  Physics-informed machine learning (PIML) has emerged as a promising new
approach for simulating complex physical and biological systems that are
governed by complex multiscale processes for which some data are also
available. In some instances, the objective is to discover part of the hidden
physics from the available data, and PIML has been shown to be particularly
effective for such problems for which conventional methods may fail. Unlike
commercial machine learning where training of deep neural networks requires big
data, in PIML big data are not available. Instead, we can train such networks
from additional information obtained by employing the physical laws and
evaluating them at random points in the space-time domain. Such
physics-informed machine learning integrates multimodality and multifidelity
data with mathematical models, and implements them using neural networks or
graph networks. Here, we review some of the prevailing trends in embedding
physics into machine learning, using physics-informed neural networks (PINNs)
based primarily on feed-forward neural networks and automatic differentiation.
For more complex systems or systems of systems and unstructured data, graph
neural networks (GNNs) present some distinct advantages, and here we review how
physics-informed learning can be accomplished with GNNs based on graph exterior
calculus to construct differential operators; we refer to these architectures
as physics-informed graph networks (PIGNs). We present representative examples
for both forward and inverse problems and discuss what advances are needed to
scale up PINNs, PIGNs and more broadly GNNs for large-scale engineering
problems.
