# Universal Approximation

The macroscopic constitutive law of an inelastic heterogeneous material depends on both loading history and microstructure, and computational homogenization must repeatedly solve a unit-cell problem at every macroscopic quadrature point. For Kelvin–Voigt viscoelasticity, a recurrent neural operator model takes the microstructure as input, and it is proved to approximate the homogenized constitutive law uniformly in time; the sufficient internal-variable dimension grows logarithmically with the inverse relative stress error and with each material contrast. Crystal-plasticity experiments show accuracy improving and then plateauing as that dimension grows.

The macroscopic constitutive law of an inelastic heterogeneous material depends on both loading history and microstructure, and computational homogenization must repeatedly solve a unit-cell problem at every macroscopic quadrature point. For Kelvin–Voigt viscoelasticity, a recurrent neural operator model takes the microstructure as input, and it is proved to approximate the homogenized constitutive law uniformly in time; the sufficient internal-variable dimension grows logarithmically with the inverse relative stress error and with each material contrast. Crystal-plasticity experiments show accuracy improving and then plateauing as that dimension grows.

For a class of equivariant graph-neural-network interatomic potentials, a universal approximation theorem now covers atomic energies: provided every atom's neighbors are reached within a common number of layers and no more than a given number of neighbors sit at a common distance inside the layer cutoff, a multilayer network whose layer cutoff is shorter than the physical range still approximates the atomic energy over that range, with spherical harmonic filter degrees no larger than that neighbor bound. The production DPA4 architecture inherits the result by simulating the constructed layers and realizing the bilinear readout.

For a class of equivariant graph-neural-network interatomic potentials, a universal approximation theorem now covers atomic energies: provided every atom's neighbors are reached within a common number of layers and no more than a given number of neighbors sit at a common distance inside the layer cutoff, a multilayer network whose layer cutoff is shorter than the physical range still approximates the atomic energy over that range, with spherical harmonic filter degrees no larger than that neighbor bound. The production DPA4 architecture inherits the result by simulating the constructed layers and realizing the bilinear readout.