Stochastic differential equations (SDEs) are used to describe a wide variety of complex stochastic dynamical systems. Learning the hidden physics within SDEs is crucial for unraveling fundamental understanding of these systems' stochastic and nonlinear behavior. We propose a flexible and scalable framework for training artificial neural networks to learn constitutive equations that represent hidden physics within SDEs. The proposed stochastic physics-informed neural ordinary differential equation framework (SPINODE) propagates stochasticity through the known structure of the SDE (i.e., the known physics) to yield a set of deterministic ODEs that describe the time evolution of statistical moments of the stochastic states. SPINODE then uses ODE solvers to predict moment trajectories. SPINODE learns neural network representations of the hidden physics by matching the predicted moments to those estimated from data. Recent advances in automatic differentiation and mini-batch gradient descent with adjoint sensitivity are leveraged to establish the unknown parameters of the neural networks. We demonstrate SPINODE on three benchmark in-silico case studies and analyze the framework's numerical robustness and stability. SPINODE provides a promising new direction for systematically unraveling the hidden physics of multivariate stochastic dynamical systems with multiplicative noise.

随机微分方程 (SDE) 用于描述各种复杂的随机动力系统。学习 SDE 中隐藏的物理特性对于解开对这些系统的随机和非线性行为的基本理解至关重要。我们提出了一个灵活且可扩展的框架，用于训练人工神经网络以学习代表 SDE 中隐藏物理的本构方程。所提出的基于随机物理的神经常微分方程框架 (SPINODE) 通过 SDE 的已知结构（即已知物理）传播随机性，以产生一组确定性 ODE，这些 ODE 描述了随机状态的统计矩的时间演化。然后，SPINODE 使用 ODE 求解器来预测矩轨迹。SPINODE 通过将预测的矩与从数据中估计的矩相匹配来学习隐藏物理的神经网络表示。利用自动微分和具有伴随敏感性的小批量梯度下降的最新进展来建立神经网络的未知参数。我们在三个基准上演示了 SPNODE计算机案例研究并分析框架的数值鲁棒性和稳定性。SPINODE 为系统地揭示具有乘性噪声的多元随机动力系统的隐藏物理学提供了一个有前途的新方向。