发表机构
Pohang University of Science and Technology; Samsung Electronics; Konkuk University（浦项科技大学; 三星电子; 建国大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
提出一种无网格神经JKO格式，通过空间映射与质量变化因子联合处理HK几何中的平流-反应-扩散梯度流，建立离散能量耗散与收敛性，数值验证了其与PDE的一致性和能量耗散。
AI 中文摘要
我们针对具有Hellinger-Kantorovich (HK)非平衡最优传输几何中梯度流结构的平流-反应-扩散方程，发展了一种无网格的神经JKO格式。每一步更新由一个空间映射和一个质量变化因子参数化，使得空间重新分布和局部质量产生或损失能够在单个变分步骤中联合处理。它们的锥作用从上方界定平方HK距离，通过与恒等对比较得出离散能量耗散的充分条件。当源和极小化子具有正密度时，在所有可容许对上最小化对目标可恢复精确的JKO最小值。我们建立了JKO极小化子的存在性和质量界，并在额外假设下，连同离散Euler-Lagrange方程和度量-耗散恒等式，获得正性和正则性。自洽化学势随后沿最优映射非增。存在参数对，其端点密度和目标值收敛到精确JKO极小化子的那些值，前提是正则对逼近假设成立。最后，我们证明原始-对偶间隙控制目标次优性，并且对于Boltzmann熵，在精确步正则性、半正定相互作用和全局对偶可行性假设下，控制$L^1$密度误差。数值实验检验了与PDE的逐点一致性、能量耗散以及传输、反应和全隐式相互作用的作用。
英文摘要
We develop a mesh-free neural JKO scheme for advection-reaction-diffusion equations with a gradient-flow structure in the Hellinger-Kantorovich (HK) geometry of unbalanced optimal transport. Each update is parametrized by a spatial map and a mass-changing factor, allowing spatial redistribution and local mass creation or loss to be treated jointly within a single variational step. Their cone action bounds the squared HK distance from above, yielding a sufficient condition for discrete energy dissipation through comparison with the identity pair. Minimizing the pair objective over all admissible pairs recovers the exact JKO minimum when the source and a minimizer have positive densities. We establish existence and mass bounds for JKO minimizers and, under additional assumptions, obtain positivity and regularity together with a discrete Euler-Lagrange equation and a metric-dissipation identity. The self-consistent chemical potential is then nonincreasing along an optimal map. There exist parametric pairs whose endpoint densities and objective values converge to those of an exact JKO minimizer, provided a regular-pair approximation hypothesis holds. Finally, we show that a primal-dual gap controls objective suboptimality and, for Boltzmann entropy, the $L^1$ density error, assuming exact-step regularity, positive-semidefinite interactions, and global dual feasibility. Numerical experiments examine pointwise agreement with the PDE, energy dissipation, and the roles of transport, reaction, and fully implicit interactions.
Comments55 pages, 10 figures
