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We propose and study the evolving neural network (ENN) method for solvingone-dimensional scalar hyperbolic conservation laws with linear and quadraticspatial fluxes. The ENN method first represents the initial data and the inflowboundary data by neural networks. Then, it evolves the neural networkrepresentation of the initial data along the temporal direction. The evolutionis 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 therepresentation errors of the initial and boundary condition. For the quadraticflux, an error estimate is studied in a companion paper. Finally, numericalresults for the linear advection equation and the inviscid Burgers equation arepresented to show that the ENN method is more accurate and cost efficient thantraditional mesh-based methods.

Evolving Neural Network (ENN) Method for One-Dimensional Scalar Hyperbolic Conservation Laws: I Linear and Quadratic Fluxes