Ruixuan Deng

Wei Xiong, Xiaomeng Huang, Ziyang Zhang et al.

The lacking of analytic solutions of diverse partial differential equations (PDEs) gives birth to a series of computational techniques for numerical solutions. Although numerous latest advances are accomplished in developing neural operators, a kind of neural-network-based PDE solver, these solvers become less accurate and explainable while learning long-term behaviors of non-linear PDE families. In this paper, we propose the Koopman neural operator (KNO), a new neural operator, to overcome these challenges. With the same objective of learning an infinite-dimensional mapping between Banach spaces that serves as the solution operator of the target PDE family, our approach differs from existing models by formulating a non-linear dynamic system of equation solution. By approximating the Koopman operator, an infinite-dimensional operator governing all possible observations of the dynamic system, to act on the flow mapping of the dynamic system, we can equivalently learn the solution of a non-linear PDE family by solving simple linear prediction problems. We validate the KNO in mesh-independent, long-term, and5zero-shot predictions on five representative PDEs (e.g., the Navier-Stokes equation and the Rayleigh-B{é}nard convection) and three real dynamic systems (e.g., global water vapor patterns and western boundary currents). In these experiments, the KNO exhibits notable advantages compared with previous state-of-the-art models, suggesting the potential of the KNO in supporting diverse science and engineering applications (e.g., PDE solving, turbulence modelling, and precipitation forecasting).

Zehao Jin, Ruixuan Deng, Junran Wang et al.

Activation steering has emerged as a promising alternative for controlling language-model behavior at inference time by modifying intermediate representations while keeping model parameters frozen. However, large-scale evaluations such as AxBench show that existing steering methods are often outperformed by simple in-context prompting and generalize poorly to unseen concepts. We hypothesize that these limitations arise from unvalidated simplifying assumptions shared across prior methods, which typically restrict steering interventions to fixed, single-step, position-invariant transforms. We propose FLAS (Flow-based Activation Steering), which learns a general, concept-conditioned velocity field $v_t(h,t,c)$ that transports unsteered activations to steered ones without relying on these assumptions. On AxBench, FLAS is the first learned method to consistently outperform prompting, reaching held-out harmonic means of $1.015$ on Gemma-2-2B-IT and $1.113$ on Gemma-2-9B-IT without per-concept tuning. Analysis of the learned flow shows curved, multi-step, token-varying trajectories, which suggests that previous hypotheses on activation space geometry might be incomplete.

Ruixuan Deng, Xiaoyang Hu, Miles Gilberti et al.

We identify semantically coherent, context-consistent network components in large language models (LLMs) using coactivation of sparse autoencoder (SAE) features collected from just a handful of prompts. Focusing on concept-relation prediction tasks, we show that ablating these components for concepts (e.g., countries and words) and relations (e.g., capital city and translation language) changes model outputs in predictable ways, while amplifying these components induces counterfactual responses. Notably, composing relation and concept components yields compound counterfactual outputs. Further analysis reveals that while most concept components emerge from the very first layer, more abstract relation components are concentrated in later layers. Lastly, we show that extracted components more comprehensively capture concepts and relations than individual features while maintaining specificity. Overall, our findings suggest a modular organization of knowledge accessed through compositional operations, and advance methods for efficient, targeted LLM manipulation.