# A general physics-constrained method for the modelling of equation's closure terms with sparse data

arXiv: https://arxiv.org/abs/2505.03783 · HTML: https://mlsanity.com/paper/2505.03783/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 10 of its 49 references that are in the map.
- First posted on arXiv 30 Apr 2025.

## Builds on (most cited first, 10 of 10)

- [Scientific Machine Learning through Physics-Informed Neural Networks: Where we are and What's next](https://mlsanity.com/paper/2201.05624/) (2022)
- [Gradient-enhanced physics-informed neural networks for forward and inverse PDE problems](https://mlsanity.com/paper/2111.02801/) (2021)
- [Hidden Physics Models: Machine Learning of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1708.00588/) (2017)
- [Physics-informed neural networks for inverse problems in supersonic flows](https://mlsanity.com/paper/2202.11821/) (2022)
- [Numerical Gaussian Processes for Time-dependent and Non-linear Partial Differential Equations](https://mlsanity.com/paper/1703.10230/) (2017)
- [Inferring solutions of differential equations using noisy multi-fidelity data](https://mlsanity.com/paper/1607.04805/) (2016)
- [Deep learning observables in computational fluid dynamics](https://mlsanity.com/paper/1903.03040/) (2019)
- [A review on data-driven constitutive laws for solids](https://mlsanity.com/paper/2405.03658/) (2024)
- [Neural-net-induced Gaussian process regression for function approximation and PDE solution](https://mlsanity.com/paper/1806.11187/) (2018)
- [Gradient-free online learning of subgrid-scale dynamics with neural emulators](https://mlsanity.com/paper/2310.19385/) (2023)

## Abstract (arXiv)

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.