# GlueNN: gluing patchwise analytic solutions with neural networks

arXiv: https://arxiv.org/abs/2601.05889 · HTML: https://mlsanity.com/paper/2601.05889/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 8 of its 35 references that are in the map.
- First posted on arXiv 9 Jan 2026.

## Builds on (most cited first, 8 of 8)

- [Augmented Physics-Informed Neural Networks (APINNs): A gating network-based soft domain decomposition methodology](https://mlsanity.com/paper/2211.08939/) (2022)
- [NeuralStagger: Accelerating Physics-constrained Neural PDE Solver with Spatial-temporal Decomposition](https://mlsanity.com/paper/2302.10255/) (2023)
- [Asymptotic-Preserving Neural Networks for multiscale hyperbolic models of epidemic spread](https://mlsanity.com/paper/2206.12625/) (2022)
- [AdS/Deep-Learning made easy: simple examples](https://mlsanity.com/paper/2011.13726/) (2020)
- [Better Neural PDE Solvers Through Data-Free Mesh Movers](https://mlsanity.com/paper/2312.05583/) (2023)
- [ASPINN: An asymptotic strategy for solving singularly perturbed differential equations](https://mlsanity.com/paper/2409.13185/) (2024)
- [PVD-ONet: A Multi-scale Neural Operator Method for Singularly Perturbed Boundary Layer Problems](https://mlsanity.com/paper/2507.21437/) (2025)
- [AdS/Deep-Learning made easy II: neural network-based approaches to holography and inverse problems](https://mlsanity.com/paper/2511.22522/) (2025)

## Abstract (arXiv)

In the analysis of complex physical systems, the objective often extends beyond merely computing a numerical solution to capturing the precise crossover between different regimes and extracting parameters containing meaningful information. However, standard numerical solvers and conventional deep learning approaches, such as Physics-Informed Neural Networks (PINNs), typically operate as black boxes that output solution fields without disentangling the solution into its interpretable constituent parts. In this work, we propose GlueNN, a physics-informed learning framework that decomposes the global solution into interpretable, patchwise analytic components. Rather than approximating the solution directly, GlueNN promotes the integration constants of local asymptotic expansions to learnable, scale-dependent coefficient functions. By constraining these coefficients with the differential equation, the network effectively performs regime transition, smoothly interpolating between asymptotic limits without requiring ad hoc boundary matching. We demonstrate that this coefficient-centric approach reproduces accurate global solutions in various examples and thus directly extracts physical information that is not explicitly available through standard numerical integration.