MACE-PINNs: Multi-Network Driven Decoupling of Interdependent Physics in Coupled PDE Systems

Physics-Informed Neural Networks (PINNs) provide an innovative framework for solving complex nonlinear Partial Differential Equations (PDEs) by embedding the governing equations directly into neural networks. Recent advancements have sought to improve their performance, yet standard ("vanilla") PINNs frequently encounter instabilities and inaccuracies, particularly for PDEs with coupled variables or dynamic constraints. These limitations stem from stiff gradient dynamics and multi-scale nonlinear interactions. Traditional strategies, such as time marching and curriculum training, have been employed to mitigate these issues but often yield error magnitudes higher than anticipated, reducing their effectiveness for certain PDE classes. To address these challenges, the Multi-network Architecture for Coupled Equations Physics-Informed Neural Networks (MACE-PINNs) is introduced. This approach employs parallel subnetworks to independently approximate coupled variables, interconnected via iterative residual constraints. Inspired by classical numerical solvers, this decoupled training enhances stability and learning efficiency, particularly for PDEs with sensitive initial conditions and strong parameter dependencies. MACE-PINNs is evaluated on the Gray-Scott-2D reaction-diffusion system (RDS) and the Ginzburg-Landau-2D equation—canonical examples of spatiotemporal pattern formation and intrinsic instabilities. This method integrates Fourier feature embeddings to enhance diffusion dynamics representation and adaptive gradient-norm weighting to balance residual loss with data-driven soft temporal regularization. Experimental results demonstrate robust pattern reproduction spanning 5 parametric variations for each RDS, with L2 errors ranging from $10^{-3}$ to $10^{-2}$. This approach, inspired by classical numerical solvers, employs structured decoupling to achieve stable and physically meaningful neural approximations of complex PDE systems.