ICML 2026 论文列表

Tabular data is one of the most fundamental and widely used formats for representing structured information. Classical machine learning algorithms continue to achieve substantial success in extracting predictive patterns and constructing accurate models from structured data; however, representation learning approaches that extend language-model-based methods to the tabular setting have opened new opportunities. Nevertheless, conventional tokenization procedures and token embedding mechanisms are not well-suited to numerical variables, as they fail to preserve key numerical properties, including proximity structure and ordinal relationships. To address this limitation, we propose TabularBERT, a Transformer-based model that discretizes numerical variables via binning-based tokenization and learns representations that account for numerical proximity and ordinal information while capturing conditional dependencies among variables through masked self-supervised pretraining. We empirically demonstrate the effectiveness and interpretability of the proposed approach, highlighting the benefits of language-model-based representation learning in the tabular domain.

Neural operators provide a powerful framework for learning discretization invariant mappings between function spaces, but standard deterministic models do not capture predictive uncertainty. We introduce *diffusion last layer* (DLL), a modular probabilistic output head for neural operator backbones. DLL represents target fields through an input dependent low rank expansion inspired by the Karhunen--Lo\`eve expansion and learns a conditional diffusion model over the corresponding coefficient space. This design enables efficient distributional modeling while preserving the structural advantages of operator learning. On stochastic PDE benchmarks with random forcing, DLL achieves strong distributional fidelity and performs competitively with pixel space and conventional latent diffusion baselines. In deterministic long horizon rollout tasks, DLL improves rollout stability over the underlying backbone and provides useful estimates of predictive uncertainty under compounding autoregressive errors. These results suggest that diffusion modeling in learned coefficient spaces offers a practical route to uncertainty aware neural operators. Code is available at https://github.com/sungwpark/dll-no

Supervised classification has a theoretical optimum, Neural Collapse (NC), yet neither of its two dominant paradigms reaches it in practice. Cross entropy (CE) leaves radial degrees of freedom unconstrained and converges to a degenerate geometry, while supervised contrastive learning (SCL) drives features toward NC during pretraining but discards this structure in a post hoc linear probing phase. We show that both paradigms are different appearances of the same method that contrasts prototypes on the unit hypersphere, and that closing the gap requires fixing each at its point of failure. From the CE side, we propose NTCE and NONL, two normalized losses that import contrastive optimization's missing ingredients into classifier learning: a large effective negative set and decoupled alignment and uniformity terms. From the SCL side, we prove that SCL's objective already optimizes throughout training for a principled classifier whose weights are the class mean embeddings, making linear probing both redundant and harmful. Empirically, on four benchmarks including ImageNet-1K, NTCE and NONL surpass CE accuracy, closely approximate NC ($\geq 95\%$), and match CE's converged NC on 4/5 metrics in under $7.5\%$ of its iterations, while SCL with fixed prototypes matches linear probing without the hours-long classifier training phase. The learned geometry yields $+5.5\%$ mean relative improvement in transfer learning, up to $+8.7\%$ under severe class imbalance, and improved robustness to corruptions on ImageNet-C. Our work recasts supervised learning as prototype learning on the hypersphere, with NC reached *by design*.

Neural operators learn mappings between function spaces, enabling fast surrogate solutions to partial differential equations. Despite remarkable architectural diversity, these methods often share a common input representation: raw coordinate-value pairs $(x, u(x))$. We ask whether inputs aligned with PDE differential structure can improve learning. Through Picard iteration on mild solutions, we show that derivatives of the input appear explicitly in the solution operator, suggesting that providing derivative features should reduce the network's implicit differentiation burden. We prove this intuition: providing derivative features improves approximation rates from $O(W^{-(s-m)/d})$ to $O(W^{-s/d})$, where $W$ is network width, $s$ is input regularity, $m$ is the PDE order, and $d$ is spatial dimension. Our central finding, however, is a surprise: the optimal derivative order $\beta^*$ is strictly less than the PDE order $m$. This gap arises from a bias-variance tradeoff in spectral space that we characterize in closed form. Learning $\beta$ from data achieves automatic spectral regularization. We introduce $\partial$-NO (del-NO), for derivative-augmented neural operators, an architecture-agnostic augmentation that provides learnable fractional derivative features to any neural operator backbone. Across benchmark problems and architectures, $\partial$-NO consistently improves prediction accuracy, with learned orders that reflect a representation of known physics modulated by noise and finite-sample constraints.

