AI 中文总结
针对带Neumann边界条件的稳态扩散种群增长模型，研究其最优数据驱动解，刻画松弛代价泛函、证明解的存在性与一致性，并验证数据集演化时解的Γ-收敛性。
AI 中文摘要
我们研究有界域Ω⊂ℝᴺ中带Neumann边界条件的稳态扩散种群增长模型-Δu = r u的最优数据驱动解。我们并未预设位置x、人均净增长率r与种群规模u之间的函数关系，而是寻找一对属于H¹(Ω)×L^∞(Ω)的(u,r)，使其以代价泛函I与额外惩罚项衡量的方式最优拟合给定数据集。我们通过证明松弛代价泛函sc⁻I的密度为关于变量r的部分下凸包，刻画了该松弛泛函，并证明了最优数据驱动解的存在性。此外，我们建立了一致性结果，将-Δu = ϱ(x,u)u的传统解与数据集源自函数关系(x,u)↦ϱ(x,u)的最优数据驱动解进行对比。最后，我们证明了当数据集演化时，最优解通过相关代价泛函的Γ-收敛实现收敛。
英文摘要
We study optimal data-driven solutions for the stationary diffusive population growth model $-Δu = r u$ in a bounded domain $Ω\subset\mathbb R^N$ with Neumann boundary conditions. Instead of prescribing a functional relation between the position $x$, the net per-capita growth rate $r$ and the population size $u$, we look for a pair $(u,r)\in H^1(Ω)\times L^\infty(Ω)$ that fits a given data set in an optimal way measured by a cost functional $I$ and an additional penalty term. We characterize the relaxed cost functional sc$^- I$ by showing that its density is given as the partial lower convex envelope with respect to the variable $r$, and prove the existence of optimal data-driven solutions. Furthermore, we establish a consistency result comparing conventional solutions of $-Δu = \varrho(x,u)u$ with optimal data-driven solutions where the data set stems from the functional relation $(x,u)\mapsto \varrho(x,u)$. Finally, as data sets evolve, we prove the convergence of optimal solutions via the $Γ$-convergence of the associated cost functionals.
