通过高阶作用量匹配，高效预测含参数的随机与平均场系统演化。
Parametric model reduction of mean-field and stochastic systems via higher-order action matching
- 基于最优传输的变分框架，学习随参数和时间变化的梯度场以逼近群体动态。
- 在多参数下准确预测等离子体不稳定性及高维混沌系统群体行为，优于现有扩散与流模型。
- 结合蒙特卡洛与高阶积分规则，提升训练稳定性和目标估计精度，适合物理系统仿真加速。
本文旨在学习包含随机与平均场效应且依赖物理参数的物理系统群体动力学模型。所学模型可作为经典数值模型的代理，高效预测系统在不同物理参数下的行为。基于最优传输中的Benamou-Brenier公式与作用量匹配，我们采用变分问题推断随参数和时间变化的梯度场，以近似群体动态。这些推断出的梯度场可用于快速生成模拟系统群体层面演化的样本轨迹。我们表明，将蒙特卡洛采样与高阶求积规则结合，对从样本数据中准确估计训练目标和稳定训练过程至关重要。在Vlasov-Poisson不稳定性以及高维粒子与混沌系统上，该方法在宽范围参数下均能准确预测群体动态，显著优于仅条件于时间和物理参数的主流扩散与流模型。
原文摘要 · Abstract (English)
The aim of this work is to learn models of population dynamics of physical systems that feature stochastic and mean-field effects and that depend on physics parameters. The learned models can act as surrogates of classical numerical models to efficiently predict the system behavior over the physics parameters. Building on the Benamou-Brenier formula from optimal transport and action matching, we use a variational problem to infer parameter- and time-dependent gradient fields that represent approximations of the population dynamics. The inferred gradient fields can then be used to rapidly generate sample trajectories that mimic the dynamics of the physical system on a population level over varying physics parameters. We show that combining Monte Carlo sampling with higher-order quadrature rules is critical for accurately estimating the training objective from sample data and for stabilizing the training process. We demonstrate on Vlasov-Poisson instabilities as well as on high-dimensional particle and chaotic systems that our approach accurately predicts population dynamics over a wide range of parameters and outperforms state-of-the-art diffusion-based and flow-based modeling that simply condition on time and physics parameters.
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