AI for fusion requires bridging a critical ``sim-to-real'' gap: simulation-trained models must generalize reliably under distribution shifts in safety-critical workflows. Focusing on the nonlinear Grad-Shafranov equation (GSE), we develop and analyze a physics-anchored operator-learning framework for fixed-boundary equilibrium prediction. The framework combines data anchors with PDE residual constraints and uses a physics-motivated Transformer-KAN Neural Operator (TKNO) to capture global elliptic coupling and nonlinear source response. Under multi-parameter distribution shifts, our analysis shows that data-only surrogates can develop severe OOD tails, while physics-only training may converge to incorrect solution branches; by combining sparse data anchors with PDE constraints, physics-anchored training reduces worst-tail errors on shape-driven and joint shifts. Mechanism diagnostics indicate that data anchors suppress branch-scale failures, whereas PDE residuals reduce OOD-induced high-frequency error amplification. Evaluated on EXL-50U discharge inputs against the device's operational equilibrium solver, the model achieves close agreement (RMSE $<$ 1.3\%) with millisecond-level inference. These results provide a practical route toward physically reliable AI surrogates for fusion workflows. Our code will be available at https://github.com/dsqzhou/physics-anchored-gse.

We study high-dimensional convex empirical risk minimization (ERM) under general non-Gaussian data designs. By heuristically extending the Convex Gaussian Min–Max Theorem (CGMT) to non-Gaussian settings, we derive an asymptotic min–max characterization of key statistics, enabling approximation of the mean $\mu_{\hat{\theta}}$ and covariance $C_{\hat{\theta}}$ of the ERM estimator $\hat{\theta}$. Specifically, under a concentration assumption on the data matrix and standard regularity conditions on the loss and regularizer, we show that for a test covariate $x$ independent of the training data, the projection $\hat{\theta}^\top x$ approximately follows the convolution of the (generally non-Gaussian) distribution of $\mu_{\hat{\theta}}^\top x$ with an independent centered Gaussian variable of variance $\mathrm{tr}\!\big(C_{\hat{\theta}}\,\mathbb{E}[xx^\top]\big)$. This result clarifies the scope and limits of Gaussian universality for ERMs. Additionally, we prove that any $\mathcal{C}^2$ regularizer is asymptotically equivalent to a quadratic form determined solely by its Hessian at zero and gradient at $\mu_{\hat{\theta}}$. Numerical simulations across diverse losses and models are provided to validate our theoretical predictions and qualitative insights.

Semi-supervised learning faces significant challenges in realistic scenarios where labeled data is scarce and unlabeled data follows unknown, arbitrary distributions. We formalize this critical yet under-explored paradigm as Universal Semi-supervised Learning (UniSSL). Existing methods typically leverage unlabeled data via pseudo-labeling. However, they often rely on the idealized assumption of a uniform unlabeled data distribution or require sufficient labeled data to estimate it. In the UniSSL setting, such dependencies lead to numerous erroneous pseudo-labels, thereby triggering representation confusion. Fortunately, we observe that inter-sample relations captured by representations are more reliable than pseudo-labels. Leveraging this insight, we shift our focus to representation-level structural inference to bypass distribution estimation. Accordingly, we propose Simplex Anchored Graph-state Equipartition (SAGE), which captures high-order inter-sample dependencies to establish structural consensus for guiding representation learning. Meanwhile, to mitigate representation confusion, we employ vectors that satisfy a simplex equiangular tight frame to serve as a coordinate frame for guiding inter-class representation separation. Finally, we introduce a weighting strategy based on distribution-agnostic metrics to prioritize reliable pseudo-labels and an auxiliary branch to isolate potentially erroneous pseudo-labels. Evaluations on five standard benchmarks show that SAGE consistently outperforms state-of-the-art methods, with an average accuracy gain of **8.52\%**.

