A Framework for Physics-Informed Deep Learning Over Freeform Domains

Deep learning is a popular approach for approximating the solutions to partial differential equations (PDEs) over different material parameters and boundary conditions. However, no work has yet been reported on learning PDE solutions over changing shapes of the underlying domain. We present a framework to train neural networks (NN) and physics-informed neural networks (PINNs) to learn the solutions to PDEs defined over varying freeform domains. This is made possible through our adoption of a parametric non-uniform rational B-Spline (NURBS) representation of the underlying physical shape. Distinct physical domains are mapped to a common parametric space via NURBS parameterization. In our approach, we formulate NNs and PINNs that learn the solutions to PDEs as a function of the shape of the domain itself through shape parameters. Under this formulation, the loss function is based on an unchanging parametric domain that maps to a variable physical domain. Residual computation in PINNs is made possible through the Jacobian of the mapping. Numerical results show that our networks can be trained to predict the solutions to a PDE defined over an entire set of shapes. We focus on the linear elasticity PDE and show how we can build a surrogate model that is able to predict displacements and stresses over a variety of freeform domains. To assess the efficacy of all networks in this work, data efficiency, network accuracy, and the capacity of networks to extrapolate are considered and compared between NNs and PINNs. The comparison includes cases where little training data is available. Transfer learning and applications to shape optimization are analyzed as well. A selection of the used codes and datasets is provided at https://github.com/fmezzadri/shape_parameterized.git.