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.

翻译：在复杂物理系统的分析中，目标通常不仅限于计算数值解，还包括精确捕捉不同区域间的交叉行为以及提取蕴含物理信息的参数。然而，标准数值求解器和传统深度学习方法（如物理信息神经网络，PINNs）通常作为黑箱运行，直接输出解场而未能将解分解为可解释的组成部分。本文提出GlueNN，一种物理信息学习框架，将全局解分解为可解释的分片解析分量。GlueNN不直接逼近解，而是将局部渐近展开中的积分常数提升为可学习的、尺度依赖的系数函数。通过用微分方程约束这些系数，网络有效地实现了区域过渡，在渐近极限之间平滑插值，无需特设的边界匹配。我们通过多个示例证明，这种以系数为中心的方法能够重构精确的全局解，从而直接提取通过标准数值积分无法显式获得的物理信息。