A general physics-constrained method for the modelling of equation's closure terms with sparse data
Tian Chen, Shengping Liu, Li Liu, Heng Yong
TL;DR
This work tackles the closure problem in multiscale PDEs under sparse data by introducing a Series-Parallel Multi-Network Architecture that couples Physics-Informed Neural Networks with traditional solvers. The learning framework is decomposed into a Constructor that isolates known terms and learns a universal unknown-closure term, and an Applicator that embeds the learned closure into high-order finite-difference solvers (e.g., WENO-Z with RK3) to yield accurate forward predictions. The approach is validated on two systems: a simple PDE with an unknown source term and the compressible Euler equations with an unknown EOS (Noble–Abel), demonstrating strong cross-case generalization and high fidelity to reference solutions under both smooth and discontinuous regimes. By leveraging physics constraints (e.g., Rankine–Hugoniot) and sparse data, the method achieves low $L^2$ errors (e.g., $0.24\%-0.27\%$) while maintaining stability, suggesting substantial practical impact for engineering simulations with limited data.
Abstract
Accurate modeling of closure terms is a critical challenge in engineering and scientific research, particularly when data is sparse (scarse or incomplete), making widely applicable models difficult to develop. This study proposes a novel approach for constructing closure models in such challenging scenarios. We introduce a Series-Parallel Multi-Network Architecture that integrates Physics-Informed Neural Networks (PINNs) to incorporate physical constraints and heterogeneous data from multiple initial and boundary conditions, while employing dedicated subnetworks to independently model unknown closure terms, enhancing generalizability across diverse problems. These closure models are integrated into an accurate Partial Differential Equation (PDE) solver, enabling robust solutions to complex predictive simulations in engineering applications.