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.

翻译：本研究旨在学习具有随机效应与均值场效应且依赖于物理参数的实际系统群体动力学模型。所习得的模型可作为经典数值模型的替代品，以高效预测系统在物理参数范围内的行为。基于最优传输理论中的Benamou-Brenier公式与作用量匹配方法，我们通过变分问题推断参数依赖且时间依赖的梯度场，该梯度场可近似表征群体动力学。推断得到的梯度场随后可用于快速生成样本轨迹，这些轨迹能在群体层面上模拟物理系统随物理参数变化的动力学行为。研究表明，将蒙特卡洛采样与高阶数值积分规则相结合，对于从样本数据中准确估计训练目标函数及稳定训练过程至关重要。通过在Vlasov-Poisson不稳定性系统、高维粒子系统及混沌系统中的实验验证，我们证明该方法能准确预测宽参数范围内的群体动力学，其性能优于当前仅以时间与物理参数为条件的先进扩散模型与流模型。