  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 function space 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 two examples: a heat equation model
problem that demonstrates the benefits of the function space formulation in
terms of consistency with the underlying continuous truth, and 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 five orders of magnitude and model runtime reduction
of up to nine orders of magnitude.
