用神经网络分解物理解的局部解析部分，精准捕捉不同物理区间的过渡。
GlueNN: gluing patchwise analytic solutions with neural networks
- 将解拆分为局部解析项，通过可学习系数函数连接
- 无需人为匹配边界，自然实现渐近极限间的平滑过渡
- 适合需要解读物理机制与参数的复杂系统研究
在复杂物理系统分析中，目标常不仅限于数值求解，还涉及精确捕捉不同物理区间的转变并提取有意义的参数。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）通常作为黑箱输出解场，难以分离出可解释的组成部分。本文提出GlueNN，一种基于物理信息的学习框架，将全局解分解为可解释的分片解析成分。不同于直接逼近解，GlueNN将局部渐近展开的积分常数建模为可学习的、尺度相关的系数函数，并通过微分方程约束这些系数，使网络能有效实现区域过渡，在不依赖人为边界匹配的情况下，平滑衔接渐近极限。我们在多个例子中验证了该系数中心方法能准确重建全局解，从而直接提取出标准数值积分无法显式获得的物理信息。
原文摘要 · Abstract (English)
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.
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