GlueNN: gluing patchwise analytic solutions with neural networks

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.

Dynamic neurons: A statistical physics approach for analyzing deep neural networks

Deep neural network architectures often consist of repetitive structural elements. We introduce a new approach that reveals these patterns and can be broadly applied to the study of deep learning. Similar to how a power strip helps untangle and organize complex cable connections, this approach treats neurons as additional degrees of freedom in interactions, simplifying the structure and enhancing the intuitive understanding of interactions within deep neural networks. Furthermore, it reveals the translational symmetry of deep neural networks, which simplifies the application of the renormalization group transformation - a method that effectively analyzes the scaling behavior of the system. By utilizing translational symmetry and renormalization group transformations, we can analyze critical phenomena. This approach may open new avenues for studying deep neural networks using statistical physics.