NeurIPS 2026 Papers

We introduce the Z-Domain Neural Operator (ZNO), a causal neural operator whose layers are stable low-rank multiple-input multiple-output (MIMO) rational filters parameterized directly in the $z$-plane. This operator is designed to address a critical limitation of existing operator learning methods, as most of these methods are primarily tailored for continuous-time problems, while a large class of system-identification problems is intrinsically discrete-time. The $z$-domain form expresses stability as a unit-disk pole constraint and makes learned discrete-time poles directly readable. The model combines low-rank channel mixing, smooth stable pole reparameterization, causal recurrence, and an optional short finite impulse response (FIR) branch in a single $z$-domain rational recurrent layer. Across controlled discrete system-identification experiments, ZNO's advantage is most evident when the target dynamics are stable rational systems with lightly damped poles near the unit circle. Under matched parameter budgets, ZNO is not uniformly dominant; however, with validation-selected configurations, the same architecture can achieve the lowest mean error across the controlled tasks. A five-bin difficulty sweep over near-unit-circle / long-memory dynamics further shows that ZNO has the lowest mean error across all memory regimes, from short ($\approx 10$ steps) to long ($\approx 100-200$ steps). On five public nonlinear system-identification benchmarks, ZNO is competitive with neural operator and state-space baselines, achieving the lowest mean error on benchmarks whose dynamics align with stable rational discrete-time filters, while classical or state-space baselines remain preferable on some systems. These results position ZNO as a strong model for stable rational discrete-time dynamics, especially in near-unit-circle and long-memory regimes, but not as a universal replacement for specialized system-identification methods.

The effectiveness of modern deep learning models in forecasting *groups of time series* is often attributed to their ability to capture (long-range) dependencies across long observation windows. In this paper, we show that this forecasting task involves two objectives: (i) *generative process identification* (GPI), i.e., inferring the specific process generating the input sequence, and (ii) *conditional forecasting* (CF), i.e., predicting future values given input observations. From this perspective, optimal predictions can be interpreted as an average over plausible data-generating processes, weighted by their likelihood given the input window. This suggests a different explanation for the benefits of long context windows: they reduce the uncertainty about which specific process is generating the input time series during operation. We prove that even for processes with memory length $P$, an input window size strictly larger than $P$ is *necessary* to achieve the minimum attainable error. Finally, we show how decoupling GPI and CF can improve computational scalability without compromising accuracy. Experiments on synthetic and real-world data validate our insights and their relevance for designing forecasting architectures.

Spatiotemporal forecasting plays an important role in real-world applications, where probabilistic approaches provide uncertainty estimates for decision-making. Real-world systems are inherently non-stationary, leading to distribution shifts between training and test data. However, existing probabilistic methods mainly focus on modeling data distributions under standard settings, while distribution shift methods often handle such shifts in a coarse manner, failing to characterize shift degree and adapt prediction strategy accordingly. As a result, they often produce miscalibrated and unreliable forecasts. To address this issue, we propose REMIND, a retrieval-guided probabilistic framework for spatiotemporal forecasting under distribution shift. REMIND leverages retrieval to estimate how well current observation are supported by historical patterns, enabling adaptive prediction under different shift regimes. It employs spatiotemporal retrieval to construct a memory bank and introduces a dual branch architecture, combining a memory-calibrated probabilistic branch for mild shifts with a deviation-aware probabilistic branch for severe shifts. A shift-adaptive gate, guided by retrieval similarity and memory dispersion, balances the two branches according to shift degree. We further design a hierarchical Gaussian mixture to capture heterogeneous uncertainty over the dual branch predictions, and introduce a memory perturbation training scheme to improve generalization. Experiments on real-world datasets with distribution shifts demonstrate that REMIND consistently outperforms state-of-the-art methods in both forecasting accuracy and probabilistic prediction quality. Code is available at https://anonymous.4open.science/r/REMIND-A46E/.

