Forecasting the evolution of complex dynamical systems remains a fundamentally challenging task, primarily due to pronounced nonlinear interactions, high-dimensional state spaces, and the concomitant requirement for rigorous and reliable uncertainty quantification. Contemporary reduced-order modelling (ROM) frameworks frequently exhibit inherent trade-offs among predictive accuracy, numerical stability, and interpretability, and thus often fail to achieve an optimal balance among these competing objectives. To address these limitations, we propose a framework for forecasting complex dynamical systems via a kernel autonomous ordinary differential equation approach based on Gaussian Processes and Quadratic Order Model Reduction. Our base method, the Gaussian Process Ordinary Differential Equations model, allows accurate short-term forecasting with uncertainty quantification, and it provably converges to the real autonomous equation in the smooth case. We integrate it with quadratic order reduced-order modelling and sphere projection for learning the latent dynamics efficiently while preserving stability. Numerical experiments demonstrate that our full model outperforms ROM forecasting methods such as Extended Dynamic Mode Decomposition, Bagging Optimised Dynamic Mode Decomposition and Linear and Nonlinear Disambiguation Optimisation in terms of accuracy or computational costs. These results demonstrate the potential of the framework as a robust and stable tool for forecasting complex dynamical systems with rigorous uncertainty quantification.

We investigate limitations of learning $\tanh$ neural networks from point evaluations under finite-precision computations and $L^p$ accuracy guarantees, building on Berner, Grohs, and Voigtländer (2023). Our approach is based on a novel construction of sharply localized bump functions via iterated $\tanh$ activations. Using this mechanism, we show that, in a finite-precision setting, no adaptive randomized algorithm based on $m$ samples can achieve a convergence rate higher than the Monte Carlo rate $O(m^{-1/p})$ in the $L^p$ norm, unless the sampling budget grows exponentially with the size of the network parameters and architecture. The results reveal fundamental limitations imposed by finite precision on the learnability of classes containing localized bump functions, extending previous results for ReLU networks to the $\tanh$ setting.

Designing optimal prompts and reasoning processes for large language models (LLMs) on domain-specific tasks is both necessary and challenging in real-world applications. Determining how to integrate domain knowledge, enhance reasoning efficiency, and even provide domain experts with refined knowledge integration hints are particularly crucial yet unresolved tasks. In this research, we propose Evolutionary Graph Optimization for Prompting (EGO-Prompt), an automated framework to designing better prompts, efficient reasoning processes and providing enhanced causal-informed process. EGO-Prompt begins with a general prompt and fault-tolerant initial Semantic Causal Graph (SCG) descriptions, constructed by human experts, which is then automatically refined and optimized to guide LLM reasoning. Recognizing that expert-defined SCGs may be partial or imperfect and that their optimal integration varies across LLMs, EGO-Prompt integrates a novel causal-guided textual gradient process in two steps: first, generating nearly deterministic reasoning guidance from the SCG for each instance, and second, adapting the LLM to effectively utilize the guidance alongside the original input.

We present a systematic theoretical framework that interprets masked diffusion models (MDMs) as solutions to energy minimization problems in discrete optimal transport. Specifically, we prove that three distinct energy formulations--kinetic, conditional kinetic, and geodesic energy--are mathematically equivalent under the structure of MDMs, and that MDMs minimize all three when the mask schedule satisfies a closed-form optimality condition. This unification not only clarifies the theoretical foundations of MDMs, but also motivates practical improvements in sampling. By parameterizing interpolation schedules via Beta distributions, we reduce the schedule design space to a tractable 2D search, enabling efficient post-training tuning without model modification. Experiments on synthetic and real-world benchmarks demonstrate that our energy-inspired schedules outperform hand-crafted baselines, particularly in low-step sampling settings.

The bisimulation metric (BSM) is a powerful tool for computing state similarities within a Markov decision process (MDP), revealing that states closer in BSM have more similar optimal value functions. While BSM has been successfully utilized in reinforcement learning (RL) for tasks like state representation learning and policy exploration, its application to multiple-MDP scenarios, such as policy transfer, remains challenging. Prior work has attempted to generalize BSM to pairs of MDPs, but a lack of rigorous analysis of its mathematical properties has limited further theoretical progress. In this work, we formally establish a generalized bisimulation metric (GBSM) between pairs of MDPs, which is rigorously proven with the three fundamental properties: GBSM symmetry, inter-MDP triangle inequality, and the distance bound on identical states. Leveraging these properties, we theoretically analyse policy transfer, state aggregation, and sampling-based estimation in MDPs, obtaining explicit bounds that are strictly tighter than those derived from the standard BSM. Additionally, GBSM provides a closed-form sample complexity for estimation, improving upon existing asymptotic results based on BSM. Numerical results validate our theoretical findings and demonstrate the effectiveness of GBSM in multi-MDP scenarios.

