GlueNN: gluing patchwise analytic solutions with neural networks

TL;DR

GlueNN tackles the challenge of solving differential equations with strongly scale-dependent terms by constructing a global solution from patchwise analytic forms whose integration constants become learnable, scale-dependent coefficient functions. The method uses a head–trunk neural network to produce these coefficients and a composite loss that combines data fitting, differential-equation residuals, and out-of-patch suppression, eliminating the need for ad hoc boundary matching. It demonstrates accurate, extrapolatable solutions in chemical kinetics and inflationary cosmology, and provides an interpretable decomposition that reveals which asymptotic regime dominates where. This approach offers a robust, physically principled alternative to traditional matching and extends the reach of physics-informed learning to multiregime problems.

Abstract

In many problems in physics and engineering, one encounters complicated differential equations with strongly scale-dependent terms for which exact analytical or numerical solutions are not available. A common strategy is to divide the domain into several regions (patches) and simplify the equation in each region. When approximate analytic solutions can be obtained in each patch, they are then matched at the interfaces to construct a global solution. However, this patching procedure can fail to reproduce the correct solution, since the approximate forms may break down near the matching boundaries. In this work, we propose a learning framework in which the integration constants of asymptotic analytic solutions are promoted to scale-dependent functions. By constraining these coefficient functions with the original differential equation over the domain, the network learns a globally valid solution that smoothly interpolates between asymptotic regimes, eliminating the need for arbitrary boundary matching. We demonstrate the effectiveness of this framework in representative problems from chemical kinetics and cosmology, where it accurately reproduces global solutions and outperforms conventional matching procedures.

Figures (3)

• Figure 1: Schematic of the GlueNN framework. The input coordinate $x$ is processed by a shared head network and multiple trunk branches, each producing a coefficient function $\{c_j^{(i)}(x)\}$. These coefficients parameterize the patchwise analytic forms $f_j\bigl(x;\{c_j^{(i)}(x)\}\bigr)$, whose sum defines the global ansatz. Training minimizes a composite loss function, $\mathrm{MSE}_{\mathrm{data}} + \mathrm{MSE}_{\mathcal{D}}$, enforcing agreement with data and the differential-equation residual over the full domain. The learned coefficients and functions provide an interpretable view of how the solution transitions between asymptotic regimes.
• Figure 2: Training results for the chemical reaction example. (a): The ground truth value of the field amplitude, the prediction from the GlueNN, and the curve obtained from the $C^0$ matching procedure. The shaded band indicates the variation obtained by shifting the matching point by $\pm 20\%$. GlueNN closely reproduces the true solution. (b): The normalized coefficients of the asymptotic solutions, which approach constant values toward the end of the domain. Here, each term in Eq. \ref{['eq:sol_chem']} is normalized by $y(x)$ itself.
• Figure 3: Training results for the production example. (a): The ground truth value of the field amplitude, the prediction from the GlueNN, and the curve obtained from the $C^1$ matching procedure. The shaded band indicates the variation obtained by shifting the matching point by $\pm 20\%$. GlueNN closely reproduces the true solution, whereas the $C^{1}$ matching shows a clear deviation. (b): The normalized terms of the asymptotic solutions, which approach constant values toward each end of the domain. Here, each term in Eq. \ref{['eq:sol_inf_prod']} is normalized by $y(a)$ itself.