  Quantum computing promises to speed up some of the most challenging problems
in science and engineering. Quantum algorithms have been proposed showing
theoretical advantages in applications ranging from chemistry to logistics
optimization. Many problems appearing in science and engineering can be
rewritten as a set of differential equations. Quantum algorithms for solving
differential equations have shown a provable advantage in the fault-tolerant
quantum computing regime, where deep and wide quantum circuits can be used to
solve large linear systems like partial differential equations (PDEs)
efficiently. Recently, variational approaches to solving non-linear PDEs also
with near-term quantum devices were proposed. One of the most promising general
approaches is based on recent developments in the field of scientific machine
learning for solving PDEs. We extend the applicability of near-term quantum
computers to more general scientific machine learning tasks, including the
discovery of differential equations from a dataset of measurements. We use
differentiable quantum circuits (DQCs) to solve equations parameterized by a
library of operators, and perform regression on a combination of data and
equations. Our results show a promising path to Quantum Model Discovery (QMoD),
on the interface between classical and quantum machine learning approaches. We
demonstrate successful parameter inference and equation discovery using QMoD on
different systems including a second-order, ordinary differential equation and
a non-linear, partial differential equation.
