← 研踪 PaperTrailarXiv:2611.11805 · v1
Operator Learning of Microstructure-Dependent Internal-Variable Models for Inelastic Homogenization: Approximation and Complexity
首次提交 2026-11-09 11:08
版本更新 2026-11-09 11:08
雷达观测 2026-11-10 13:18
摘要
We study the learning of homogenized constitutive laws for inelastic heterogeneous materials, aiming to replace costly concurrent unit-cell simulations with surrogates that generalize across loading histories and microstructures. We focus on the recurrent neural operator framework, motivated by the explicit form of homogenized Kelvin--Voigt linear viscoelasticity. In this framework, internal variables enable a Markovian approximation of loading-history dependence, and neural operators represent the microstructure-dependent constitutive response. For periodic Kelvin--Voigt viscoelasticity, we prove a universal approximation theorem showing that the model can approximate the homogenized constitutive law to any prescribed accuracy, uniformly in time and over a wide set of admissible strain trajectories and microstructural property fields, in two and three dimensions. This comprises a significant extension of the existing, somewhat limited, one-dimensional theory that uses explicit solutions for piecewise constant materials. Instead, in two and three dimensions, we adopt a more versatile approximation based on the spectral representation of the memory kernel that arises from homogenization. This also enables us to give an explicit bound on the internal-variable dimension, sufficient for approximation, exhibiting logarithmic growth in the inverse relative stress error and in material contrast. Numerical experiments on viscoelasticity and crystal plasticity complement the theory, illustrating the roles of internal-variable dimension and sample complexity in learning homogenized responses from unit-cell data. Theoretical results are formally verified in Lean 4.
作者备注：52 pages, 12 figures, including appendices
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v1 · 2026-11-09 11:08 · 最新观测版本
Operator Learning of Microstructure-Dependent Internal-Variable Models for Inelastic Homogenization: Approximation and Complexity
We study the learning of homogenized constitutive laws for inelastic heterogeneous materials, aiming to replace costly concurrent unit-cell simulations with surrogates that generalize across loading histories and microstructures. We focus on the recurrent neural operator framework, motivated by the explicit form of homogenized Kelvin--Voigt linear viscoelasticity. In this framework, internal variables enable a Markovian approximation of loading-history dependence, and neural operators represent the microstructure-dependent constitutive response. For periodic Kelvin--Voigt viscoelasticity, we prove a universal approximation theorem showing that the model can approximate the homogenized constitutive law to any prescribed accuracy, uniformly in time and over a wide set of admissible strain trajectories and microstructural property fields, in two and three dimensions. This comprises a significant extension of the existing, somewhat limited, one-dimensional theory that uses explicit solutions for piecewise constant materials. Instead, in two and three dimensions, we adopt a more versatile approximation based on the spectral representation of the memory kernel that arises from homogenization. This also enables us to give an explicit bound on the internal-variable dimension, sufficient for approximation, exhibiting logarithmic growth in the inverse relative stress error and in material contrast. Numerical experiments on viscoelasticity and crystal plasticity complement the theory, illustrating the roles of internal-variable dimension and sample complexity in learning homogenized responses from unit-cell data. Theoretical results are formally verified in Lean 4.
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