  We propose and study the evolving neural network (ENN) method for solving
one-dimensional scalar hyperbolic conservation laws with linear and quadratic
spatial fluxes. The ENN method first represents the initial data and the inflow
boundary data by neural networks. Then, it evolves the neural network
representation of the initial data along the temporal direction. The evolution
is computed using a combination of characteristic and finite volume methods.
For the linear spatial flux, the method is not subject to any time step size,
and it is shown theoretically that the error at any time step is bounded by the
representation errors of the initial and boundary condition. For the quadratic
flux, an error estimate is studied in a companion paper. Finally, numerical
results for the linear advection equation and the inviscid Burgers equation are
presented to show that the ENN method is more accurate and cost efficient than
traditional mesh-based methods.
