NeurIPS 2025 Papers

Neural population activity exhibits complex, nonlinear dynamics, varying in time, over trials, and across experimental conditions. Here, we develop *Conditionally Linear Dynamical System* (CLDS) models as a general-purpose method to characterize these dynamics. These models use Gaussian Process priors to capture the nonlinear dependence of circuit dynamics on task and behavioral variables. Conditioned on these covariates, the data is modeled with linear dynamics. This allows for transparent interpretation and tractable Bayesian inference. We find that CLDS models can perform well even in severely data-limited regimes (e.g. one trial per condition) due to their Bayesian formulation and ability to share statistical power across nearby task conditions. In example applications, we apply CLDS to model thalamic neurons that nonlinearly encode heading direction and to model motor cortical neurons during a cued-reaching task.

Analysing how neural networks represent data features in their activations can help interpret how they perform tasks. Hence, a long line of work has focused on mathematically characterising the geometry of such "neural representations." In parallel, machine learning has seen a surge of interest in understanding how dynamical systems perform computations on time-varying input data. Yet, the link between computation-through-dynamics and representational geometry remains poorly understood. Here, we hypothesise that recurrent neural networks (RNNs) perform computations by dynamically warping their representations of task variables. To test this hypothesis, we develop a Riemannian geometric framework that enables the derivation of the manifold topology and geometry of a dynamical system from the manifold of its inputs. By characterising the time-varying geometry of RNNs, we show that dynamic warping is a fundamental feature of their computations.

Data-driven approaches, particularly those based on deep learning, are rapidly advancing Earth system modeling. However, their application to ocean forecasting remains limited despite the ocean's pivotal role in climate regulation and marine ecosystems. To address this gap, we present OceanBench, a benchmark designed to evaluate and accelerate global short-range (1–10 days) data-driven ocean forecasting. OceanBench is constructed from a curated dataset comprising first-guess trajectories, nowcasts, and atmospheric forcings from operational physical ocean models, typically unavailable in public datasets due to assimilation cycles. Matched observational data are also included, enabling realistic evaluation in an operational-like forecasting framework. The benchmark defines three complementary evaluation tracks: (i) Model-to-Reanalysis, where models are compared against the reanalysis dataset commonly used for training; (ii) Model-to-Analysis, assessing generalization to a higher-resolution physical analysis; and (iii) Model-to-Observations, Intercomparison and Validation (IV-TT) CLASS-4 evaluation against independent observational data. The first two tracks are further supported by process-oriented diagnostics to assess the dynamical consistency and physical plausibility of forecasts. OceanBench includes key ocean variables: sea surface height, temperature, salinity, and currents, along with standardized metrics grounded in physical oceanography. Baseline comparisons with operational systems and state-of-the-art deep learning models are provided. All data, code, and evaluation protocols are openly available at https://github.com/mercator-ocean/oceanbench, establishing OceanBench as a foundation for reproducible and rigorous research in data-driven ocean forecasting.

Uncovering cause-effect relationships from observational time series is fundamental to understanding complex systems. While many methods infer static causal graphs, real-world systems often exhibit *dynamic causality*—where relationships evolve over time. Accurately capturing these temporal dynamics requires time-resolved causal graphs. We propose UnCLe, a novel deep learning method for scalable dynamic causal discovery. UnCLe employs a pair of Uncoupler and Recoupler networks to disentangle input time series into semantic representations and learns inter-variable dependencies via auto-regressive Dependency Matrices. It estimates dynamic causal influences by analyzing datapoint-wise prediction errors induced by temporal perturbations. Extensive experiments demonstrate that UnCLe not only outperforms state-of-the-art baselines on static causal discovery benchmarks but, more importantly, exhibits a unique capability to accurately capture and represent evolving temporal causality in both synthetic and real-world dynamic systems (e.g., human motion). UnCLe offers a promising approach for revealing the underlying, time-varying mechanisms of complex phenomena.

We study the effect of normalization schemes on token representations in deep transformers. Modeling their evolution as interacting particles on the sphere, we show that normalization acts as a form of speed regulation. This perspective enables a unified analysis of several schemes---including **Post-LN**, **Pre-LN**, **Mix-LN**, **Peri-LN**, **nGPT**, and **LN-scaling**---revealing how they influence clustering dynamics and representation collapse. Our framework clarifies how different schemes shape token representations across layers and provides a principled basis for comparing them, identifying **Peri-LN** as a particularly effective choice.