For $K=2$ regimes, scalar thresholding of the Bayesian posterior is Bayes-optimal. For $K \geq 3$, the posterior evolves on a $(K-1)$-dimensional simplex $\Delta^{K-1}$, yet virtually all deployed detection systems reduce it to a scalar score without theoretical justification. We resolve this question completely. We introduce **Extended Decision Sufficiency (EDS)**—three linear constraints on emission densities and transition dynamics (Rank-One Emissions, Markov Factorisation, Normal Factorisation), verifiable in $O(K^2)$ operations from model parameters alone—and prove the following exact characterisation: **Main result.** EDS holds if and only if the Bayes-optimal stopping value function is constant on every level set of a scalar statistic $\phi : \Delta^{K-1} \to \mathbb{R}$. Sufficiency follows from algebraic closure of the Bellman operator $\mathcal{B}$ under EDS. Necessity is proved by a coupling-based fixed-point contradiction: for each failure mode we exhibit an explicit belief pair with equal $\phi$-values but a strictly positive total-variation gap on predictive densities, which propagates to a value-function gap via the Lipschitz bound on $\mathcal{B}$ and the $\rho$-contraction of $\mathcal{T}$, without assuming any structure on $V$ beyond continuity. Three consequences are sharp and quantitative: * **(i)** When EDS fails, *every* scalar rule—regardless of architecture, capacity, or training data—incurs a strictly positive, computable, algorithm-independent lead-time loss. * **(ii)** When EDS holds, optimal expected lead time is given in closed form by a hitting-time functional of a scalar Markov chain. * **(iii)** Under approximate EDS with deviation $\eta$, performance degrades at rate $O(\eta/(1-\rho))$, making $\eta$ a practical model-diagnostic with direct performance implications. Across 72 synthetic configurations and the CICIDS2017 benchmark, predicted and observed lead times agree within 0.4 steps (MAE). A controlled experiment with matched parameter counts confirms that the lead-time advantage is structural, not a regularisation artefact, thereby validating the theory as the sole causal mechanism.

Large language models are increasingly deployed with test-time strategies: sample $N$ responses, score them with a reward model or verifier, and return the best. This deployment rule exposes a mismatch in post-training: standard objectives optimize the mean reward of a single response, whereas best-of-$N$ performance is governed by the upper tail of the reward distribution. Recent test-time-aware objectives partly address this mismatch, but typically assume that training can use the same per-prompt rollout budget as deployment, which is impractical when post-training must cover many prompts while deployment can allocate much larger per-prompt test-time compute. We study this budget-mismatch regime, where only $m\ll N$ per-prompt rollouts are available during training but the target objective is best-of-$N$ deployment. Under structural assumptions on the reward tails, we show that the policy gradient of the best-of-$N$ objective can be approximated from a much smaller rollout group by extrapolating upper-tail statistics. This yields a family of Tail-Extrapolated estimators for best-of-$N$-oriented post-training: a simple direct estimator, Tail-Extrapolated Advantage (TEA), and a fixed-order debiased Prefix-TEA estimator based on moment cancellation. Experiments on instruction-following tasks show that TEA and Prefix-TEA improve best-of-$N$ performance across different language models, reward models and datasets under various training and test-time budget settings.

Recent work has shown that flow matching models can be trained without explicit time conditioning, challenging the standard view that the interpolation time is needed to disambiguate velocity targets. But why should a time-blind model work at all? Decomposing the time-blind flow matching loss, we identify two sources of irreducible error: a *coupling variance*, which arises from ambiguous velocity targets induced by how noise and data points are paired, and the *time-blindness gap*, which is the additional error caused by ignoring time. This gap shows that time-blind training is strictly harder than conventional training, reinforcing the puzzle that time-blind models work so well in practice. We resolve this tension by showing that the geometry of high-dimensional data makes time identifiable directly from noisy observations. When data concentrates near a $k$-dimensional subspace, time can be recovered from the statistical structure of noisy interpolants in directions orthogonal to the data; under a spiked-covariance model, this yields a closed-form estimator that recovers $t$ from a single observation $z$ at rate $O(1/\sqrt{d-k})$ for ambient dimension $d$. As a consequence, we prove that the time-blindness gap is asymptotically negligible relative to the coupling variance. We empirically demonstrate our identifiability result on real-world data and show that changing the coupling has a much larger effect on loss and sample quality than removing time conditioning across CIFAR-10, CelebA-HQ, and FFHQ. These results explain why time-blind flow matching works and show that the main practical lever is the choice of coupling, not explicit time conditioning.

