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.

Optimizing Soft Prompt Tuning via Structural Evolution

Soft prompt tuning leverages continuous embeddings to capture task-specific information in large pre-trained language models (LLMs), achieving competitive performance in few-shot settings. However, soft prompts rely on high-dimensional, implicit representations and lack explicit semantics and traceable training behaviors, which limits their interpretability. To address this limitation, we propose a soft prompt tuning optimization method based on topological morphological evolution. Specifically, we employ persistent homology from topological data analysis (TDA) to quantify the structural representations of soft prompts in continuous parameter space and their training process evolution. Quantitative analysis shows that topologically stable and compact soft prompts achieve better downstream performance. Based on this empirical observation, we construct a loss function for optimizing soft prompt tuning, termed Topological Soft Prompt Loss (TSLoss). TSLoss guides the model to learn structurally stable adaptations by quantifying inter-parameter connectivity and redundancy. Extensive experiments show that training with TSLoss accelerates convergence and improves tuning performance, providing an interpretable method to understand and optimize soft prompt tuning from structural and topological perspectives.

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.

DefFiller: Mask-Conditioned Diffusion for Salient Steel Surface Defect Generation

Current saliency-based defect detection methods show promise in industrial settings, but the unpredictability of defects in steel production environments complicates dataset creation, hampering model performance. Existing data augmentation approaches using generative models often require pixel-level annotations, which are time-consuming and resource-intensive. To address this, we introduce DefFiller, a mask-conditioned defect generation method that leverages a layout-to-image diffusion model. DefFiller generates defect samples paired with mask conditions, eliminating the need for pixel-level annotations and enabling direct use in model training. We also develop an evaluation framework to assess the quality of generated samples and their impact on detection performance. Experimental results on the SD-Saliency-900 dataset demonstrate that DefFiller produces high-quality defect images that accurately match the provided mask conditions, significantly enhancing the performance of saliency-based defect detection models trained on the augmented dataset.

Defect Image Sample Generation With Diffusion Prior for Steel Surface Defect Recognition

The task of steel surface defect recognition is an industrial problem with great industry values. The data insufficiency is the major challenge in training a robust defect recognition network. Existing methods have investigated to enlarge the dataset by generating samples with generative models. However, their generation quality is still limited by the insufficiency of defect image samples. To this end, we propose Stable Surface Defect Generation (StableSDG), which transfers the vast generation distribution embedded in Stable Diffusion model for steel surface defect image generation. To tackle with the distinctive distribution gap between steel surface images and generated images of the diffusion model, we propose two processes. First, we align the distribution by adapting parameters of the diffusion model, adopted both in the token embedding space and network parameter space. Besides, in the generation process, we propose image-oriented generation rather than from pure Gaussian noises. We conduct extensive experiments on steel surface defect dataset, demonstrating state-of-the-art performance on generating high-quality samples and training recognition models, and both designed processes are significant for the performance.