TY - JOUR
T1 - Physics-informed machine learning in asymptotic homogenization of elliptic equations
AU - Soyarslan, Celal
AU - Pradas, Marc
N1 - Publisher Copyright:
© 2024 The Author(s)
PY - 2024/7
Y1 - 2024/7
N2 - We apply physics-informed neural networks (PINNs) to first-order, two-scale, periodic asymptotic homogenization of the property tensor in a generic elliptic equation. The problem of lack of differentiability of property tensors at the sharp phase interfaces is circumvented by making use of a diffuse interface approach. Periodic boundary conditions are incorporated strictly through the introduction of an input-transfer layer (Fourier feature mapping), which takes the sine and cosine of the inner product of position and reciprocal lattice vectors. This, together with the absence of Dirichlet boundary conditions, results in a lossless boundary condition application. Consequently, the sole contributors to the loss are the locally-scaled differential equation residuals. We use crystalline arrangements that are defined via Bravais lattices to demonstrate the formulation's versatility based on the reciprocal lattice vectors. We also show that considering integer multiples of the reciprocal basis in the Fourier mapping leads to improved convergence of high-frequency functions. We consider applications in one, two, and three dimensions, including periodic composites, composed of embeddings of monodisperse inclusions in the form of disks/spheres, and stochastic monodisperse disk arrangements.
AB - We apply physics-informed neural networks (PINNs) to first-order, two-scale, periodic asymptotic homogenization of the property tensor in a generic elliptic equation. The problem of lack of differentiability of property tensors at the sharp phase interfaces is circumvented by making use of a diffuse interface approach. Periodic boundary conditions are incorporated strictly through the introduction of an input-transfer layer (Fourier feature mapping), which takes the sine and cosine of the inner product of position and reciprocal lattice vectors. This, together with the absence of Dirichlet boundary conditions, results in a lossless boundary condition application. Consequently, the sole contributors to the loss are the locally-scaled differential equation residuals. We use crystalline arrangements that are defined via Bravais lattices to demonstrate the formulation's versatility based on the reciprocal lattice vectors. We also show that considering integer multiples of the reciprocal basis in the Fourier mapping leads to improved convergence of high-frequency functions. We consider applications in one, two, and three dimensions, including periodic composites, composed of embeddings of monodisperse inclusions in the form of disks/spheres, and stochastic monodisperse disk arrangements.
KW - UT-Hybrid-D
KW - Bravais lattice
KW - Diffuse interface
KW - Elliptic equations
KW - Machine learning
KW - Physics-based neural network
KW - Asymptotic homogenization
UR - http://www.scopus.com/inward/record.url?scp=85193818198&partnerID=8YFLogxK
U2 - 10.1016/j.cma.2024.117043
DO - 10.1016/j.cma.2024.117043
M3 - Article
AN - SCOPUS:85193818198
SN - 0045-7825
VL - 427
JO - Computer methods in applied mechanics and engineering
JF - Computer methods in applied mechanics and engineering
M1 - 117043
ER -