Physics-informed machine learning in asymptotic homogenization of elliptic equations

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.</p

Physics-informed machine learning in asymptotic homogenization of elliptic equations

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

On a Variational Norm Tailored to Variable-Basis Approximation Schemes

A variational norm associated with sets of computational units and used in function approximation, learning from data, and infinite-dimensional optimization is investigated. For sets Gk obtained by varying a vector y of parameters in a fixed-structure computational unit K(-,y) (e.g., the set of Gaussians with free centers and widths), upper and lower bounds on the GK -variation norms of functions having certain integral representations are given, in terms of the £1-norms of the weighting functions in such representations. Families of functions for which the two norms are equal are described