Physics-informed artificial intelligence with splines for modeling advection–diffusion–reaction under dynamic boundaries

This paper proposes Physics-Informed Deep Operator Networks (PI-DeepONets) for rapid simulation of Advection–Diffusion–Reaction (ADR) systems with time-variable boundary conditions. These are viewed as inputs of the system and each family of inputs can be represented by a set of parameters. It is shown that, in practice, PI-DeepONets may not be able to model accurately the system of Partial Differential Equations (PDEs) for any combination of those parameterized inputs. Therefore, a new distributed architecture is proposed, which combines specialized PI-DeepONets, each being dedicated to represent one PDE. Furthermore, a mixture of experts allows to simulate multiple PI-DeepONets for which the input is a combination of several parameterized base functions. A model of an adsorption column is used to evaluate the relevance of the method. The Mixture of Experts PI-DeepONets with complex inputs such as splines, reaches a Mean Arctangent Absolute Percentage Error around 0.2 comparable to that of physics informed neural networks with constant inputs. The accuracy is improved by 20 % compared to distributed DeepONets, whereas a conventional DeepONet is unable to converge to the true solution. The results highlight the ability of a mixture of experts PI-DepeOnets to deliver an accurate and fast simulations of chemical reactors described by partial differential equations under complex and time-variable boundary conditions. The simulations can be completed in less than 10 ms compared to several minutes for traditional simulations methods, therefore opening the door to real time predictions and optimization.