Yu Qiao

Hengguang Li, Victor Nistor, Yu Qiao

Let $Ω\subset \RR^d$, $d \geqslant 1$, be a bounded domain with piecewise smooth boundary $\partial Ω$ and let $U$ be an open subset of a Banach space $Y$. Motivated by questions in "Uncertainty Quantification," we consider a parametric family $P = (P_y)_{y \in U}$ of uniformly strongly elliptic, second order partial differential operators $P_y$ on $Ω$. We allow jump discontinuities in the coefficients. We establish a regularity result for the solution $u: Ω\times U \to \RR$ of the parametric, elliptic boundary value/transmission problem $P_y u_y = f_y$, $y \in U$, with mixed Dirichlet-Neumann boundary conditions in the case when the boundary and the interface are smooth and in the general case for $d=2$. Our regularity and well-posedness results are formulated in a scale of broken weighted Sobolev spaces $\hat\maK^{m+1}_{a+1}(Ω)$ of Babuška-Kondrat'ev type in $Ω$, possibly augmented by some locally constant functions. This implies that the parametric, elliptic PDEs $(P_y)_{y \in U}$ admit a shift theorem that is uniform in the parameter $y\in U$. In turn, this then leads to $h^m$-quasi-optimal rates of convergence (i.e. algebraic orders of convergence) for the Galerkin approximations of the solution $u$, where the approximation spaces are defined using the "polynomial chaos expansion" of $u$ with respect to a suitable family of tensorized Lagrange polynomials, following the method developed by Cohen, Devore, and Schwab (2010).

Enshen Zhou, Yiran Qin, Zhenfei Yin et al.

It is a long-lasting goal to design a generalist-embodied agent that can follow diverse instructions in human-like ways. However, existing approaches often fail to steadily follow instructions due to difficulties in understanding abstract and sequential natural language instructions. To this end, we introduce MineDreamer, an open-ended embodied agent built upon the challenging Minecraft simulator with an innovative paradigm that enhances instruction-following ability in low-level control signal generation. Specifically, MineDreamer is developed on top of recent advances in Multimodal Large Language Models (MLLMs) and diffusion models, and we employ a Chain-of-Imagination (CoI) mechanism to envision the step-by-step process of executing instructions and translating imaginations into more precise visual prompts tailored to the current state; subsequently, the agent generates keyboard-and-mouse actions to efficiently achieve these imaginations, steadily following the instructions at each step. Extensive experiments demonstrate that MineDreamer follows single and multi-step instructions steadily, significantly outperforming the best generalist agent baseline and nearly doubling its performance. Moreover, qualitative analysis of the agent's imaginative ability reveals its generalization and comprehension of the open world.