Time-series foundation models (TSFMs) are increasingly deployed as zero-shot forecasters in observability platforms, yet the operational concepts encoded within their internal representations remain largely un-audited. To address this gap, we introduce $\texttt{toto-interp}$, a control-first interpretability protocol that pairs linear probes with four rigorous baselines: raw-feature, shuffled-label, randomized-backbone, and supervised raw-window Fourier Neural Operator (FNO) controls. Applying this harness to TOTO on BOOM, we demonstrate that three structural telemetry concepts, such as cadence bucket ("frequency"), metric type, and domain, are linearly decodable, layer-localized, and specifically tied to pretraining. A budget-matched replication on MOMENT-base successfully recovers these same axes, confirming these representations are not isolated artifacts of the TOTO architecture. Furthermore, our matched-null and on-manifold interventions reveal a critical two-way dissociation between linear decodability and forecast-sensitivity. For instance, concepts like future burstiness are highly decodable yet completely inert under intervention, whereas other weakly decodable directions can severely disrupt forecasts off-manifold. By releasing the audit harness and detailing negative control cases (such as cardinality and shift risk), we demonstrate that linear decodability alone is insufficient for reliable post-training steering or deployment auditing in TSFMs.

An important characteristic of temporal graphs is how the directed arrow of time influences their causal topology, i.e., which nodes can possibly influence each other causally via time-respecting paths. The resulting patterns are often neglected by temporal graph neural networks (TGNNs). To formally analyze the expressive power of TGNNs, we lack a generalization of graph isomorphism to temporal graphs that captures their causal topology. Addressing this gap, we introduce consistent event graph isomorphism, which utilizes a time-unfolded representation of time-respecting paths in temporal graphs. We compare this definition with existing notions of temporal graph isomorphisms. We highlight the advantages of our approach and develop a temporal generalization of the Weisfeiler-Leman algorithm to heuristically distinguish non-isomorphic temporal graphs. Building on this theoretical foundation, we derive a novel message passing scheme for temporal graph neural networks that operates on the event graph representation of temporal graphs and experimentally evaluate it in a temporal graph classification setting.

Dynamic scene reconstruction from monocular videos remains highly challenging, as existing methods often struggle to balance global structural coherence and local fine-grained details under limited multi-view cues. To address this challenge, we propose WebSpline, a novel dynamic 3D Gaussian framework that enables structurally coherent and high-fidelity reconstruction from monocular videos with fast rendering. The core of WebSpline is the Structure-Informed Spline (SIS) representation, which models each dynamic Gaussian trajectory using a learnable cubic Hermite spline whose motion is structurally organized with an auxiliary Structural Proxy Graph (SPG). The proposed framework is optimized in two stages: (i) in the first stage, the SPG is initialized from 2D point tracks and refined with temporal rigidity regularization to establish structural coherence for moving objects across the sequence; and (ii) in the second stage, the SIS representation is initialized from the refined SPG and optimized under both spatial and structural neighborhood constraints. At inference, Gaussian motion is obtained solely by evaluating the learned SIS, enabling fast rendering. Extensive experiments on the challenging monocular dynamic scene benchmarks, iPhone and NVIDIA, demonstrate that our WebSpline achieves state-of-the-art rendering quality while rendering over 10x faster than WorldTree, the second-best method on the iPhone dataset.

