Learning nonlocal constitutive models with neural networks

Constitutive and closure models play important roles in computational mechanics physics general. Classical constitutive for solid fluid materials are typically local, algebraic equations or flow rules describing the dependence of stress on local strain and/or strain-rate. Closure such as those Reynolds turbulent flows laminar--turbulent transition can involve transport PDEs (partial differential equations). Such similar to relation, but they often more challenging develop calibrate describe nonlocal mappings contain many submodels. Inspired by structure exact solutions linear PDEs, we propose a neural network representing region-to-point mapping models. The range convolution derived from formal solution equations. network-based model is trained with data. Numerical experiments demonstrate predictive capability proposed method. Moreover, learned embedded submodel without using data that level, thanks its interpretable mathematical structure, which makes it promising alternative traditional