Many scientific and geometric problems exhibit general linear symmetries, yet most equivariant neural networks are built for compact groups or simple vector features, limiting their reuse on matrix-valued data such as covariances, inertias, or shape tensors. We introduce \textbf{Reductive Lie Neurons (ReLNs)}, an exactly $\mathrm{GL}(n)$-equivariant architecture that natively supports matrix-valued and Lie-algebraic features. ReLNs resolve a central stability issue for reductive Lie algebras by introducing a non-degenerate adjoint (conjugation)-invariant bilinear form, enabling principled nonlinear interactions and invariant feature construction in a single architecture that \textit{transfers across subgroups without redesign}. We demonstrate ReLNs on algebraic tasks with $\mathfrak{sl}(3)$ and $\mathfrak{sp}(4)$ symmetries, Lorentz-equivariant particle physics, uncertainty-aware drone state estimation via joint velocity--covariance processing, learning from 3D Gaussian-splat representations, and EMLP double-pendulum benchmark spanning multiple symmetry groups. ReLNs consistently match or outperform strong equivariant and self-supervised baselines while using substantially fewer parameters and compute, improving the accuracy–efficiency trade-off and providing a practical, reusable backbone for learning with broad linear symmetries.

Learning fluid PDE dynamics has increasingly benefited from neural operators and Transformer-based models, but their latent states often remain opaque, and sample-wise attention can be costly at high resolutions. We propose the \emph{Gaussian Particle Operator} (GPO), a basis-to-basis neural operator that represents fields with a learned \emph{Gaussian basis}. Each atom carries explicit geometric parameters, including centers, anisotropic scales, and weights, yielding a compact, mesh-agnostic, and directly visualizable intermediate representation. GPO operates in modal space: learned Gaussian modal windows perform Petrov--Galerkin measurements, and a PG Gaussian Attention module couples the resulting modes globally before scattering them back to the spatial domain. This design is resolution-agnostic, scales near-linearly with the number of samples for a fixed modal budget, and naturally applies to irregular geometries and 3D domains. We separately evaluate Gaussian-basis reconstruction and operator prediction. Across standard PDE benchmarks and real reanalysis datasets, GPO achieves competitive accuracy against neural-operator and Transformer baselines, ranking best on most tasks and close to the best on the remainder. The learned particles and modal couplings further provide representation-level interpretability, positioning GPO as a useful intermediate representation rather than a one-to-one decomposition of physical structures. The code is available at \url{https://github.com/lizhihao2022/GPO}.

We introduce Flower, a neural architecture for learning PDE solution operators built entirely from multihead warps. Aside from pointwise channel mixing and a multiscale scaffold, Flowers use no Fourier multipliers, no dot-product attention, and no convolutional mixing. Each head predicts a displacement field and warps the mixed input features. Motivated by physics and computational efficiency, displacements are predicted pointwise, without any spatial aggregation, and nonlocality enters *only* through sparse sampling at source coordinates, *one* per head. Stacking warps in multiscale residual blocks yields Flowers, which implement adaptive, global interactions at linear cost. We theoretically motivate this design through three complementary lenses: flow maps for conservation laws, waves in inhomogeneous media, and a kinetic-theoretic continuum limit. Flowers achieve excellent performance on a broad suite of 2D and 3D time-dependent PDE benchmarks, particularly flows and waves. A compact 17M-parameter model consistently outperforms Fourier, convolution, and attention-based baselines of similar size, while a 150M-parameter variant improves over recent transformer-based foundation models with much more parameters, data, and training compute.

