Physics-Informed Neural Network with PDE Soft Constraint Regularization Invariance

Neural networks are increasingly used to solve for partial differential equations (PDEs). One of the most popular ones for solving PDEs is the physics-informed neural network (PINN) which is a neural network that incorporates physical domain knowledge as soft constraints on an empirical loss function. However, PINNs learn to approximate the solution of a PDE with specific (e.g., fixed) physical conditions (e.g., PDE-based differential operators or soft constraint regularizations) and fail to generalize on other different physical conditions. In this paper, we are interested in solving PDEs given any set of physical conditions. As such, we propose the “Physics-Informed Neural Network with physical conditions invariance”, a variant of the PINN which, given any PDE, generalizes to any problem setup involving physical conditions. We implement and test our model on several different situations of widespread physical interest, especially differential equations with reaction and diffusion operators. Corresponding results show the effectiveness of our algorithm in approximating the solution of PDEs given any set/combination of physical conditions.