Heterogeneous Peridynamic Neural Operators: Discover Biotissue
  Constitutive Law and Microstructure From Digital Image Correlation
  Measurements
Siavash Jafarzadeh Stewart Silling Lu Zhang Colton Ross Chung-Hao Lee S. M. Rakibur Rahman Shuodao Wang Yue Yu
Human tissues are highly organized structures with specific collagen fiber
arrangements varying from point to point. The effects of such heterogeneity
play an important role for tissue function, and hence it is of critical to
discover and understand the distribution of such fiber orientations from
experimental measurements, such as the digital image correlation data. To this
end, we introduce the heterogeneous peridynamic neural operator (HeteroPNO)
approach, for data-driven constitutive modeling of heterogeneous anisotropic
materials. The goal is to learn both a nonlocal constitutive law together with
the material microstructure, in the form of a heterogeneous fiber orientation
field, from loading field-displacement field measurements. To this end, we
propose a two-phase learning approach. Firstly, we learn a homogeneous
constitutive law in the form of a neural network-based kernel function and a
nonlocal bond force, to capture complex homogeneous material responses from
data. Then, in the second phase we reinitialize the learnt bond force and the
kernel function, and training them together with a fiber orientation field for
each material point. Owing to the state-based peridynamic skeleton, our
HeteroPNO-learned material models are objective and have the balance of linear
and angular momentum guaranteed. Moreover, the effects from heterogeneity and
nonlinear constitutive relationship are captured by the kernel function and the
bond force respectively, enabling physical interpretability. As a result, our
HeteroPNO architecture can learn a constitutive model for a biological tissue
with anisotropic heterogeneous response undergoing large deformation regime.
Moreover, the framework is capable to provide displacement and stress field
predictions for new and unseen loading instances.