Physics-Informed Neural Network for Discovering Systems with
  Unmeasurable States with Application to Lithium-Ion Batteries
Yuichi Kajiura Jorge Espin Dong Zhang
Combining machine learning with physics is a trending approach for
discovering unknown dynamics, and one of the most intensively studied
frameworks is the physics-informed neural network (PINN). However, PINN often
fails to optimize the network due to its difficulty in concurrently minimizing
multiple losses originating from the system's governing equations. This problem
can be more serious when the system's states are unmeasurable, like lithium-ion
batteries (LiBs). In this work, we introduce a robust method for training PINN
that uses fewer loss terms and thus constructs a less complex landscape for
optimization. In particular, instead of having loss terms from each
differential equation, this method embeds the dynamics into a loss function
that quantifies the error between observed and predicted system outputs. This
is accomplished by numerically integrating the predicted states from the neural
network(NN) using known dynamics and transforming them to obtain a sequence of
predicted outputs. Minimizing such a loss optimizes the NN to predict states
consistent with observations given the physics. Further, the system's
parameters can be added to the optimization targets. To demonstrate the ability
of this method to perform various modeling and control tasks, we apply it to a
battery model to concurrently estimate its states and parameters.