ICML 2026 Papers

Learning mappings between infinite-dimensional function spaces, or operator learning, is essential for many machine learning applications. Although transformer-based operators are popular, they often rely on token-wise attention. These methods treat continuous fields as discrete tokens and usually ignore the global functional structure. We introduce *Functional Attention*, which reinterprets attention as a functional correspondence between adaptive bases. Inspired by geometric functional maps, our method replaces softmax affinities with structured linear operators. This yields a compact, generalizable, resolution-invariant representation that explicitly captures global dependencies. Experiments demonstrate that *Functional Attention* can match state-of-the-art performance in many operator learning tasks, including solving PDEs, 3D segmentation, and regression, while remaining robust to varying discretizations. Project page is available at https://github.com/xjffff/FUNCATTN.

For low-dimensional problems ($d\leq3$), spectral methods can achieve exceptionally high accuracy. For middle-dimensional problems ($4 \leq d \lesssim 10$), spectral methods remain feasible through specific techniques such as sparse grids or hyperbolic cross. However, for high-dimensional problems ($d\gg 10$), spectral methods suffer frome the curse of dimensionality. Physics-informed neural networks (PINNs) have emerged as a promising approach to overcome this challenge, offering scalability to high dimensions, but often suffer from limited accuracy and efficiency. Recently proposed spectral-informed neural networks (SINNs) combine spectral methods with PINNs, operating directly in the spectral domain to avoid spatial derivative computations and to reduce memory consumption. In this work, we introduce Modified SINNs, which integrate coefficient decay scaling and basis embeddings motivated by harmonic analysis to enhance accuracy in high-dimensional problems and enable accurate approximation of unknown spectral coefficients. Numerical experiments on steady and time-dependent partial differential equations demonstrate that Modified SINNs outperform sparse grid spectral methods on middle-dimensional problems with incomplete spectral information and achieve superior accuracy compared to PINNs on high-dimensional problems.

Programming adaptive behaviors at the cellular level is a long-standing goal that raises the question of how probabilistic computation can be implemented in biochemical systems. Chemical reaction networks (CRNs) provide such a substrate and have been shown to realize probabilistic models, including hidden Markov models and factor graphs, with dynamics reproducing Bayesian inference and belief propagation. However, encoding these algorithms typically requires prohibitively large reaction networks, and classical CRN reduction techniques do not directly apply. By recovering the factor graph structure encoded in Napp--Adams-compiled CRNs, we transport recent factor-graph reduction results to their chemical implementations, obtaining significantly smaller CRNs while preserving the belief-propagation fixed points on surviving variables.

Navigation in complex environments relies on internal spatial representations that guide action. While the brain employs a diverse repertoire of spatial tuning cells—including grid, place, and head-direction cells—a normative theory linking these static neural codes to the dynamic process of navigation remains elusive. In this work, we propose a Unified Laplacian Framework derived from first principles of representational smoothness and efficiency. We first demonstrate that diverse spatial codes emerge naturally as spectral decompositions of the Laplace operator. Crucially, bridging the gap from representation to action, we derive a computational-level navigation policy based on the Green's function potential. We show that this potential encodes the environment's intrinsic geometry to enable geometry-aware gradient ascent, achieving improved sample efficiency and generalization in goal-reaching tasks. Furthermore, we demonstrate that these spectral representations can be learned directly from high-dimensional visual inputs, supporting their learnability from sensory experience. Our results suggest that the ``cognitive map" can be viewed as a spectral embedding of the Laplacian, providing a normative computational account that is biologically consistent with observed spatial-code phenomenology and useful for artificial agents.

Unlike most static material properties widely studied in the machine learning literature, ionic transport properties are inherently dynamic, making their fast and accurate prediction from static atomic structures challenging. The current standard approach, molecular dynamics (MD) simulations, suffers from prohibitively high computational cost. Recent autoregressive learning-based MD acceleration methods requiring sequential inference remain slow and prone to error accumulation; in contrast, existing non-autoregressive material property prediction models are less accurate because they fail to exploit dynamics. Moreover, existing methods typically benefit from datasets either with or without atomic trajectories, but not both. To overcome these limitations, we propose a non-autoregressive learning framework based on auxiliary modality learning, which treats atomic trajectories as an auxiliary modality during training but does not require them at inference. This enables the predictor to learn dynamics without sequential inference while benefiting from both types of datasets. As a result, our framework achieves over 200 times speedup compared to autoregressive models on the dataset with atomic trajectories while substantially reducing prediction error relative to non-autoregressive benchmarks across both types of datasets. Our code is available at https://github.com/jykim-git/MD.

