Geometric Neural Operators via Lie Group-Constrained Latent Dynamics

Neural operators offer an effective framework for learning solutions of partial differential equations for many physical systems in a resolution-invariant and data-driven manner. Existing neural operators, however, often suffer from instability in multi-layer iteration and long-horizon rollout, which stems from the unconstrained Euclidean latent space updates that violate the geometric and conservation laws. To address this challenge, we propose to constrain manifolds with low-rank Lie algebra parameterization that performs group action updates on the latent representation. Our method, termed Manifold Constraining based on Lie group (MCL), acts as an efficient \emph{plug-and-play} module that enforces geometric inductive bias to existing neural operators. Extensive experiments on various partial differential equations, such as 1-D Burgers and 2-D Navier-Stokes, over a wide range of parameters and steps demonstrate that our method effectively lowers the relative prediction error by 30-50\% at the cost of 2.26\% of parameter increase. The results show that our approach provides a scalable solution for improving long-term prediction fidelity by addressing the principled geometric constraints absent in the neural operator updates.

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.

Thinking with Frames: Generative Video Distortion Evaluation via Frame Reward Model

Recent advances in video reward models and post-training strategies have improved text-to-video (T2V) generation. While these models typically assess visual quality, motion quality, and text alignment, they often overlook key structural distortions, such as abnormal object appearances and interactions, which can degrade the overall quality of the generative video. To address this gap, we introduce REACT, a frame-level reward model designed specifically for structural distortions evaluation in generative videos. REACT assigns point-wise scores and attribution labels by reasoning over video frames, focusing on recognizing distortions. To support this, we construct a large-scale human preference dataset, annotated based on our proposed taxonomy of structural distortions, and generate additional data using a efficient Chain-of-Thought (CoT) synthesis pipeline. REACT is trained with a two-stage framework: ((1) supervised fine-tuning with masked loss for domain knowledge injection, followed by (2) reinforcement learning with Group Relative Policy Optimization (GRPO) and pairwise rewards to enhance reasoning capability and align output scores with human preferences. During inference, a dynamic sampling mechanism is introduced to focus on frames most likely to exhibit distortion. We also present REACT-Bench, a benchmark for generative video distortion evaluation. Experimental results demonstrate that REACT complements existing reward models in assessing structutal distortion, achieving both accurate quantitative evaluations and interpretable attribution analysis.

Understanding Chain-of-Thought in Large Language Models via Topological Data Analysis

With the development of large language models (LLMs), particularly with the introduction of the long reasoning chain technique, the reasoning ability of LLMs in complex problem-solving has been significantly enhanced. While acknowledging the power of long reasoning chains, we cannot help but wonder: Why do different reasoning chains perform differently in reasoning? What components of the reasoning chains play a key role? Existing studies mainly focus on evaluating reasoning chains from a functional perspective, with little attention paid to their structural mechanisms. To address this gap, this work is the first to analyze and evaluate the quality of the reasoning chain from a structural perspective. We apply persistent homology from Topological Data Analysis (TDA) to map reasoning steps into semantic space, extract topological features, and analyze structural changes. These changes reveal semantic coherence, logical redundancy, and identify logical breaks and gaps. By calculating homology groups, we assess connectivity and redundancy at various scales, using barcode and persistence diagrams to quantify stability and consistency. Our results show that the topological structural complexity of reasoning chains correlates positively with accuracy. More complex chains identify correct answers sooner, while successful reasoning exhibits simpler topologies, reducing redundancy and cycles, enhancing efficiency and interpretability. This work provides a new perspective on reasoning chain quality assessment and offers guidance for future optimization.

Precise, Fast, and Low-cost Concept Erasure in Value Space: Orthogonal Complement Matters

The success of text-to-image generation enabled by diffuion models has imposed an urgent need to erase unwanted concepts, e.g., copyrighted, offensive, and unsafe ones, from the pre-trained models in a precise, timely, and low-cost manner. The twofold demand of concept erasure requires a precise removal of the target concept during generation (i.e., erasure efficacy), while a minimal impact on non-target content generation (i.e., prior preservation). Existing methods are either computationally costly or face challenges in maintaining an effective balance between erasure efficacy and prior preservation. To improve, we propose a precise, fast, and low-cost concept erasure method, called Adaptive Vaule Decomposer (AdaVD), which is training-free. This method is grounded in a classical linear algebraic orthogonal complement operation, implemented in the value space of each cross-attention layer within the UNet of diffusion models. An effective shift factor is designed to adaptively navigate the erasure strength, enhancing prior preservation without sacrificing erasure efficacy. Extensive experimental results show that the proposed AdaVD is effective at both single and multiple concept erasure, showing a 2- to 10-fold improvement in prior preservation as compared to the second best, meanwhile achieving the best or near best erasure efficacy, when comparing with both training-based and training-free state of the arts. AdaVD supports a series of diffusion models and downstream image generation tasks, the code is available on the project page: https://github.com/WYuan1001/AdaVD

Densely Distilling Cumulative Knowledge for Continual Learning

Continual learning, involving sequential training on diverse tasks, often faces catastrophic forgetting. While knowledge distillation-based approaches exhibit notable success in preventing forgetting, we pinpoint a limitation in their ability to distill the cumulative knowledge of all the previous tasks. To remedy this, we propose Dense Knowledge Distillation (DKD). DKD uses a task pool to track the model's capabilities. It partitions the output logits of the model into dense groups, each corresponding to a task in the task pool. It then distills all tasks' knowledge using all groups. However, using all the groups can be computationally expensive, we also suggest random group selection in each optimization step. Moreover, we propose an adaptive weighting scheme, which balances the learning of new classes and the retention of old classes, based on the count and similarity of the classes. Our DKD outperforms recent state-of-the-art baselines across diverse benchmarks and scenarios. Empirical analysis underscores DKD's ability to enhance model stability, promote flatter minima for improved generalization, and remains robust across various memory budgets and task orders. Moreover, it seamlessly integrates with other CL methods to boost performance and proves versatile in offline scenarios like model compression.