-Lifshitz equation with rough coefficients, including non-periodicity and nonseparable scales. Direct numerical resolution of such multiscale problems on fine meshes incurs prohibitive computational costs. To address this challenge, we propose an efficient coarse scale approximation through localized basis functions derived from energy minimization within the Generalized Rough Polyharmonic Splines (GRPS) framework. These basis functions preserve critical multiscale features while operating on a computationally tractable coarse mesh. The nonlinear, vectorial, and non-symmetric nature of the Landau-Lifshitz equation necessitates careful design of variational formulations for basis construction. We introduce several such formulations, each tailored to specific structural aspects of the problem. Through systematic numerical experiments, we demonstrate that our approach achieves significant computational savings without compromising accuracy, offering a robust framework for simulating multiscale magnetic systems with complex microstructures.

Jingrun Chen, Rui Du, Panchi Li et al.

Solving partial differential equations in high dimensions by deep neural network has brought significant attentions in recent years. In many scenarios, the loss function is defined as an integral over a high-dimensional domain. Monte-Carlo method, together with the deep neural network, is used to overcome the curse of dimensionality, while classical methods fail. Often, a deep neural network outperforms classical numerical methods in terms of both accuracy and efficiency. In this paper, we propose to use quasi-Monte Carlo sampling, instead of Monte-Carlo method to approximate the loss function. To demonstrate the idea, we conduct numerical experiments in the framework of deep Ritz method proposed by Weinan E and Bing Yu. For the same accuracy requirement, it is observed that quasi-Monte Carlo sampling reduces the size of training data set by more than two orders of magnitude compared to that of MC method. Under some assumptions, we prove that quasi-Monte Carlo sampling together with the deep neural network generates a convergent series with rate proportional to the approximation accuracy of quasi-Monte Carlo method for numerical integration. Numerically the fitted convergence rate is a bit smaller, but the proposed approach always outperforms Monte Carlo method. It is worth mentioning that the convergence analysis is generic whenever a loss function is approximated by the quasi-Monte Carlo method, although observations here are based on deep Ritz method.