ICLR 2026 论文列表

Neural operators offer powerful approaches for solving parametric partial differential equations, but extending them to spherical domains remains challenging due to the need to preserve intrinsic geometry while avoiding distortions that break rotational consistency. Existing spherical operators rely on rotational equivariance but often lack the flexibility for real-world complexity. We propose a generalized operator-design framework based on \textbf{designable Green’s function} and its harmonic expansion, establishing a solid operator-theoretic foundation for spherical learning. Based on this, we propose an absolute and relative position-dependent Green’s function that enables flexible balance of equivariance and invariance for real-world modeling. The resulting operator, \textit{Green's-function Spherical Neural Operator} (GSNO) with a novel spectral learning method, can adapt to non-equivariant systems while retaining spherical geometry, spectral efficiency and grid invariance. To exploit GSNO, we develop SHNet, a hierarchical architecture that combines multi-scale spectral modeling with spherical up-down sampling, enhancing global feature representation. Evaluations on diffusion MRI, shallow water dynamics, and global weather forecasting, GSNO and SHNet consistently outperform state-of-the-art methods. The theoretical and experimental results position GSNO as a principled and generalized framework for spherical operator learning, bridging rigorous theory with real-world complexity. The code is available at: \url{https://github.com/haot2025/GSNO}.

Transformers robustly exhibit the ability to perform in-context learning, whereby their predictive accuracy on a task can increase not by parameter updates but merely with the placement of training samples in their context windows. Recent works have shown that transformers achieve this by implementing gradient descent in their forward passes. Such results, however, are restricted to standard transformer architectures, which handle finite-dimensional inputs. In the space of PDE surrogate modeling, a generalization of transformers to handle infinite-dimensional function inputs, known as "continuum transformers," has been proposed and similarly observed to exhibit in-context learning. Despite impressive empirical performance, such in-context learning has yet to be theoretically characterized. We herein demonstrate that continuum transformers perform in-context operator learning by performing gradient descent in an operator RKHS. We demonstrate this using novel proof strategies that leverage a generalized representer theorem for Hilbert spaces and gradient flows over the space of functionals of a Hilbert space. We additionally show the operator learned in context is the Bayes Optimal Predictor in the infinite depth limit of the transformer. We then provide empirical validations of this optimality result and demonstrate that the parameters under which such gradient descent is performed are recovered through the continuum transformer training.

Local non-stationarity is pivotal to solving partial differential equations (PDEs). However, in operator learning, the spatially local information inherent in the data is often overlooked. Even when explicitly modeled, it is usually collapsed into local superpositions within the model architecture, preventing full exploitation of local features in physical phenomena. To address this limitation, our paper proposes a novel Riesz Neural Operator (RNO) based on the spectral derivative representation. Since PDEs are fundamentally governed by local derivatives, RNO leverages the Riesz transform, a natural spectral representation of derivatives, to mix global spectral information with local directional variations. This approach allows the RNO to outperform existing operators in complex scenarios that require sensitivity to local detail. Our design bridges the gap between physical interpretability and local dynamics. Experimental results demonstrate that the RNO consistently achieves superior prediction accuracy and generalization performance compared to existing approaches across various benchmark PDE problems and complex real-world datasets, presenting superior non-linear reconstruction capability in model analysis.

Few-shot text classification is a critical real-world task for which Large Language Models (LLMs) have shown great promise. However, their high inference costs and lack of interpretability limit their practical use. While Concept Bottleneck Models (CBMs) offer an efficient and interpretable alternative, their reliance on training surrogate models makes them incompatible with few-shot scenarios. To bridge this gap, we introduce a novel CBM paradigm that relies solely on sample-concept similarity to make predictions. We ensure the effectiveness of our concepts through a prototypical-discriminative dual-level architecture and a dynamic concept refinement mechanism. Extensive experiments show that with as few as 10 training samples, our method surpasses prior CBMs and even achieves performance comparable to LLMs. The code is available at https://anonymous.4open.science/r/StructCBM-EB1E.

