GlueNN: gluing patchwise analytic solutions with neural networks

Abstract

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.