Recovering the state and dynamics of autonomous system with partial states solution using neural networks
Vijay Kag
TL;DR
This work tackles recovering state trajectories and their governing dynamics for autonomous ODEs from partial state data using the Deep Hidden Physics Model (DHPM). It couples two neural networks to predict the state solution $u$ and the hidden dynamics $\mathcal{N}$, enforcing physics through a residual $r(x,t)=\frac{\partial u}{\partial t}-\mathcal{N}$ and optimizing $\mathcal{L}_{total}=\mathcal{L}_{data}+\mathcal{L}_{eq}$. Through experiments on 2D linear, nonlinear, and Lorenz systems, the authors show that when all observed states are provided, both states and dynamics are learned with high accuracy; with partial observability, the model still accurately learns dynamics for the observed states, while unobserved states become poorly determined. The results highlight DHPM's potential for data-efficient physics discovery under partial observability and point to theoretical questions about why learned physics emerges for observed variables even when some states are missing.
Abstract
In this paper we explore the performance of deep hidden physics model (M. Raissi 2018) for autonomous systems. These systems are described by set of ordinary differential equations which do not explicitly depend on time. Such systems can be found in nature and have applications in modeling chemical concentrations, population dynamics, n-body problems in physics etc. In this work we consider dynamics of states, which explain how the states will evolve are unknown to us. We approximate state and dynamics both using neural networks. We have considered examples of 2D linear/nonlinear and Lorenz systems. We observe that even without knowing all the states information, we can estimate dynamics of certain states whose state information are known.