Soft Constraint in Local Structure Approximation-PINN

We previously presented a novel loss formulation for efficient learning of complex dynamics from governing physics, typically described by partial differential equations (PDEs), using physics-informed neural networks (PINNs). In these experiments, the incorporation of a Boundary Connectivity (BCXN) loss function was shown to greatly improve physics-informed learning across many problems, especially those with complex geometries. However, this imposition may not always be ideal, as the previously presented BCXN-loss strongly enforces a linear local structure at the boundary. While this assumption helps facilitate faster learning with an order of magnitude sparser training samples, this can adversely impact convergence in other situations. Hence, we propose a modification of this BCXN-loss that reduces the imposed structure to a soft constraint, allowing for more flexible learning and convergence. We further demonstrate the potential for this method to improve the convergence and performance of LSA-PINN across additional numerical experiments, with much smaller errors than existing methods in terms of the standard L2-norm metric. In particular, we have applied this method to the modelling of flow past complex geometries such as airfoils, which serve as the basic building block for various applications in fluid dynamics and renewable energy (e.g., in wind and tidal turbine design).