Inelastic homogenization stops solving a unit cell at every point: internal-variable dimension grows only logarithmically with error and material contrast

The macroscopic law no longer requires re-solving the unit-cell problem at every time step; a recurrent neural operator with a finite-dimensional internal variable supplies it, storing only the current internal state instead of the full loading history. The prior route is computational homogenization, which repeatedly solves the unit-cell problem at macroscopic quadrature points and time steps at high cost, while existing learned surrogates mostly fit input-output relations on discretized loading histories and microstructure descriptions. On two-dimensional plane-stress Kelvin–Voigt viscoelasticity trained with 4096 samples, the memoryless model with zero internal-variable dimension has a clearly higher relative error than the internal-variable model, showing that memory effects cannot be ignored in this experimental setup, and the mean error of the internal-variable model falls as the dimension increases.

Both the stress response and the kinetic response are written as operators acting on the microstructure field, so the internal-variable dimension sets both representational capacity and macroscopic simulation cost. The two responses are parameterized by Fourier neural mappings, which lift the input pointwise, apply linear transforms on retained Fourier modes, and project back to a vector output. Training data come from numerical unit-cell solutions of periodic Voronoi viscoelastic microstructures, with four scalar fields as input: elastic and viscous Young's moduli and Poisson ratios.

The model is proved to approximate the homogenized constitutive law uniformly in time, with an explicit sufficient bound on the internal-variable dimension. Earlier one-dimensional analysis relied on exact Prony-series representations for piecewise-constant materials, and two and three dimensions require new arguments. The proof builds an intermediate Prony-series model by local Chebyshev interpolation of the memory kernel's spectral representation, then approximates its responses with neural operators; the theoretical results are formally verified in Lean 4.

The theory covers periodic Kelvin–Voigt viscoelasticity, with admissible microstructures whose viscosity and elasticity tensor fields are uniformly bounded, coercive and of bounded spatial variation, and admissible strain trajectories with uniformly bounded magnitudes and rates; idealized step-strain tests are explicitly excluded. The numerical part trains and tests on two-dimensional plane-stress viscoelasticity and three-dimensional finite-strain face-centered-cubic crystal plasticity, the latter outside the theory's scope.

The internal-variable dimension bound assumes the neural operator model can emulate the intermediate Prony-series model, and its constant depends only on the given parameters. In crystal plasticity the test error plateaus once the dimension is large enough, and mean relative errors on normal components are clearly smaller than on shear components. After a linear change of coordinates the learned internal variables admit a tensorial interpretation, but their fitted dynamics remain coupled and need not be dissipative.