Recent work on Neural Network-based methods for nonlinear control use Lyapunov Functions to obtain controllers with guarantees of stability. However, Lyapunov-based methods are fundamentally limited: they cannot be used for smooth blending with formal Region of Attraction (RoA) expansion guarantees, and also fail to certify stability when unstable equilibria or saddle points are present. Density functions provide an alternate stability certificate, and address these limitations by certifying almost everywhere stability, and enable smooth blending of controllers. Learning valid density certificates is challenging due to integrability constraints, and the effect of density-based blending controllers on RoAs is not well understood. In this work, we provide the first guarantee that controllers blended with density functions yield RoAs containing the union of the RoAs achieved by the constituent controllers. Then, we propose a novel exponential characterization of density functions that provably satisfies the integrability condition, and introduce Neural Control Density Functions (NCDFs), that leverage this new parameterization. We also extend NCDFs for synthesizing safe-stable controllers by combining NCDFs with control barrier functions (NCDF-CBFs). Our experiments show that blended controllers obtain superior RoAs to state-of-the-art methods like Neural Lyapunov Control and Sum-of-Squares based techniques.

This work studies the convergence of two-timescale stochastic approximations (SA), a class of iterative algorithms that update two sets of parameters in fast and slow timescales respectively. Notable examples of two-timescale SA in reinforcement learning (RL) include temporal difference learning with gradient correction (TDC) and actor-critic methods. Previously, the stability (i.e., boundedness) and convergence of two-timescale SA were only established under i.i.d. noise. This work instead establishes the stability and convergence of two-timescale SA under Markovian noise, a setup that is more realistic in RL. Notably, we do not need to use any projection operator and the noise does not need to live in a compact space. Our key technical novelty is to control the fast timescale parameter with the running max of the slow timescale parameter, instead of with the current slow timescale parameter, as most prior works do. As a key application, we establish the first almost sure convergence of TDC with eligibility traces under off-policy learning with linear function approximation.

Scheduling problems arise from repeatedly selecting one item from a set of candidates based on their states. These problems often reduce to assigning priority scores and choosing the highest-ranked item. In this work, we propose a factorized scheduling principle (FSP) framework to learn interpretable and transferable scheduling rules. The FSP framework represents system states as condition distributions and decomposes a global scheduling principle into additive univariate and pairwise components with identifiability constraints. The scheduling principle enables the framework to maintain a simple priority-based structure during deployment. This principle is learned by using a policy-based objective combined with a temporal-difference signal defined on the condition distribution. Experiments on synthetic and realistic scheduling tasks demonstrate the FSP framework's strong performance, interpretability, and zero-shot generalization across different system scales.

Commercial Microwave Links (CMLs) offer dense spatial coverage for rainfall sensing but produce path-integrated measurements that make accurate ground-level reconstruction challenging. Existing methods typically oversimplify CMLs as point sensors and neglect the physical power-law relating rainfall to signal attenuation, resulting in degraded performance under heterogeneous precipitation. In this work, we view rain field reconstruction as a Bayesian inverse problem with Diffusion Models (DMs) as high-fidelity spatial priors. We show that diffusion models better preserve key rainfall statistics compared to censored Gaussian processes. Framing rainfall estimation as a Bayesian inverse problem with a DM prior enables training-free posterior sampling using a broad family of methods, including Plug-and-Play, Sequential Monte Carlo, and Replica Exchange methods. Experiments on synthetic and real-world datasets demonstrate consistent improvements over established CML-based reconstruction baselines.

Coupled spatiotemporal forecasting is important for predicting the future evolution of multiple interacting dynamical systems, such as in climate models. However, existing methods are severely constrained by the persistent bottleneck of compounding errors. In coupled systems, errors from each subsystem simulator propagate and amplify one another, a phenomenon we term Reciprocal Error Amplification leading to a rapid collapse of long-range predictions. To address this challenge, we propose a universal framework called PnP-Corrector (Plug-and-Play Corrector). The core idea of our framework is to decouple the physical simulation from the error correction process: it freezes pre-trained physics simulation engines and exclusively trains a correction agent to proactively counteract the systematic biases emerging from the coupled system. Furthermore, we design an efficient predictive model architecture, DSLCast, to serve as the backbone of this framework. Extensive experiments demonstrate that our method significantly enhances the long-term stability and accuracy of coupled forecasting systems. For instance, in the challenging task of a 300-day global ocean-atmosphere coupled forecast, our PnP-Corrector framework reduces the prediction error of the baseline model by 28% and surpasses state-of-the-art models on several key metrics. Codes link: https://github.com/Alexander-wu/PnP-Corrector.