Multi-objective optimization (MOO) arises in many real-world applications where trade-offs between competing objectives must be carefully balanced. In the offline setting, where only a static dataset is available, the main challenge is generalizing beyond observed data. We introduce Pareto-Conditioned Diffusion (PCD), a novel framework that formulates offline MOO as a conditional sampling problem. By conditioning directly on desired trade-offs, PCD avoids the need for explicit surrogate models. To effectively explore the Pareto front, PCD employs a reweighting strategy that focuses on high-performing samples and a reference-direction mechanism to guide sampling towards novel, promising regions beyond the training data. Experiments on standard offline MOO benchmarks show that PCD achieves highly competitive performance and, importantly, demonstrates greater consistency across diverse tasks than existing offline MOO approaches.

Modern AI agents such as large language models are trained on diverse tasks---translation, code generation, mathematical reasoning, and text prediction---simultaneously. A key question is to quantify how each individual training task influences performance on a target task, a problem we refer to as task attribution. The direct approach, leave-one-out retraining, measures the effect of removing each task, but is computationally infeasible at scale. An alternative approach that builds surrogate models to predict a target task's performance for any subset of training tasks has emerged in recent literature. Prior work focuses on linear surrogate models, which capture first-order relationships, but miss nonlinear interactions such as synergy, antagonism, or XOR-type effects. In this paper, we first consider a unified task weighting framework for analyzing task attribution methods, and show a new connection between linear surrogate models and influence functions through a second-order analysis. Then, we introduce kernel surrogate models, which more effectively represent second-order task interactions. To efficiently learn the kernel surrogate, we develop a gradient-based estimation procedure that leverages a first-order approximation of pretrained models; empirically, this yields accurate surrogate estimates with less than 2% relative error without repeated retraining. Experiments across multiple domains---including mathematical reasoning in transformers, in-context learning, and multi-objective reinforcement learning---demonstrate the effectiveness of kernel surrogate models. They achieve a 25% higher correlation with the leave-one-out ground truth than linear surrogates and influence-function baselines, enabling more accurate and scalable task attribution. When used for downstream task selection, kernel surrogate models further yield a 40% improvement in demonstration selection for in-context learning and multi-objective reinforcement learning benchmarks.

Neural networks (NNs) often have critical behavioural trade-offs that are set at design time with hyperparameters—such as reward weighting in reinforcement learning or quantile targets in regression. Post-deployment, however, user preferences can evolve, making initially optimal settings undesirable, necessitating expensive retraining. To circumvent this, we introduce the task of Hyperparameter Trajectory Inference (HTI), to learn, from observed data, how a NN's conditional output distribution changes as a function of its hyperparameters, such that a surrogate model can approximate the NN at unobserved hyperparameter settings. HTI requires extending existing trajectory inference approaches to incorporate conditions, posing key challenges to ensure meaningful inferred conditional probability paths. We propose an approach grounded in conditional Lagrangian optimal transport theory, jointly learning the Lagrangian function governing hyperparameter-induced dynamics along with the associated optimal transport maps and geodesics, which form the surrogate model. We incorporate inductive biases based on the manifold hypothesis and least-action principles into the learned Lagrangian, improving surrogate model feasibility. We empirically demonstrate that our approach reconstructs NN behaviour across hyperparameter spectrums better than other alternatives, enabling effective inference-time adaptation of NNs.

