In this paper, we consider Ricci flow with curvature $L^p$ bound for $p>n/4+1$ and bounded scalar curvature. We show an isoperimetric inequality along such Ricci flow, which generalizes Tian-Zhang's result \cite{TZq14}. We also consider the convergence of this flow and show that there exists a sequence of time slice metrics converging to a Ricci soliton outside a closed singular set with codimension $2p$, and that the convergence is smooth outside the singular set. Moreover, in the K\"ahler-Ricci flow case, we can estimate the Hausdorff measure of this singular set.
成果 2
数学学院教师陈秀琼与清华大学孙泽钜博士、陶飏天泽博士，清华大学和北京雁栖湖应用数学研究院的丘成栋教授合作的论文《A uniform framework of Yau-Yau algorithm based on deep learning with the capability of overcoming the curse of dimensionality》在IEEE Transactions on Automatic Control 发表
In numerous application areas, high-dimensional nonlinear filtering is still a challenging problem. The introduction of deep learning and neural networks has improved the efficiency of classical algorithms and they perform well in many practical tasks. However, a theoretical interpretation of their feasibility is still lacking. In this article, we exploit the representational ability of recurrent neural networks (RNNs) and provide a computationally efficient and optimal framework for nonlinear filter design based on the Yau–Yau algorithm and RNNs. Theoretically, it can be proved that the size of the neural network required in this algorithm increases only polynomially rather than exponentially with dimension, which implies that the Yau–Yau algorithm based on RNNs has the ability to overcome the curse of dimensionality. Numerical results also show that our method is more competitive than classical algorithms for high-dimensional problems.
成果 3
数学学院教师陈秀琼与清华大学孙泽钜博士，清华大学和北京雁栖湖应用数学研究院的丘成栋教授合作的论文《Recurrent neural network spectral method and its application in stable filtering problems》在Automatica发表
Solving high-dimensional filtering problems with high nonlinearity is essential for controlling complicated systems and for data assimilation. In this paper, we propose a novel recurrent neural network spectral method (RNNSM) to address this kind of problems, especially for the systems with additional stability properties. As a combination of modern deep learning strategy and classical spectral method, the proposed algorithm integrates the advantages of both. On the one hand, by exploiting the approximation capability of recurrent neural networks, RNNSM can overcome the obstacles that classical spectral methods face in high-dimensional problems, and obtain a heuristic approach to finding the optimal orthonormal basis in spectral methods; on the other hand, with the theoretical foundation of spectral methods, RNNSM provides a more reasonable mathematical interpretation of neural network-based filtering algorithms and bridges the gap between practical performance and theoretical convergence. Finally, the efficiency of RNNSM is also verified by numerical experiments.