Multimodal Emotion Recognition in Conversations (MERC) enhances emotional understanding through the fusion of multimodal signals. However, unpredictable modality absence in real-world scenarios significantly degrades the performance of existing methods. Conventional missing-modality recovery approaches, which depend on training with complete multimodal data, often suffer from semantic distortion under extreme data distributions, such as fixed-modality absence. To address this, we propose the Federated Dialogue-guided and Semantic-Consistent Diffusion (FedDISC) framework, pioneering the integration of federated learning into missing-modality recovery. By federated aggregation of modality-specific diffusion models trained on clients and broadcasting them to clients missing corresponding modalities, FedDISC overcomes single-client reliance on modality completeness. Additionally, the DISC-Diffusion module ensures consistency in context, speaker identity, and semantics between recovered and available modalities, using a Dialogue Graph Network to capture conversational dependencies and a Semantic Conditioning Network to enforce semantic alignment. We further introduce a novel Alternating Frozen Aggregation strategy, which cyclically freezes recovery and classifier modules to facilitate collaborative optimization. Extensive experiments on the IEMOCAP, CMUMOSI, and CMUMOSEI datasets demonstrate that FedDISC achieves superior emotion classification performance across diverse missing modality patterns, outperforming existing approaches.

The Koopman operator provides a principled framework for analyzing nonlinear dynamical systems through linear operator theory. Recent advances in dynamic mode decomposition (DMD) have shown that trajectory data can be used to identify dominant modes of a system in a data-driven manner. Building on this idea, deep learning methods such as VAMPnet and DPNet have been proposed to learn the leading singular subspaces of the Koopman operator. However, these methods require backpropagation through potentially numerically unstable operations on empirical second moment matrices, such as singular value decomposition and matrix inversion, during objective computation, which can introduce biased gradient estimates and hinder scalability to large systems. In this work, we propose a scalable and conceptually simple method for learning the top-$k$ singular functions of the Koopman operator for stochastic dynamical systems based on the idea of low-rank approximation. Our approach eliminates the need for unstable linear-algebraic operations and integrates easily into modern deep learning pipelines. Empirical results demonstrate that the learned singular subspaces are both reliable and effective for downstream tasks such as eigen-analysis and multi-step prediction.

The sparse Mixture-of-Experts (MoE) architecture is increasingly favored for scaling Large Language Models (LLMs) efficiently, but it depends on heterogeneous compute and memory resources. These factors jointly affect system Cost, Accuracy, and Performance (CAP), making trade-offs inevitable. Existing benchmarks often fail to capture these trade-offs accurately, complicating practical deployment decisions. To address this, we introduce MoE-CAP, a benchmark specifically designed for MoE systems. Our analysis reveals that achieving an optimal balance across CAP is difficult with current hardware; MoE systems typically optimize two of the three dimensions at the expense of the third—a dynamic we term the MoE-CAP trade-off. To visualize this, we propose the CAP Radar Diagram. We further introduce sparsity-aware performance metrics—Sparse Memory Bandwidth Utilization (S-MBU) and Sparse Model FLOPS Utilization (S-MFU)—to enable accurate performance benchmarking of MoE systems across diverse hardware platforms and deployment scenarios. This benchmark is available on Github: https://github.com/sparse-generative-ai/MoE-CAP.

Latent stochastic differential equation (SDE) models are important tools for the unsupervised discovery of dynamical systems from data, with applications ranging from engineering to neuroscience. In these complex domains, exact posterior inference of the latent state path is typically intractable, motivating the use of approximate methods such as variational inference (VI). However, existing VI methods for inference in latent SDEs often suffer from slow convergence and numerical instability. Here, we propose SDE Inference via Natural Gradients (SING), a method that leverages natural gradient VI to efficiently exploit the underlying geometry of the model and variational posterior. SING enables fast and reliable inference in latent SDE models by approximating intractable integrals and parallelizing computations in time. We provide theoretical guarantees that SING will approximately optimize the intractable, continuous-time objective of interest. Moreover, we demonstrate that better state inference enables more accurate estimation of nonlinear drift functions using, for example, Gaussian process SDE models. SING outperforms prior methods in state inference and drift estimation on a variety of datasets, including a challenging application to modeling neural dynamics in freely behaving animals. Altogether, our results illustrate the potential of SING as a tool for accurate inference in complex dynamical systems, especially those characterized by limited prior knowledge and non-conjugate structure.

