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.
翻译:用于描述各种复杂的随机动态系统(SDEs)的Stochanic 差异方程式(SDEs) 。 在 SDEs 中学习隐藏的物理物理普通差异方程式(SPINODE), 通过SDE(即已知物理学)的已知结构来传播随机性, 以产生一套确定性代码, 描述这些系统的随机和非线性行为的时间变化。 我们提出一个灵活和可扩展的框架, 用于培训人造神经网络, 以学习代表SDEs内隐藏的物理的构形方程式。 拟议的Stochindical Squal diffical complical 等式框架(SPINODE), 通过已知的SDE(即已知物理学)结构, 以生成一套确定性代码, 描述这些系统的统计时刻演变过程。 SPINODE然后使用ODS 解析器预测时的瞬时轨图。 SPINODE通过将预测的时间和数据估计的时间相匹配的时段进行神经网络的描述。 将利用最新的自动区分和微梯度梯态的梯度梯度脱脱脱的梯度的精度脱的浮度导系统, 。