Machine learning is transforming molecular sciences by accelerating property prediction, simulation, and the discovery of new molecules and materials. Acquiring labeled data in these domains is often costly and time-consuming, whereas large collections of unlabeled molecular data are readily available. Standard semi-supervised learning methods often rely on label-preserving augmentations, which are challenging to design in the molecular domain, where minor changes can drastically alter properties. In this work, we show that semi-supervised methods that rely on an ensemble consensus can boost predictive accuracy across a diverse range of molecular datasets, task types, and graph neural network architectures. We find that training with an ensemble consensus objective increases robustness in models and exhibits an effect similar to knowledge distillation; an individual member of an ensemble trained this way outperforms a full ensemble trained in a traditional supervised fashion in almost all cases. In addition, this type of semi-supervised training reduces calibration error.

Implicit neural representations (INRs) encode data as continuous coordinate-based functions parameterized by neural networks, shifting downstream tasks such as image recognition to operate on functional rather than discrete representations. Despite their increasing adoption, the adversarial robustness of INR-based classification pipelines remain largely underexplored. In this work, we present the first systematic study of adversarial robustness in INR-based classifiers. A key challenge is that generating an INR requires $\text{\emph{training}}$ a neural network for each input sample, resulting in an optimization-in-the-loop forward pass that renders standard gradient-based attacks computationally prohibitive. To address this, we design surrogate models that amortizes the INR-generation process, serving as a practical proxy for attacking INR-based classifiers. We also develop speed-up techniques that substantially reduce the training cost of the surrogate. We show that in contrast to recent work, INR-based classifiers are vulnerable: under adversarial input perturbations, classification accuracy collapses to near zero. Moreover, existing countermeasures designed to operate on discrete representations offer limited protection.

Large reasoning models (LRMs) achieve remarkable reasoning performance by generating long chains-of-thought (CoT). However, standard supervised fine-tuning (SFT) treats all tokens uniformly, indiscriminately minimizing loss across both essential reasoning steps and those that are noisy, redundant, or instance-specific. This often leads student models to memorize superficial patterns rather than acquire generalizable reasoning capabilities. To better understand this limitation, we introduce \textit{Loss Subspace Attribution}, a gradient decomposition analysis approach that uncovers a striking geometric structure: Gradients corresponding to effective reasoning predominantly lie within a low-rank consensus subspace, while conflicting or unstructured signals dominate the residual subspace. Guided by this insight, we propose \textbf{\textit{Spectral-guided Learning}}, a step-level distillation strategy that uses spectral strength to identify reasoning steps aligned with the consensus subspace and prioritizes their contribution to parameter updates, while suppressing gradients from the residual subspace. Experiments across various LRMs and diverse complex reasoning tasks consistently demonstrate that focusing optimization on the consensus subspace yields more robust and generalizable student models.

Label ambiguity/polysemy is pervasive in supervised learning, motivating a variety of representations beyond the traditional single-label setting. While label distribution (LD) provides a probabilistic description and has attracted increasing attention, we reveal its inherent limitations, including inconsistency with raw data, distortion of inter-sample order, and limited applicability. To address these issues, we introduce generalized label distribution (GLD), a unified representation that can perfectly recover raw data while preserving inter-sample order consistency, transform into existing forms of label representations without information loss, and capture out-of-distribution samples as well as negative label correlations. We further develop GLD learning algorithms and demonstrate their effectiveness through both theoretical analysis and extensive experiments.

Counterfactual explanations (CFs) help understand machine learning models by identifying minimal input changes that would lead to alternative model outcomes. Recent work demonstrates their utility for reconstructing black-box models, enabling third-party auditing of opaque decision systems for fairness and accountability. Still, CF-based reconstruction may suffer from decision boundary shifts, overfitting, and restrictive assumptions requiring online query access to target platforms. We propose **REconstruction via Counterfactual-Aware waSserstein opTimization (RECAST)** under limited data and restricted access, a behavioral surrogate model based on Wasserstein barycenteric prototypes. Our approach addresses decision boundary shifts by incorporating CFs as informative, though less representative, samples for both classes, maintaining high surrogate fidelity in low-sample regimes without requiring online access during reconstruction. To enhance fairness auditing, our method enables systematic group fairness diagnostics. Experiments on real-world datasets and various setups show that **RECAST** effectively achieves high fidelity and query efficiency, as well as stable results even when the access is limited and noisy.