Modeling complex dynamics from observational data is fundamental to scientific discovery and artificial intelligence. However, existing approaches are often plagued by the entanglement of static state representations and instantaneous motion, leading to accumulated errors and off-manifold hallucinations where predicted trajectories violate intrinsic geometric constraints. To address this, we propose Geometric Flow Grounding, a unified framework that enforces dynamic evolution strictly along the tangent bundle of the learned data manifold via a differentiable Neural Tangent Projection Layer. By geometrically decoupling state representation from tangential dynamics, our method generalizes across diverse data regimes. In scientific discovery, GFG reduces numerical aliasing and improves long-horizon stability in sparse dynamical systems, while recovering interpretable gene regulatory motifs from single-cell data. For trustworthy AI, the projection residual provides a zero-shot metric for deepfake video detection by revealing inconsistencies with the implicit flow of pre-trained world models. Our results establish manifold-constrained projection as a universal operator for both discovering natural laws and verifying synthetic content. Code will be available at \url{https://github.com/yuchang97/GFG-public}

Recent progress in geospatial foundation models highlights the importance of learning general-purpose representations for real-world locations, particularly points-of-interest (POIs) where human activity concentrates. Existing approaches, however, focus primarily on place identity derived from static textual metadata, or learn representations tied to trajectory context, which capture movement regularities rather than how places are actually used (i.e., POI's function). We argue that POI function is a missing but essential signal for general POI representations. We introduce Mobility-Embedded POIs (ME-POIs), a framework that augments POI embeddings derived, from language models with large-scale human mobility data to learn POI-centric, context-independent representations grounded in real-world usage. ME-POIs encodes individual visits as temporally contextualized embeddings and aligns them with learnable POI representations via contrastive learning to capture usage patterns across users and time. To address long-tail sparsity, we propose a novel mechanism that propagates temporal visit patterns from nearby, frequently visited POIs across multiple spatial scales. We evaluate ME-POIs on five newly proposed map enrichment tasks, testing its ability to capture both the identity and function of POIs. Across all tasks, augmenting text-based embeddings with ME-POIs consistently outperforms both text-only and mobility-only baselines. Notably, ME-POIs trained on mobility data alone can surpass text-only models on certain tasks, highlighting that POI function is a critical component of accurate and generalizable POI representations.

The hippocampus encodes space through a striking gradient of place field sizes along its dorsal-ventral axis, yet the principles generating this continuous gradient from discrete grid cell inputs remain unclear. We propose a unified theoretical framework establishing how multiscale hippocampal place fields arise from the frequency-dependent decay of grid cell projections. Functionally, this organization establishes an inductive bias in the population code, managing a fundamental trade-off between spatial precision and sample efficiency. Translating this insight to artificial neural networks, we incorporate a hippocampus-inspired positional embedding (HIPE) into the Transformer architecture to induce multi-scale representation. Experimental results confirm that this mechanism effectively improves data efficiency. Our work establishes a link between neural connectivity, activity patterns, and learning, suggesting a principled approach to utilizing multi-scale representations for sample-efficiency learning. Our codes are available at https://github.com/AIogry/relative_PE.

Decoding fMRI into natural language is challenging because strong, pre-trained language priors can dominate autoregressive generation, obscuring whether a model truly utilizes neural evidence. We introduce BIT-LLM, which exposes fMRI-derived tokens as a persistent key–value memory through interleaved cross-attention adapters, enabling repeated neural access throughout decoding. BIT-LLM is trained with a three-stage pipeline: (i) multimodal contrastive learning to obtain semantically aligned fMRI representations, (ii) supervised fine-tuning to learn the brain-LLM interface while freezing the encoder and backbone LLM, and (iii) reward-based finetuning to optimize sequence-level caption quality directly. On the NSD subject-heldout benchmark (S1-7 train, S8 test), BIT-LLM yields substantially improved captioning quality over prior baselines under greedy decoding. In addition to standard captioning metrics, we perform several complementary evaluations to assess the robustness of brain–language grounding. Specifically, we conduct perturbation-based sanity checks by zeroing fMRI inputs or shuffling voxel values, and examine whether internal representations and generated outputs change accordingly. BIT-LLM exhibits clear sensitivity to these perturbations, indicating effective utilization of voxel values and their spatial correspondence.

