An approach to encode divergence-free stress fields in neural approximations based on stress potentials
The purpose of the current work is the development of an approach to account for quasi-static mechanical equilibrium in empirical (i.e., data-based) models for the stress field employing neural approximations (NAs), which include neural networks (NNs) and neural operators (NOs), in particular Fourier NOs (FNOs). Rather than including such constraints from physics in the loss function as done in the (now standard) physics-informed approach, the current approach incorporates or "encodes" such constraints directly into the architecture of the NA. As a result, both NA training and output are physically constrained in the physics-encoded approach, in contrast to the physics-informed approach, in which only training is physically constrained. For the current constraint of divergence-free stress, a novel encoding approach based on a stress potential is proposed. As a "proof-of-concept" example application of the current approach, a physics-encoded FNO (PeFNO) is developed for a heterogeneous polycrystalline material consisting of isotropic elastic grains and subject to uniaxial extension. Stress field data for this purpose are obtained from the numerical solution of corresponding boundary-value problems for quasi-static mechanical equilibrium. For comparison with the PeFNO, this data is also employed to develop an analogous physics-guided FNO (PgFNO) and physics-informed FNO (PiFNO). As expected theoretically, and confirmed by this computational comparison, for comparable accuracy of the stress field itself as compared to the data, the stress field output by the trained and tested PeFNO is significantly more accurate in satisfying mechanical equilibrium than the output of either the PgFNO or the PiFNO.