  We show that machine learning can improve the accuracy of simulations of
stress waves in one-dimensional composite materials. We propose a data-driven
technique to learn nonlocal constitutive laws for stress wave propagation
models. The method is an optimization-based technique in which the nonlocal
kernel function is approximated via Bernstein polynomials. The kernel,
including both its functional form and parameters, is derived so that when used
in a nonlocal solver, it generates solutions that closely match high-fidelity
data. The optimal kernel therefore acts as a homogenized nonlocal continuum
model that accurately reproduces wave motion in a smaller-scale, more detailed
model that can include multiple materials. We apply this technique to wave
propagation within a heterogeneous bar with a periodic microstructure. Several
one-dimensional numerical tests illustrate the accuracy of our algorithm. The
optimal kernel is demonstrated to reproduce high-fidelity data for a composite
material in applications that are substantially different from the problems
used as training data.