We propose a new method for the problem of controlling linear dynamical systems under partial observation and adversarial disturbances. Our new algorithm, Double Spectral Control (DSC), matches the best known regret guarantees while exponentially improving runtime complexity over previous approaches in its dependence on the system's stability margin. Our key innovation is a two-level spectral approximation strategy, leveraging double convolution with a universal basis of spectral filters, enabling efficient and accurate learning of the best linear dynamical controllers.

We study the dynamics of gradient flow with small weight decay on general training losses $F: \mathbb{R}^d \to \mathbb{R}$. Under mild regularity assumptions and assuming convergence of the unregularised gradient flow, we show that the trajectory with weight decay $\lambda$ exhibits a two-phase behaviour as $\lambda \to 0$. During the initial fast phase, the trajectory follows the unregularised gradient flow and converges to a manifold of critical points of $F$. Then, at time of order $1/\lambda$, the trajectory enters a slow drift phase and follows a Riemannian gradient flow minimising the $\ell_2$-norm of the parameters. This purely optimisation-based phenomenon offers a natural explanation for the \textit{grokking} effect observed in deep learning, where the training loss rapidly reaches zero while the test loss plateaus for an extended period before suddenly improving. We argue that this generalisation jump can be attributed to the slow norm reduction induced by weight decay, as explained by our analysis. We validate this mechanism empirically on several synthetic regression tasks.

In recent years, nonlinear dynamic system identification using artificial neural networks has garnered attention due to its broad potential applications across science and engineering. However, purely data-driven approaches often struggle with extrapolation and may yield physically implausible forecasts. Furthermore, the learned dynamics can exhibit instabilities, making it difficult to apply such models safely and robustly. This article introduces stable port-Hamiltonian neural networks, a machine learning architecture that incorporates physical biases of energy conservation and dissipation while ensuring global Lyapunov stability of the learned dynamics. Through illustrative and real-world examples, we demonstrate that these strong inductive biases facilitate robust learning of stable dynamics from sparse data, while avoiding instability and surpassing purely data-driven approaches in accuracy and physically meaningful generalization. Furthermore, the model's applicability and potential for data-driven surrogate modeling are showcased on multi-physics simulation data.

Inspired by recent findings on the fractal geometry of language, we introduce Recursive INference Scaling (RINS) as a complementary, plug-in recipe for scaling inference time in language and multimodal systems. RINS is a particular form of recursive depth that significantly outperforms +55 other variants, including the recent "repeat-all-over" (RAO) strategy in Mobile LLM (Liu et al., 2024) and latent recurrent thinking (Geiping et al., 2025). Unlike prior works, we carry out our comparisons on a compute-matched regime, and demonstrate that for a fixed model size and training compute budget, RINS substantially improves language modeling performance. It also generalizes beyond pure language tasks, delivering gains in multimodal systems, including a +2% improvement in 0-shot ImageNet accuracy for SigLIP-B/16. Additionally, by deriving data scaling laws, we show that RINS improves both the asymptotic performance limits and the scaling exponents. More importantly, with light-weight (linear) adapters (comprising <1% of model parameters) and stochastic dropout, RINS offers a no-regret strategy, meaning that RINS-enabled pretraining improves performance in language modeling even when recursive depth is not applied at inference time. This corresponds to improving performance on a training compute-, parameter-, and inference-matched regime, suggesting its potential as a viable component of LLM pretraining!

Heterogeneity poses a fundamental challenge for many real-world large-scale decision-making problems but remains largely understudied. In this paper, we study the _fully heterogeneous_ setting of a prominent class of such problems, known as weakly-coupled Markov decision processes (WCMDPs). Each WCMDP consists of $N$ arms (or subproblems), which have distinct model parameters in the fully heterogeneous setting, leading to the curse of dimensionality when $N$ is large. We show that, under mild assumptions, an efficiently computable policy achieves an $O(1/\sqrt{N})$ optimality gap in the long-run average reward per arm for fully heterogeneous WCMDPs as $N$ becomes large. This is the _first asymptotic optimality result_ for fully heterogeneous average-reward WCMDPs. Our main technical innovation is the construction of projection-based Lyapunov functions that certify the convergence of rewards and costs to an optimal region, even under full heterogeneity.