Transformer-based time series forecasting models often suffer from block-wise attention collapse, where temporal smoothness and cross-variable correlations concentrate attention locally and restrict global modeling capacity. We argue that this collapse is partly a pre-attention representation-conditioning problem: attention logits inherit the spectral concentration and low effective rank of the input representation. We propose Widely Linear Frequency-Domain (WILD) conditioning, a lightweight, model-agnostic module inserted before self-attention. WILD maps inputs to the frequency domain, applies data-dependent spectral preconditioning to redistribute over-dominant frequency energy, and then uses asymmetric real/imaginary projections to realize a widely linear transform beyond fixed time-domain channel mixing. The preconditioning step is essential: it constructs the usable normalized frequency basis on which the widely linear branch can operate. Thus, WILD uses frequency-domain modeling to condition an existing attention layer, not to replace it with a new forecasting architecture. Extensive experiments on eight benchmark datasets show positive gains on 105 of 112 model-dataset-metric configurations across seven backbones. Attention diagnostics show that WILD substantially increases effective rank over the baseline, and a W_r=W_i ablation identifies spectral preconditioning as the primary mechanism, with widely linear mixing providing a secondary, dataset-dependent gain.

Compared with deterministic methods that output only point estimates, probabilistic forecasting can characterize both future trends and their uncertainty simultaneously, making it more suitable for decision-making in complex real-world scenarios. In recent years, diffusion models, owing to their powerful generative modeling capability, have been introduced into the field of time series forecasting. However, we find that most existing diffusion-based forecasting methods directly construct the diffusion process in the original future space, which is easily affected by heteroscedastic scale imbalance, causing highly volatile dimensions to dominate the denoising learning and leading to unstable corrections near observation boundaries. Motivated by these issues, we propose VolaRM, a volatility-whitened probabilistic residual modeling framework for long-term time series forecasting. By constructing a conditional probabilistic base distribution, it represents future targets as whitened residuals relative to this base distribution and performs conditional modeling in the normalized residual space, thereby alleviating scale imbalance across variables and forecasting horizons. Meanwhile, a specific gating mechanism is introduced to enhance boundary continuity and the stability of long-term forecasting. Extensive experiments on eight real-world datasets from different application domains demonstrate the effectiveness of VolaRM.

Flow matching trains a continuum of velocity regression problems but usually reports a loss averaged over interpolation time, obscuring which regions are limited by intrinsic ambiguity and which remain data-limited. We introduce the Velocity Ambiguity Profile (VAP), $\mathcal{A}_v(t)$, the time-local Bayes-risk floor of velocity prediction, and decompose time-resolved loss as $L_N(t)=\mathcal{A}_v(t)+R(t;N)$, where only the excess term $R(t;N)$ is reducible by more data or a different estimator. We prove two statistical benchmarks. In an intrinsic local-regression benchmark, $\mathcal{A}_v(t)$ enters the time-dependent noise factor of the standard nonparametric rate. In separated isotropic Gaussian mixtures, the Bayes velocity becomes locally affine, and a label-oracle residual-regression benchmark has parametric excess-risk scale $K\mathcal{A}_v(t)/N$, matched within the same separated residual Gaussian-location subproblem. Controlled Gaussian-mixture experiments validate this diagnostic through a dense transition sweep, calibrated oracle-to-blind and empirical-proxy audits, and a local controlled time-sampler stress test in which a frozen reducible-error proxy reduces reducible loss while cross-cell sensitivity delimits aggressive sampling. VAP turns an averaged flow-matching objective into a time-resolved diagnostic for locating ambiguity-limited and data-limited regions.

