Zhenzhen Huang
Publications
Rethinking Input Domains in Physics-Informed Neural Networks via Geometric Compactification Mappings
Several complex physical systems are governed by multi-scale partial differential equations (PDEs) that exhibit both smooth low-frequency components and localized high-frequency structures. Existing physics-informed neural network (PINN) methods typically train with fixed coordinate system inputs, where geometric misalignment with these structures induces gradient stiffness and ill-conditioning that hinder convergence. To address this issue, we introduce a mapping paradigm that reshapes the input coordinates through differentiable geometric compactification mappings and couples the geometric structure of PDEs with the spectral properties of residual operators. Based on this paradigm, we propose Geometric Compactification (GC)-PINN, a framework that introduces three mapping strategies for periodic boundaries, far-field scale expansion, and localized singular structures in the input domain without modifying the underlying PINN architecture. Extensive empirical evaluation demonstrates that this approach yields more uniform residual distributions and higher solution accuracy on representative 1D and 2D PDEs, while improving training stability and convergence speed.
GHS-TDA: A Synergistic Reasoning Framework Integrating Global Hypothesis Space with Topological Data Analysis
Chain-of-Thought (CoT) has been shown to significantly improve the reasoning accuracy of large language models (LLMs) on complex tasks. However, due to the autoregressive, step-by-step generation paradigm, existing CoT methods suffer from two fundamental limitations. First, the reasoning process is highly sensitive to early decisions: once an initial error is introduced, it tends to propagate and amplify through subsequent steps, while the lack of a global coordination and revision mechanism makes such errors difficult to correct, ultimately leading to distorted reasoning chains. Second, current CoT approaches lack structured analysis techniques for filtering redundant reasoning and extracting key reasoning features, resulting in unstable reasoning processes and limited interpretability. To address these issues, we propose GHS-TDA. GHS-TDA first constructs a semantically enriched global hypothesis graph to aggregate, align, and coordinate multiple candidate reasoning paths, thereby providing alternative global correction routes when local reasoning fails. It then applies topological data analysis based on persistent homology to capture stable multi-scale structures, remove redundancy and inconsistencies, and extract a more reliable reasoning skeleton. By jointly leveraging reasoning diversity and topological stability, GHS-TDA achieves self-adaptive convergence, produces high-confidence and interpretable reasoning paths, and consistently outperforms strong baselines in terms of both accuracy and robustness across multiple reasoning benchmarks.