KV cache eviction has emerged as an effective solution to alleviate resource constraints faced by LLMs in long-context scenarios. However, existing token-level eviction methods often overlook two critical aspects: (1) their irreversible eviction strategy fails to adapt to dynamic attention patterns during decoding (the saliency shift problem), and (2) they treat both marginally important tokens and truly unimportant tokens uniformly, despite the collective significance of marginal tokens to model performance (the marginal information over-compression problem). To address these issues, we design two compensation mechanisms based on the high similarity of attention matrices between LLMs with different scales. We propose SmallKV, a small model assisted compensation method for KV cache compression. SmallKV can maintain attention matching between different-scale LLMs to: 1) assist the larger model in perceiving globally important information of attention; and 2) use the smaller model's attention scores to approximate those of marginal tokens in the larger model. Extensive experiments on benchmarks including GSM8K, BBH, MT-Bench, and LongBench demonstrate the effectiveness of SmallKV. Moreover, efficiency evaluations show that SmallKV achieves 1.75 - 2.56 times higher throughput than baseline methods, highlighting its potential for efficient and performant LLM inference in resource constrained environments.

Large Language Models (LLMs) have yet to effectively leverage the vast amounts of edge-device data, and Federated Learning (FL) offers a promising paradigm to collaboratively fine-tune LLMs without transferring private edge data to the cloud. To operate within the computational and communication constraints of edge devices, recent literature on federated fine-tuning of LLMs proposes the use of low-rank adaptation (LoRA) and similar parameter-efficient methods. However, LoRA-based methods suffer from accuracy degradation in FL settings, primarily because of data and computational heterogeneity across clients. We propose Ravan, an adaptive multi-head LoRA method that balances parameter efficiency and model expressivity by reparameterizing the weight updates as the sum of multiple LoRA heads, $s_i\textbf{B}_i\textbf{H}_i\textbf{A}_i$, in which only the $\textbf{H}_i$ parameters and their lightweight scaling factors $s_i$ are trained. These trainable scaling factors let the optimization focus on the most useful heads, recovering a higher-rank approximation of the full update without increasing the number of communicated parameters since clients upload $s_i\textbf{H}_i$ directly. Experiments on vision and language benchmarks show that Ravan improves test accuracy by 2-8\% over prior parameter-efficient baselines, making it a robust and scalable solution for federated fine-tuning of LLMs.

In this work, we demonstrate that affine mappings between residual streams of language models is a cheap way to effectively transfer represented features between models. We apply this technique to transfer the \textit{weights} of Sparse Autoencoders (SAEs) between models of different sizes to compare their representations. We find that small and large models learn highly similar representation spaces, which motivates training expensive components like SAEs on a smaller model and transferring to a larger model at a FLOPs savings. For example, using a small-to-large transferred SAE as initialization can lead to 50% cheaper training runs when training SAEs on larger models. Next, we show that transferred probes and steering vectors can effectively recover ground truth performance. Finally, we dive deeper into feature-level transferability, finding that semantic and structural features transfer noticeably differently while specific classes of functional features have their roles faithfully mapped. Overall, our findings illustrate similarities and differences in the linear representation spaces of small and large models and demonstrate a method for improving the training efficiency of SAEs.

Despite the impressive performance of general-purpose large language models (LLMs), they often require fine-tuning or post-training to excel at specific tasks. For instance, large reasoning models (LRMs), such as the DeepSeek-R1 series, demonstrate strong reasoning capabilities after post-training different general large language models on diverse chain-of-thought (CoT) datasets. However, this additional training frequently comes at the cost of reduced safety, as the fine-tuned or post-trained models tend to exhibit more harmful behaviors compared with the regular LLMs before post-training or fine-tuning, potentially leading to harmful outcomes due to their enhanced capabilities. Taking LRMs as an example, we first investigate the underlying cause of this safety degradation in this paper. Our analysis reveals that post-training can mask the original safety mechanisms of the base LLM, while over-amplifying representations related to their post-training ability. But luckily, we also find that LRMs' safety mechanisms still exist instead of being removed during their post-training. Based on these findings, we propose a lightweight and cost-effective solution called SafeReAct that restores the suppressed safety behaviors by aligning with LoRA adapters on a few layers. Experiments on four state-of-the-art LRMs show that our method significantly improves safety on harmful prompts without compromising reasoning performance. Besides LRMs, additional results on other domain-specific LLMs, like medical models, further confirm the generality and effectiveness of our approach.

Short videos are an effective tool for promoting contents and improving knowledge accessibility. While existing extractive video summarization methods struggle to produce a coherent narrative, existing abstractive methods cannot `quote' from the input videos, i.e., inserting short video clips in their outputs. In this work, we explore novel video editing models for generating shorts that feature a coherent narrative with embedded video insertions extracted from a long input video. We propose a novel retrieval-embedded generation framework that allows a large language model to quote multimodal resources while maintaining a coherent narrative. Our proposed REGen system first generates the output story script with quote placeholders using a finetuned large language model, and then uses a novel retrieval model to replace the quote placeholders by selecting a video clip that best supports the narrative from a pool of candidate quotable video clips. We examine the proposed method on the task of documentary teaser generation, where short interview insertions are commonly used to support the narrative of a documentary. Our objective evaluations show that the proposed method can effectively insert short video clips while maintaining a coherent narrative. In a subjective survey, we show that our proposed method outperforms existing abstractive and extractive approaches in terms of coherence, alignment, and realism in teaser generation.