We propose a new method for inferring the governing stochastic ordinary differential equations (SODEs) by observing particle ensembles at discrete and sparse time instants, i.e., multiple "snapshots". Particle coordinates at a single time instant, possibly noisy or truncated, are recorded in each snapshot but are unpaired across the snapshots. By training a physics-informed generative model that generates "fake" sample paths, we aim to fit the observed particle ensemble distributions with a curve in the probability measure space, which is induced from the inferred particle dynamics. We employ different metrics to quantify the differences between distributions, e.g., the sliced Wasserstein distances and the adversarial losses in generative adversarial networks (GANs). We refer to this method as generative "ensemble-regression" (GER), in analogy to the classic "point-regression", where we infer the dynamics by performing regression in the Euclidean space. We illustrate the GER by learning the drift and diffusion terms of particle ensembles governed by SODEs with Brownian motions and Levy processes up to 100 dimensions. We also discuss how to treat cases with noisy or truncated observations. Apart from systems consisting of independent particles, we also tackle nonlocal interacting particle systems with unknown interaction potential parameters by constructing a physics-informed loss function. Finally, we investigate scenarios of paired observations and discuss how to reduce the dimensionality in such cases by proving a convergence theorem that provides theoretical support.
翻译:我们提出一种新的方法,通过在离散和稀少的时间瞬间观测粒子集合,即多个“光照”。粒子坐标在一次瞬间记录下来,可能吵闹或脱节,在每次快照中可能发生,但在整个快照中没有变化。通过培训产生“假”样本路径的物理知情的基因化模型,我们的目标是将观察到的粒子集合分布与概率测量空间的曲线相匹配,这是从推断的粒子动态中引出的。我们使用不同的指标来量化分布之间的差别,例如,切片瓦瑟斯坦距离和在基因对抗性对抗网络(GANs)中的对抗性损失。我们把这种方法称为“共振-倒退”(GER),类似于典型的“点-倒退”模型,我们通过在 Eucliidean 空间中进行回归,我们通过进行折叠合,我们通过了解粒子的流和扩散参数的分布,我们用不易变的参数来量化。我们用SROD-ROD系统来解释,我们用不易变的顺序来解释,我们用不易变的顺序分析,我们用SROD-ROD-ral deal deal deal deal deal dec ex deal deal deal dec ex dec ex decredududududustrismismismismismismismismismismismism ex vic) ex ex ex ex vic.