The LLM-as-a-judge paradigm, in which a judge LLM system replaces human raters in rating the outputs of other generative AI (GenAI) systems, plays a critical role in scaling and standardizing GenAI evaluations. To validate such judge systems, evaluators assess human--judge agreement by first collecting multiple human ratings for each item in a validation corpus, then aggregating the ratings into a single, per-item gold label rating. For many items, however, rating criteria may admit multiple valid interpretations, so a human or LLM rater may deem multiple ratings "reasonable" or "correct." We call this condition rating indeterminacy. Problematically, many rating tasks that contain rating indeterminacy rely on forced-choice elicitation, whereby raters are instructed to select only one rating for each item. In this paper, we introduce a framework for validating LLM-as-a-judge systems under rating indeterminacy. We draw theoretical connections between different measures of judge system performance under different human--judge agreement metrics, and different rating elicitation and aggregation schemes. We demonstrate that differences in how humans and LLMs resolve rating indeterminacy when responding to forced-choice rating instructions can heavily bias LLM-as-a-judge validation. Through extensive experiments involving 11 real-world rating tasks and 9 commercial LLMs, we show that standard validation approaches that rely upon forced-choice ratings select judge systems that are highly suboptimal, performing as much as 31% worse than judge systems selected by our approach that uses multi-label "response set" ratings to account for rating indeterminacy. We conclude with concrete recommendations for more principled approaches to LLM-as-a-judge validation.

Causal discovery methods are powerful tools for uncovering the structure of relationships among variables, yet they face significant challenges in scalability and interpretability, especially in high-dimensional settings. In many domains, researchers are not only interested in causal links between individual variables, but also in relationships among sets or clusters of variables. Learning causal structure at the cluster level can both reveal higher-order relationships of interest and improve scalability. In this work, we introduce an approach for causal discovery over clusters in Markov causal systems. We propose a new graphical model that encodes knowledge of relationships between user-defined clusters while fully representing independencies and dependencies over clusters, faithful to a given distribution. We then define and characterize a graphical equivalence class of these models that share cluster-level independence information. Lastly, we present a sound and complete algorithm for causal discovery to represent learnable causal relationships between clusters of variables.

Real world evolves in continuous time but computations are done from finite samples. Therefore, we study algorithms using finite observations in continuous-time linear dynamical systems. We first study the system identification problem, and propose a first non-asymptotic error analysis with finite observations. Our algorithm identifies system parameters without needing integrated observations over certain time intervals, making it more practical for real-world applications. Further we propose a lower bound result that shows our estimator is provably optimal up to constant factors. Moreover, we apply the above algorithm to online control regret analysis for continuous-time linear system. Our system identification method allows us \textcolor{blue}{to} explore more efficiently, enabling the swift detection of ineffective policies. We achieve a regret of $\mathcal{O}(\sqrt{T})$ over a single $T$-time horizon in a controllable system, requiring only $\mathcal{O}(T)$ observations of the system.

In AI research and practice, rigor remains largely understood in terms of methodological rigor---such as whether mathematical, statistical, or computational methods are correctly applied. We argue that this narrow conception of rigor has contributed to the concerns raised by the responsible AI community, including overblown claims about the capabilities of AI systems. Our position is that a broader conception of what rigorous AI research and practice should entail is needed. We believe such a conception---in addition to a more expansive understanding of 1) methodological rigor---should include aspects related to 2) what background knowledge informs what to work on (epistemic rigor); 3) how disciplinary, community, or personal norms, standards, or beliefs influence the work (normative rigor); 4) how clearly articulated the theoretical constructs under use are (conceptual rigor); 5) what is reported and how (reporting rigor); and 6) how well-supported the inferences from existing evidence are (interpretative rigor). In doing so, we also provide useful language and a framework for much needed dialogue about the AI community's work by researchers, policymakers, journalists, and other stakeholders.

The function of biomolecules such as proteins depends on their ability to interconvert between a wide range of structures or conformations. Researchers have endeavored for decades to develop computational methods to predict the distribution of conformations, which is far harder to determine experimentally than a static folded structure. We present ConforMix, an inference-time algorithm that enhances sampling of conformational distributions using a combination of classifier guidance, filtering, and free energy estimation. Our approach upgrades diffusion models---whether trained for static structure prediction or conformational generation---to enable more efficient discovery of conformational variability without requiring prior knowledge of major degrees of freedom. ConforMix is orthogonal to improvements in model pretraining and would benefit even a hypothetical model that perfectly reproduced the Boltzmann distribution. Remarkably, when applied to a diffusion model trained for static structure prediction, ConforMix captures structural changes including domain motion, cryptic pocket flexibility, and transporter cycling, while avoiding unphysical states. Case studies of biologically critical proteins demonstrate the scalability, accuracy, and utility of this method.