Inference-time reward guidance is a simple way to improve generative models when completed outputs can be checked or scored by an external verifier. Best-of-N sampling is a particularly strong baseline: it is parallelizable and often highly improves with more samples. However, when the reference model assigns low probability to reward-satisfying samples, selection among independent full rollouts becomes sample-inefficient. We introduce MDM-VGB, a reward-guided discrete diffusion sampler that augments masked generation with value-guided re-masking. MDM-VGB extends the autoregressive VGB backtracking chain from fixed left-to-right prefix trees to any-order masked-state graphs, allowing the sampler to reveal and revise tokens at any position. The resulting Markov chain favors local reveal and re-mask moves that lead to higher-value partial masked states, enabling both root-start reward tilting and leaf-start editing of low-reward samples. We further introduce a flow-cancelled momentum lift of MDM-VGB that preserves the projected target law while reducing oscillatory reveal/re-mask dynamics under finite budgets. We prove that the sampler has the correct reward-tilted leaf law, places non-negligible stationary mass on complete outputs, and remains robust to partial-state verifier error. Across structured reward satisfaction and repair tasks, MDM-VGB improves the quality--cost frontier over reference sampling, best-of-N, and forward-only value-guided rollout.

Time-series anomaly detection (TSAD) requires identifying both immediate Point Anomalies and long-range Context Anomalies. However, existing zero-shot foundation models face a fundamental trade-off: 1D temporal models provide fine-grained pointwise localization but lack a global contextual perspective, while 2D vision-based models capture global patterns but suffer from information bottlenecks due to a lack of temporal alignment and coarse-grained pointwise detection. To resolve this dilemma, we propose VETime, the first TSAD framework that unifies temporal and visual modalities through fine-grained visual-temporal alignment and dynamic fusion. VETime introduces a Reversible Image Conversion and a Patch-Level Temporal Alignment module to establish a shared visual-temporal timeline, preserving discriminative details while maintaining temporal sensitivity. Furthermore, we design an Anomaly Window Contrastive Learning mechanism and a Task-Adaptive Multi-Modal Fusion to adaptively integrate the complementary perceptual strengths of both modalities. Extensive experiments demonstrate that VETime significantly outperforms state-of-the-art models in zero-shot scenarios, achieving superior localization precision with lower computational overhead than current vision-based approaches.

Raman spectroscopy is often hindered by strong autofluorescence backgrounds, which are typically handled by waiting for fluorescence to bleach before recording the Raman spectrum. Here, we instead show that short, incomplete recordings of the bleaching process already contain enough information to recover the underlying Raman signal. We present the first unsupervised model that operates on short, incomplete, variable-length sequences of time-resolved spectra, decomposing them into Raman spectra and autofluorescence components while simultaneously removing measurement noise. Our VAE-based method simultaneously learns (i) a bank of fluorophore spectra comprising the baseline, (ii) a model of the measurement noise, and (iii) the distribution of Raman spectra and decay behavior. We evaluate our method on spectral reconstruction and downstream peak detection, where it outperforms state-of-the-art approaches on multiple simulated benchmarks as well as on a newly collected real-world dataset of spectral time series. We further show that fluorescence decay dynamics themselves contain discriminative information that may be exploited in future work.

Recent work on the sequence universality of State Space Models (SSMs) has introduced efficient, maximally expressive continuous-time approaches for time-series modelling. While these works focus on discriminative settings, we extend this perspective to generative time-series modelling by proving that maximally expressive Structured Linear Controlled Differential Equations (SLiCEs) are universal time-series generators, in the sense that they can approximate the induced path laws of continuous causal pushforwards on compact latent sets in $W_\infty$. Building on these theoretical results, we propose Generative SLiCEs (G-SLiCEs), a maximally expressive continuous-time model for flow matching on path-space. Empirically, we show that expressivity improves performance in probabilistic forecasting and downstream tasks, while retaining the advantages of continuous-time models such as generalising to arbitrary observation grids. This is particularly beneficial for irregular grids, where fixed-grid models often struggle.

Universal approximation theorems establish the expressive capacity of neural network architectures. For dynamical systems, existing results are limited to finite time horizons or systems with a globally stable equilibrium, leaving multistability and limit cycles unaddressed. We prove that Neural ODEs achieve $\varepsilon$-$\delta$ closeness, i.e., trajectories within error $\varepsilon$ except for initial conditions of measure $< \delta$, over the \emph{infinite} time horizon $[0,\infty)$ for three target classes: (1) Morse-Smale systems (a structurally stable class) with hyperbolic fixed points, (2) Morse-Smale systems with hyperbolic limit cycles via exact period matching, and (3) systems with normally hyperbolic continuous attractors via discretization. We further establish a temporal generalization bound: $\varepsilon$-$\delta$ closeness implies $L^p$ error $\leq \varepsilon^p + \delta \cdot D^p$ for all $t \geq 0$, bridging topological guarantees to training metrics. These results provide the first universal approximation framework for multistable infinite-horizon dynamics.