Deep neural networks (DNNs) are highly vulnerable to adversarial attacks, where small, carefully crafted perturbations are added to input images to cause misclassification. These perturbations are particularly effective when concentrated in sensitive regions of an image that strongly influence the model’s prediction. However, in decision-based black-box settings, where only the top-1 predicted label is observable and query budgets are strictly limited, identifying sensitive regions becomes extremely challenging. This issue is critical because without accurate region information, decision-based attacks cannot refine adversarial examples effectively, limiting both their efficiency and accuracy. We propose Sensitive Region Identification, SeRI, the first decision-based method that assigns a continuous sensitivity score to each image pixel. It enables fine-grained region discovery and substantially improves the efficiency of adversarial attacks, all without access to gradients, confidence scores, or surrogate models. SeRI progressively partitions the image into finer sub-regions and refines a continuous sensitivity score to capture their true importance. At each iteration, it generates two perturbation variants of the selected region by scaling its magnitude up or down, and compares their decision boundaries to derive an accurate, continuous characterization of pixel sensitivity. SeRI further divides selected region into smaller sub-regions, recursively refining the search for sensitive areas. This recursive refinement process enables more precise sensitivity estimation through fine-grained analysis, distinguishing SeRI from prior binary or one-shot region selection approaches. Experiments on two benchmark datasets show that SeRI significantly enhances state-of-the-art decision-based attacks in both targeted and non-targeted attack scenarios. Additionally, SeRI generates precise heatmaps that identify sensitive image regions. The code is available at https://anonymous.4open.science/r/SeRI-5310.

We introduce an end-to-end approach to learn the evolution operators of large-scale non-linear dynamical systems, such as those describing complex natural phenomena. Evolution operators are particularly well-suited for analyzing systems that exhibit spatio-temporal patterns and have become a key analytical tool across various scientific communities. As terabyte-scale weather datasets and simulation tools capable of running millions of molecular dynamics steps per day are becoming commodities, our approach provides an effective tool to make sense of them from a data-driven perspective. The core of it lies in a remarkable connection between self-supervised representation learning methods and the recently established learning theory of evolution operators. We deploy our approach across multiple scientific domains: explaining the folding dynamics of small proteins, the binding process of drug-like molecules in host sites, and autonomously finding patterns in climate data. Our code is available open-source at: https://anonymous.4open.science/r/encoderops-5F67.

Neural operators have become increasingly popular in solving partial differential equations (PDEs) due to their superior capability to capture intricate mappings between function spaces over complex domains. However, the data-hungry nature of operator learning inevitably poses a bottleneck for their widespread applications. At the core of the challenge lies the absence of transferability of neural operators to new geometries. To tackle this issue, we propose operator learning with domain decomposition, a local-to-global framework to solve PDEs on arbitrary geometries. Under this framework, we devise an iterative scheme Schwarz Neural Inference (SNI). This scheme allows for partitioning of the problem domain into smaller subdomains, on which local problems can be solved with neural operators, and stitching local solutions to construct a global solution. Additionally, we provide a theoretical analysis of the convergence rate and error bound. We conduct extensive experiments on several representative linear and nonlinear PDEs with diverse boundary conditions and achieve remarkable geometry generalization compared to alternative methods.These analysis and experiments demonstrate the proposed framework's potential in addressing challenges related to geometry generalization and data efficiency.

Neural operator surrogates for time-dependent partial differential equations (PDEs) conventionally employ autoregressive prediction schemes, which accumulate error over long rollouts and require uniform temporal discretization. We introduce the Continuous Flow Operator (CFO), a framework that learns continuous-time PDE dynamics without the computational burden of standard continuous approaches, e.g., neural ODE. The key insight is repurposing flow matching to directly learn the right-hand side of PDEs without backpropagating through ODE solvers. CFO fits temporal splines to trajectory data, using finite-difference estimates of time derivatives at knots to construct probability paths whose velocities closely approximate the true PDE dynamics. A neural operator is then trained via flow matching to predict these analytic velocity fields. This approach is inherently time-resolution invariant: training accepts trajectories sampled on arbitrary, non-uniform time grids while inference queries solutions at any temporal resolution through ODE integration. Across four benchmarks (Lorenz, 1D Burgers, 2D diffusion-reaction, 2D shallow water), CFO demonstrates superior long-horizon stability and remarkable data efficiency. CFO trained on only 25% of irregularly subsampled time points outperforms autoregressive baselines trained on complete data, with relative error reductions up to 87%. Despite requiring numerical integration at inference, CFO achieves competitive efficiency, outperforming autoregressive baselines using only 50% of their function evaluations, while uniquely enabling reverse-time inference and arbitrary temporal querying.

