从微观模拟中学习有效的随机微分方程：将随机数值与深度学习联系起来
Learning effective stochastic differential equations from microscopic simulations: linking stochastic numerics to deep learning
摘要：
我们确定有效的随机微分方程（ESTA）粗可观的细粒度粒子或代理为基础的模拟，这些ESTA然后提供有用的粗代理模型的细尺度动态。我们通过神经网络来近似这些有效网络中的漂移和扩散率函数，这些网络可以被认为是有效的随机ResNets。损失函数的灵感，并体现，建立随机数值积分器（在这里，Euler-Maruyama和Milstein）的结构;我们的近似，因此可以受益于这些基本的数值方案的向后误差分析。当近似的粗糙模型（如平均场方程）可用时，它们也自然地适用于“物理信息”灰箱识别。现有的数值积分计划Langevin型方程和随机偏微分方程（SPDE）也可以用于培训，我们证明了这一随机强迫振荡器和随机波动方程。我们的方法不需要很长的轨迹，适用于分散的快照数据，并且旨在自然地处理每个快照的不同时间步长。我们考虑的情况下，粗糙的集体观测是已知的，以及它们必须在一个数据驱动的方式被发现的情况。
英文摘要：
We identify effective stochastic differential equations (SDE) for coarse observables of fine-grained particle- or agent-based simulations; these SDE then provide useful coarse surrogate models of the fine scale dynamics. We approximate the drift and diffusivity functions in these effective SDE through neural networks, which can be thought of as effective stochastic ResNets. The loss function is inspired by, and embodies, the structure of established stochastic numerical integrators (here, Euler-Maruyama and Milstein); our approximations can thus benefit from backward error analysis of these underlying numerical schemes. They also lend themselves naturally to "physics-informed" gray-box identification when approximate coarse models, such as mean field equations, are available. Existing numerical integration schemes for Langevin-type equations and for stochastic partial differential equations (SPDE) can also be used for training; we demonstrate this on a stochastically forced oscillator and the stochastic wave equation. Our approach does not require long trajectories, works on scattered snapshot data, and is designed to naturally handle different time steps per snapshot. We consider both the case where the coarse collective observables are known in advance, as well as the case where they must be found in a data-driven manner.
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