A comprehensive analysis of physics-informed neural networks for solving one-way coupled problems

Physics-informed neural networks (PINNs) offer efficient solvers for differential equations. However, their performance often degrades when applied to complex problems, particularly in multiphysics modeling. In this study, we focus on one-way coupled models, where each equation depends only on the outputs of preceding ones. Such structures frequently arise in thermo-mechanics coupling, fluid-structure interaction, electro-thermal processes, and multispecies transport. We examine why standard PINNs struggle on one-way coupled systems. When all equations are trained simultaneously, the network tends to ignore the inherent one-way causal structure due to spectral bias, leading to biased learning and poor convergence. We analyze these issues using the neural tangent kernel, the loss landscape, the distribution of back-propagated gradients, gradient conflict, and the resulting ill-conditioning. To address these challenges, we introduce PINN strategies that explicitly preserve the one-way causal order, including sequential training, transfer learning, and other causality-aware variants. Numerical experiments on benchmark problems demonstrate that the proposed approaches preserve one-way causality and improve both accuracy and training efficiency compared to standard PINNs.

Citation: Kamil, H., Soulaïmani, A., & Beljadid, A. (2026). A comprehensive analysis of physics-informed neural networks for solving one-way coupled problems. Computers & Mathematics with Applications, 212, 166-189.
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