  We present a new scientific machine learning method that learns from data a
computationally inexpensive surrogate model for predicting the evolution of a
system governed by a time-dependent nonlinear partial differential equation
(PDE), an enabling technology for many computational algorithms used in
engineering settings. Our formulation generalizes to the PDE setting the
Operator Inference method previously developed in [B. Peherstorfer and K.
Willcox, Data-driven operator inference for non-intrusive projection-based
model reduction, Computer Methods in Applied Mechanics and Engineering, 306
(2016)] for systems governed by ordinary differential equations. The method
brings together two main elements. First, ideas from projection-based model
reduction are used to explicitly parametrize the learned model by
low-dimensional polynomial operators which reflect the known form of the
governing PDE. Second, supervised machine learning tools are used to infer from
data the reduced operators of this physics-informed parametrization. For
systems whose governing PDEs contain more general (non-polynomial)
nonlinearities, the learned model performance can be improved through the use
of lifting variable transformations, which expose polynomial structure in the
PDE. The proposed method is demonstrated on a three-dimensional combustion
simulation with over 18 million degrees of freedom, for which the learned
reduced models achieve accurate predictions with a dimension reduction of six
orders of magnitude and model runtime reduction of 5-6 orders of magnitude.
