  Classical artificial neural networks have witnessed widespread successes in
machine-learning applications. Here, we propose fermion neural networks (FNNs)
whose physical properties, such as local density of states or conditional
conductance, serve as outputs, once the inputs are incorporated as an initial
layer. Comparable to back-propagation, we establish an efficient optimization,
which entitles FNNs to competitive performance on challenging machine-learning
benchmarks. FNNs also directly apply to quantum systems, including hard ones
with interactions, and offer in-situ analysis without preprocessing or
presumption. Following machine learning, FNNs precisely determine topological
phases and emergent charge orders. Their quantum nature also brings various
advantages: quantum correlation entitles more general network connectivity and
insight into the vanishing gradient problem, quantum entanglement opens up
novel avenues for interpretable machine learning, etc.