Online matching has been a mainstay in domains such as Internet advertising and organ allocation, but practical algorithms often lack strong theoretical guarantees. We take an important step toward addressing this by developing new online matching algorithms based on a coarsening approach. Although coarsening typically implies a loss of granularity, we show that, to the contrary, aggregating offline nodes into capacitated clusters can yield near-optimal theoretical guarantees. We apply our methodology to heart transplant allocation to develop theoretically grounded policies based on structural properties of historical data. Furthermore, in simulations based on real data, our policy closely matches the performance of the omniscient benchmark, achieving competitive ratio 0.91, drastically higher than the US status quo policy's 0.51. Our work bridges the gap between data-driven heuristics and pessimistic theoretical lower bounds.

The robustness of synchronization is typically characterized by scalar, per-node stability indices whose dependence on topology is studied via network science or graph neural networks (GNNs). We propose a novel upstream task, stability landscapes, which provide deeper insights into synchronization behavior and from which many such scalar indices can be derived. Crucially, we pioneer a graph-to-image prediction paradigm: learning image-like landscapes as per-node targets directly from graph topology, a formulation we are not aware of having been established elsewhere in the literature. To support this task, we release two datasets of 10,000 graphs each at 20 and 100 nodes with per-node landscape labels, based on a conceptual oscillator model, capturing power grid synchronization behavior. A GNN encodes topology and a CNN decoder renders per-node images, learned end-to-end with good in-distribution accuracy, generalizing across graph sizes and to realistic power grid topologies. This demonstrates that stability landscapes, while beyond the reach of conventional network science, are learnable from topology and open new avenues for moving beyond scalar stability indices in biology, neuroscience, and power grids.

Attention is the critical component of a transformer. Yet the quadratic computational complexity of vanilla full attention in the input size and the inability of its linear attention variant to focus have been challenges for computer vision tasks. We provide a mathematical definition of generalized attention and formulate both vanilla softmax attention and linear attention within the general framework. We prove that generalized attention disperses, that is, as the number of keys tends to infinity, the query assigns equal weights to all keys. Motivated by the dispersion property and recent development of Mamba form of attention, we design Scalable and Efficient Mamba like Attention (SEMA) which utilizes token localization to avoid dispersion and maintain focusing, complemented by theoretically consistent arithmetic averaging to capture global aspect of attention. We support our approach on Imagenet-1k where classification results show that SEMA is a scalable and effective alternative beyond linear attention, outperforming recent vision Mamba models on increasingly larger scales of images at similar model parameter sizes. Source code can be found at: https://github.com/nhatthanhtran/SEMA.

Deep neural networks currently provide the leading quantitative models of neural responses in sensory systems. However, these networks remain implausible as models of sensory development, largely because they rely on supervised training with label efficiency far exceeding that of biological learning. Furthermore, these models are typically trained on manually curated datasets that lack the statistical properties of the natural environments to which the brain is exposed. Here, we demonstrate that models trained with unsupervised objectives on real-world data significantly outperform supervised models in predicting brain responses across both human auditory and visual cortex. We show that this performance advantage is not driven by network architecture or dataset size, but rather by the data distribution. Crucially, we find that unsupervised models trained on real-world data exhibit remarkable out-of-distribution generalization: a model trained exclusively on Mandarin speech accurately predicts English-driven brain responses, and a model trained on infant head-cam footage predicts adult visual responses to curated object images. Together, our results illustrate how deep neural networks can be used to reveal the real-world statistics that shape neural representations in the brain.

Real-world data is rarely clean; it is plagued by severe class imbalance (long-tailed distributions) and label corruption. Current solutions lean heavily on "black-box" meta-learning to re-weight samples. However, this paradigm introduces a fatal circular dependency: it relies on pristine, balanced validation sets to guide the optimization, which are essentially non-existent in the wild. We propose ProMeCD, a self-referential framework that breaks this dependency by recasting optimization as an autonomous control problem. Instead of training an opaque neural meta-learner, we employ a transparent proportional-integral controller. The system monitors ``cognitive entropy'' that is a metric derived from von Mises-Fisher gradient statistics to assess learning uncertainty. To resolve the scalar ambiguity between tail and noisy samples, ProMeCD employs a decoupled control strategy: it boosts tail classes via integral accumulation of magnitude deficits when directional consistency is high, while suppressing noise via proportional feedback when consistency collapses. Theoretically, we prove that this mechanism guarantees convergence and formally prevents the minority initial drop, ensuring monotonic improvement for rare classes. Crucially, ProMeCD is fully white-box and validation-free. Experiments on CIFAR-LT, iNaturalist, CIFAR-N, mini WebVision, and Clothing1M confirm that ProMeCD is not merely efficient; it outperforms the recent meta-learner FMW-Net by over 10\% in severe imbalance settings, proving that explicit control theory offers a superior path to handling imperfect data.