  We propose Nash Neural Networks ($N^3$) as a new type of Physics Informed
Neural Network that is able to infer the underlying utility from observations
of how rational individuals behave in a differential game with a Nash
equilibrium. We assume that the dynamics for both the population and the
individual are known, but not the payoff function, which specifies the cost per
unit time of being in any particular state. We construct our network in such a
way that the Euler-Lagrange equations of the corresponding optimal control
problem are satisfied and the optimal control is self-consistently determined.
In this way, we are able to learn the unknown payoff function in an
unsupervised manner. We have applied the $N^3$ to study the optimal behaviour
during epidemics, in which individuals can choose to socially distance
depending on the state of the pandemic and the cost of being infected. Training
our network against synthetic data for a simple SIR model, we showed that it is
possible to accurately reproduce the hidden payoff function, in such a way that
the game dynamics are respected. Our approach will have far-reaching
applications, as it allows one to infer utilities from behavioural data, and
can thus be applied to study a wide array of problems in science, engineering,
economics and government planning.