Bayesian optimization (BO) has traditionally solved black-box problems where function evaluation is expensive and, therefore, observations are few. Recently, however, there has been growing interest in applying BO to problems where function evaluation is cheaper and observations are more plentiful. In this regime, scaling to many observations $N$ is impeded by Gaussian-process (GP) surrogates: GP hyperparameter fitting scales as $\mathcal{O}(N^3)$ (reduced to roughly $\mathcal{O}(N^2)$ in modern implementations), and it is repeated at every BO iteration. Many methods improve scaling at acquisition time, but hyperparameter fitting still scales poorly, making it the bottleneck. We propose Epistemic Nearest Neighbors (ENN), a lightweight alternative to GPs that estimates function values and uncertainty (epistemic and aleatoric) from $K$-nearest-neighbor observations. ENN scales as $\mathcal{O}(N)$ for both fitting and acquisition. Our BO method, TuRBO-ENN, replaces the GP surrogate in TuRBO with ENN and its Thompson-sampling acquisition with $\mathrm{UCB} = \mu(x) + \sigma(x)$. For the special case of noise-free problems, we can omit fitting altogether by replacing $\mathrm{UCB}$ with a non-dominated sort over $\mu(x)$ and $\sigma(x)$. We show empirically that TuRBO-ENN reduces proposal time (i.e., fitting time + acquisition time) by one to two orders of magnitude compared to TuRBO at up to 50,000 observations without sacrificing solution quality.

The Gaussian graphical model (GGM) is a useful tool to represent relationships of conditional dependence among variables. In many real-world applications, datasets often contain multiple related sub-populations, whose associated GGMs may have common structure, as well as large structural differences. In such cases, it is useful to recover common hub variables, which are the highly connected variables in the GGMs of all sub-populations. In this paper, we propose the Joint Inverse Components for Hub Detection (JIC-HD) method to recover the common hubs across multiple GGMs without the need to estimate all subpopulation GGMs. To this end, we introduce joint minimax eigenspaces, and show that these can be leveraged for the recovery of common hubs. We establish theoretical guarantees for the recovery of common hubs. Additionally, our numerical simulation studies confirm superior performance of our JIC-HD in detecting common hubs compared to the existing methods in the literature. Our method is especially advantageous when the multiple GGMs have both common and individual hubs across sub-populations. Finally, we analyze cancer gene-expression datasets and identify biologically meaningful common hub genes across cancer subtypes.

Neural operators perform well on structured domains, yet their behaviour on irregular geometries remains poorly understood. We show that this limitation is not merely an encoding issue, but a depth-wise failure mode inherent to deep operator architectures. We formalise the *Geometric Forgetting Hypothesis*: due to the Markovian structure of operator layers and their reliance on global mixing mechanisms, neural operators progressively lose access to domain geometry as depth increases. Using layer-wise geometric probing, we demonstrate that both spectral and attention-based operators systematically lose geometric fidelity. We show that this geometric forgetting degrades accuracy, stability, and generalisation. To counteract it, we introduce a lightweight *geometry memory injection* mechanism that restores geometric constraints at intermediate depths with minimal architectural overhead. This simple intervention consistently mitigates forgetting and exposes a *geometric shortcut* instability in transformer-based operators, revealing that geometric retention is a structural requirement rather than a design choice.

In interactive task learning (ITL), AI agents learn new capabilities from limited human instruction provided during task execution. STAND is a new method of data-efficient rule precondition induction specifically designed for these human-in-the-loop training scenarios. A key feature of STAND is its self-awareness of its own learning—it can provide accurate metrics of training progress back to users. STAND beats popular methods like XGBoost, decision trees, random forests, and version spaces at small-data precondition induction tasks, and is highly accurate at estimating when its performance improves on holdout examples. In our evaluations, we find that STAND shows more monotonic improvement than other models with low rates of error reoccurrence. These features of STAND support a consistent training experience, enabling human instructors to estimate when they have finished training and providing active-learning support by identifying trouble spots that require more training. STAND achieves this by efficiently learning a compact space of greedy classifiers consistent with training data, rather than a finite ensemble of alternatives.