It is often possible to perform reduced order modelling by specifying linear subspace which accurately captures the dynamics of the system. This approach becomes especially appealing when linear subspace explicitly depends on parameters of the problem. A practical way to apply such a scheme is to compute subspaces for a selected set of parameters in the computationally demanding offline stage and in the online stage approximate subspace for unknown parameters by interpolation. For realistic problems the space of parameters is high dimensional, which renders classical interpolation strategies infeasible or unreliable. We propose to relax the interpolation problem to regression, introduce several loss functions suitable for subspace data, and use a neural network as an approximation to high-dimensional target function. To further simplify a learning problem we introduce redundancy: in place of predicting subspace of a given dimension we predict larger subspace. We show theoretically that this strategy decreases the complexity of the mapping for elliptic eigenproblems with constant coefficients and makes the mapping smoother for general smooth function on the Grassmann manifold. Empirical results also show that accuracy significantly improves when larger-than-needed subspaces are predicted. With the set of numerical illustrations we demonstrate that subspace regression can be useful for a range of tasks including parametric eigenproblems, deflation techniques, relaxation methods, optimal control and solution of parametric partial differential equations.

Collecting web data to train deep models has become increasingly common, raising concerns about unauthorized data usage. To mitigate this issue, unlearnable examples introduce imperceptible perturbations into data, preventing models from learning effectively. However, existing methods typically rely on deep neural networks as surrogate models for perturbation generation, resulting in significant computational costs. In this work, we propose Perturbation-Induced Linearization (PIL), a computationally efficient yet effective method that generates perturbations using only linear surrogate models. PIL achieves comparable or better performance than existing surrogate-based methods while reducing computational time dramatically. We further reveal a key mechanism underlying unlearnable examples: inducing linearization to deep models, which explains why PIL can achieve competitive results in a very short time. Beyond this, we provide an analysis about the property of unlearnable examples under percentage-based partial perturbation. Our work not only provides a practical approach for data protection but also offers insights into what makes unlearnable examples effective.

Bayesian Optimization critically depends on the choice of acquisition function, but no single strategy is universally optimal; the best choice is non-stationary and problem-dependent. Existing adaptive portfolio methods often base their decisions on past function values while ignoring richer information like remaining budget or surrogate model characteristics. To address this, we introduce LMABO, a novel framework that casts a pre-trained Large Language Model (LLM) as a zero-shot, online strategist for the BO process. At each iteration, LMABO uses a structured state representation to prompt the LLM to select the most suitable acquisition function from a diverse portfolio. In an evaluation across 50 benchmark problems, LMABO demonstrates a significant performance improvement over strong static, adaptive portfolio, and other LLM-based baselines. We show that the LLM's behavior is a comprehensive strategy that adapts to real-time progress, proving its advantage stems from its ability to process and synthesize the complete optimization state into an effective, adaptive policy.

Amortized Bayesian inference (ABI) with neural networks can solve probabilistic inverse problems orders of magnitude faster than classical methods. However, ABI is not yet sufficiently robust for widespread and safe application. When performing inference on observations outside the scope of the simulated training data, posterior approximations are likely to become highly biased, which cannot be corrected by additional simulations due to the bad pre-asymptotic behavior of current neural posterior estimators. In this paper, we propose a semi-supervised approach that enables training not only on labeled simulated data generated from the model, but also on unlabeled data originating from any source, including real data. To achieve this, we leverage Bayesian self-consistency properties that can be transformed into strictly proper losses that do not require knowledge of ground-truth parameters. We test our approach on several real-world case studies, including applications to high-dimensional time-series and image data. Our results show that semi-supervised learning with unlabeled data drastically improves the robustness of ABI in the out-of-simulation regime. Notably, inference remains accurate even when evaluated on observations far away from the labeled and unlabeled data seen during training.

