发表机构
Massachusetts Institute of Technology（麻省理工学院）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究闭流形上谱非负相互作用能量的Wasserstein梯度流，证明能量间隙以o(t^{-1})速率收敛，并通过构造线性化与非线性解验证该速率的最优性。
AI 中文摘要
我们研究了闭流形$M$上相互作用能量\\[ \mathsf E[μ] = \frac12\iint_{M\times M}K(x,y)\\,\mathrm dμ(x)\\,\mathrm dμ(y) \\]的Wasserstein梯度流的长时间行为。对于在Laplace特征基下具有非负谱系数的对角核，我们证明了相对熵与能量间隙之间满足一个微分不等式。因此，对于任意非负初始密度$u_0\in L^p(M)$，$p>1$，能量间隙在时间上可积，并且满足\\[ \mathsf E[μ_t]-\mathsf E_{\min}=o(t^{-1})。\\]若所有谱系数均为正，则该流弱收敛于常数测度。这些相互作用能量在Wasserstein空间中不必是测地凸的，且相应的流不包含扩散；因此，其全局收敛性不能由标准的Wasserstein梯度流理论推出。我们的结果所涵盖的核包括球面上的带状核、Transformer模型中出现的核、正则化Riesz核以及逆分数Laplacian核。\n\n我们还研究了$o(t^{-1})$速率的尖锐性。对于此类中任意具有无穷多个正谱系数的光滑核以及任意$δ>0$，我们构造了线性化流的一个解，其能量在趋于无穷的时间序列上可与$t^{-1-δ}$相比较。此外，对于任意$δ>0$，通过在平坦环面上选择合适的逆分数Laplacian核，我们构造了非线性Wasserstein梯度流的一个精确解，其能量可与$t^{-1-δ}$相比较。非线性构造基于二进Fourier系数块演化的时间一致估计和块状能量保持论证。这些估计还给出了非线性Wasserstein梯度流与其线性化之间的时间一致定量比较。
英文摘要
We study the long-time behavior of Wasserstein gradient flows for interaction energies \[ \mathsf E[μ] = \frac12\iint_{M\times M}K(x,y)\,\mathrm dμ(x)\,\mathrm dμ(y) \] on a closed manifold $M$. For kernels diagonal in a Laplace eigenbasis with nonnegative spectral coefficients, we prove a differential inequality relating the relative entropy to the energy gap. Consequently, for any nonnegative initial density $u_0\in L^p(M)$, $p>1$, the energy gap is integrable in time and satisfies \[ \mathsf E[μ_t]-\mathsf E_{\min}=o(t^{-1}). \] If all spectral coefficients are positive, the flow converges weakly to the constant measure. These interaction energies need not be geodesically convex in Wasserstein space, and the associated flows contain no diffusion; their global convergence therefore does not follow from standard Wasserstein gradient flow theory. The kernels covered by our results include zonal kernels on spheres, kernels arising in transformer models, regularized Riesz kernels, and inverse fractional Laplacian kernels. We also investigate the sharpness of the $o(t^{-1})$ rate. For any smooth kernel in this class with infinitely many positive spectral coefficients and any $δ>0$, we construct a solution of the linearized flow whose energy is comparable to $t^{-1-δ}$ along a sequence of times tending to infinity. Moreover, for any $δ>0$, by choosing a suitable inverse fractional Laplacian kernel on the flat torus, we construct an exact solution of the nonlinear Wasserstein gradient flow whose energy is comparable to $t^{-1-δ}$. The nonlinear construction is based on uniform-in-time estimates for the evolution of the dyadic Fourier coefficient blocks and a blockwise energy-persistence argument. These estimates also yield a uniform-in-time quantitative comparison between the nonlinear Wasserstein gradient flow and its linearization.
Comments38 pages, 1 figure
