  Given ample experimental data from a system governed by differential
equations, it is possible to use deep learning techniques to construct the
underlying differential operators. In this work we perform symbolic discovery
of differential operators in a situation where there is sparse experimental
data. This small data regime in machine learning can be made tractable by
providing our algorithms with prior information about the underlying dynamics.
Physics Informed Neural Networks (PINNs) have been very successful in this
regime (reconstructing entire ODE solutions using only a single point or entire
PDE solutions with very few measurements of the initial condition). We modify
the PINN approach by adding a neural network that learns a representation of
unknown hidden terms in the differential equation. The algorithm yields both a
surrogate solution to the differential equation and a black-box representation
of the hidden terms. These hidden term neural networks can then be converted
into symbolic equations using symbolic regression techniques like AI Feynman.
In order to achieve convergence of these neural networks, we provide our
algorithms with (noisy) measurements of both the initial condition as well as
(synthetic) experimental data obtained at later times. We demonstrate strong
performance of this approach even when provided with very few measurements of
noisy data in both the ODE and PDE regime.