Bayesian optimization (BO) has been widely used to optimize expensive and black-box functions across various domains. Existing BO methods have not addressed tensor-output functions. To fill this gap, we propose a novel tensor-output BO method. Specifically, we first introduce a tensor-output Gaussian process (TOGP) with two classes of tensor-output kernels as a surrogate model of the tensor-output function, which can effectively capture the structural dependencies within the tensor. Based on it, we develop an upper confidence bound (UCB) acquisition function to select the queried points. Furthermore, we introduce a more complex and practical problem setting, named combinatorial bandit Bayesian optimization (CBBO), where only a subset of the outputs can be selected to contribute to the objective function. To tackle this, we propose a tensor-output CBBO method, which extends TOGP to handle partially observed outputs, and accordingly design a novel CMAB-UCB2 criterion to sequentially select both the queried points and the optimal output subset. Theoretical regret bounds for the two methods are established, ensuring their sublinear performance. Extensive synthetic and real-world experiments demonstrate their superiority.

Optimizers are critical for training deep neural networks. Existing training processes rely on a single static optimizer (e.g., SGD) or a simple hybrid of two optimizers, which miss the opportunity to exploit evolving dynamics in different training states, degrading model quality and convergence. In this paper, we propose a novel dynamic optimizer switching method called **D**ynamic **O**ptimizer **I**nterleaving **T**raining (DOIT) method, which builds surrogate models to predict different optimizers' performance from current parameter states. DOIT uses an acquisition function that combines the results from surrogate models with transferability assessments and process information to select a suitable optimizer for the subsequent training. Experiments on various models and tasks (e.g., image and text classification, machine translation, and object detection) show that DOIT effectively enhances the training, achieving faster convergence (i.e., 2\% to 10\% faster) and higher accuracy (i.e., 1\% to 3\% improvement). Additional independent experiments and case studies further validate DOIT's effectiveness.

Current visual reasoning methods mainly focus on exploring specific reasoning modes. Although improvements can be achieved in particular domains, they struggle to develop general reasoning capabilities. Inspired by this, we propose a novel adaptive reasoning paradigm, $\underline{\text{M}}$ixture-$\underline{\text{o}}$f-$\underline{\text{V}}$isual-$\underline{\text{T}}$houghts (**MoVT**), which unifies different reasoning modes within a single model and guides it to select the appropriate mode based on context. To achieve this, we introduce **AdaVaR**, a two-stage $\underline{\text{Ada}}$ptive $\underline{\text{V}}$isu$\underline{\text{a}}$l $\underline{\text{R}}$easoning learning framework: different modes are unified and learned during the supervised cold-start stage, and the mode selection capability is induced via an RL process with a carefully designed AdaGRPO algorithm. Extensive experiments show that AdaVaR effectively guides the model to learn and differentiate multiple modes and perform context-adaptive mode selection, achieving consistent improvement across various scenarios, highlighting MoVT as an effective solution for building general visual reasoning models.

We study the problem of training a risk-sensitive reinforcement learning (RL) agent through imitation learning (IL). Unlike standard IL, our goal is not only to train an agent that matches the expert’s expected return (i.e., its *average performance*) but also its *risk attitude* (i.e., other features of the return distribution, such as variance). We propose a general formulation of the risk-sensitive IL problem in which the objective is to match the expert’s return distribution in Wasserstein distance. We focus on the tabular setting and assume the expert’s reward is *known*. After demonstrating the limited expressivity of Markovian policies for this task, we introduce an efficient and sufficiently expressive subclass of non-Markovian policies tailored to it. Building on this subclass, we develop two provably efficient algorithms—RS-BC and RS-KT —for solving the problem when the transition model is unknown and known, respectively. We show that RS-KT achieves substantially lower sample complexity than RS-BC by exploiting dynamics information. We further demonstrate the sample efficiency of return distribution matching in the setting where the expert’s reward is *unknown* by designing an oracle-based variant of RS-KT. Finally, we complement our theoretical analysis of RS-KT and RS-BC with numerical simulations, highlighting both their sample efficiency and the advantages of non-Markovian policies over standard sample-efficient IL algorithms.