Stochastic Physics-Informed Neural Ordinary Differential Equations

TL;DR

This paper introduces SPINODE, a scalable neural network framework that learns hidden physics in stochastic differential equations by modeling statistical moments and using ODE solvers, advancing understanding of complex stochastic systems.

Contribution

The paper presents a novel stochastic physics-informed neural ODE framework that propagates stochasticity through known physics to learn hidden dynamics from data.

Findings

Successfully applied to three benchmark cases

Demonstrates numerical robustness and stability

Provides a new approach for uncovering hidden physics in stochastic systems

Abstract

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.