We consider two-layer neural networks trained in the feature-learning regime using gradient descent, and relate the output of the finite-width network $f_{\hat{\rho}}$ to its infinite-width counterpart $f_{\rho^{MF}}$, which evolves in the mean-field dynamics. While constant-time horizon bounds for $\|f_{\rho^{MF}} -f_{\hat{\rho}}\|$ may be obtained via standard Grönwall estimates, the long-time behavior of the fluctuation is a more delicate matter. Uniform-in-time bounds often rely on (local) strong convexity in the landscape or Logarithmic Sobolev inequalities present in noisy gradient dynamics. % In this work, we study a noiseless setting and do not make assumptions on the geometry of the landscape near the optimum. Instead, we impose a weaker condition on the convergence rate of the mean-field deterministic Wasserstein-gradient-flow dynamics. In this work, we establish non-asymptotic weak propagation-of-chaos that holds uniformly in time, obtained by exploiting instead the convergence rate of the mean-field deterministic Wasserstein-gradient-flow dynamics. Specifically, denoting by $L_t$ the mean-field loss at time $t$ and $m$ the number of neurons, under standard regularity assumptions and the condition $\int_0^\infty L_t^{1/2} dt =O_d(1)$, we obtain the uniform in time bound $\|f_{\rho^{MF}} -f_{\hat{\rho}}\|^2 \lesssim \text{poly}(d) m^{-\min(1,c/6)}$ whenever $L_t \lesssim t^{-c}$. Our result holds in a noiseless setting and does not make any assumptions on the geometry of the landscape near the optimum. A key implication of our result is that whenever the convergence rate of the mean-field, population-loss dynamics is faster than $1/t^2$, we can attain a loss of $\epsilon$ with only $\text{poly}(d/\epsilon)$ neurons, training samples, and GD steps.

Time-parallel neural operators (FNO and successors) have transformed PDE surrogates on regular grids. Mesh-based operators (MeshGraphNets, Transolver) extend operator learning to irregular geometries but remain autoregressive in time and trained one geometry at a time, while classical parallel-in-time methods converge poorly on nonlinear advection-dominated PDEs. We propose space-time factorization: partition the model into local temporal operators (learned, graph-independent) and a spatial mixer that propagates information across nodes without graph-dependent learned parameters. This structural condition restores time-parallelism on irregular graphs and yields cross-graph zero-shot transfer once three information leaks—graph-dependent weights, graph-specific features, and graph-correlated training distributions—are controlled. The same protocol applies to two distinct spatial-mixer families tested here—fixed-spectral and learned-attention—suggesting the principle is not tied to a single mixer choice. We validate the principle in three domains. On simulating floodwave propagation in river networks, a 27K-parameter model trained on the Yangtze transfers zero-shot to the Mississippi at $R^2=0.998$, training $28\times$ faster than a recurrent message-passing baseline and scaling to $5\times$ larger graphs. On cross-geometry Navier–Stokes, both Chebyshev (SepFNO) and Physics-Attention (SepTransolver) variants reach within ~10% (relative $L_2$) of a locally-trained oracle using only 15 training geometries—to our knowledge the first demonstration of knowledge accumulation in this regime across fundamentally different topologies. On cross-city traffic forecasting, zero-shot transfer reaches within 12.9% of an oracle trained on the target city. A consistent pattern emerges across these settings: cross-geometry error decreases with both the number of training geometries and their relatedness to the test, with the rate of accumulation depending on the spatial mixer. The result opens the path for pooled-data, foundation-style models for graph dynamics on irregular topologies.