ICLR 2026 Papers

Equivariant Neural Networks (ENNs) have empowered numerous applications in scientific fields. Despite their remarkable capacity for representing geometric structures, ENNs suffer from degraded expressivity when processing symmetric inputs: the output representations are invariant to transformations that extend beyond the input's symmetries. The mathematical essence of this phenomenon is that a symmetric input, after being processed by an equivariant map, experiences an increase in symmetry. While prior research has documented symmetry increase in specific cases, a rigorous understanding of its underlying causes and general reduction strategies remains lacking. In this paper, we provide a detailed and in-depth characterization of symmetry increase together with a principled framework for its reduction: (i) For any given feature space and input symmetry group, we prove that the increased symmetry admits an infimum determined by the structure of the feature space; (ii) Building on this foundation, we develop a computable algorithm to derive this infimum, and propose practical guidelines for feature design to prevent harmful symmetry increases. (iii) Under standard regularity assumptions, we demonstrate that for most equivariant maps, our guidelines effectively reduce symmetry increase. To complement our theoretical findings, we provide visualizations and experiments on both synthetic datasets and the real-world QM9 dataset. The results validate our theoretical predictions.

As a critical modality for structural biology, cryogenic electron microscopy (cryo-EM) facilitates the determination of macromolecular structures at near-atomic resolution. The core computational task in single-particle cryo-EM is to reconstruct the 3D electrostatic potential of a molecule from noisy 2D projections acquired at unknown orientations. Gaussian mixture models (GMMs) provide a continuous, compact, and physically interpretable representation for molecular density and have recently gained interest in cryo-EM reconstruction. However, existing methods rely on external consensus maps or atomic models for initialization, limiting their use in self-contained pipelines. In parallel, differentiable rendering techniques such as Gaussian splatting have demonstrated remarkable scalability and efficiency for volumetric representations, suggesting a natural fit for GMM-based cryo-EM reconstruction. However, off-the-shelf Gaussian splatting methods are designed for photorealistic view synthesis and remain incompatible with cryo-EM due to mismatches in the image formation physics, reconstruction objectives, and coordinate systems. Addressing these issues, we propose cryoSplat, a GMM-based method that integrates Gaussian splatting with the physics of cryo-EM image formation. In particular, we develop an orthogonal projection-aware Gaussian splatting, with adaptations such as a view-dependent normalization term and FFT-aligned coordinate system tailored for cryo-EM imaging. These innovations enable stable and efficient homogeneous reconstruction directly from raw cryo-EM particle images using random initialization. Experimental results on real datasets validate the effectiveness and robustness of cryoSplat over representative baselines. The code will be released upon publication.

Bilevel optimization is a hierarchical framework where an upper-level optimization problem is constrained by a lower-level problem, commonly used in machine learning applications such as hyperparameter optimization. Existing bilevel optimization methods typically assume strong convexity or Polyak-Łojasiewicz (PL) conditions for the lower-level function to establish non-asymptotic convergence to a solution with a small hypergradient. However, these assumptions may not hold in practice, and recent work (Chen et al. 2024) has shown that bilevel optimization is inherently intractable for general convex lower-level functions with the goal of finding small hypergradients. In this paper, we identify a tractable class of bilevel optimization problems that interpolates between lower-level strong convexity and general convexity via lower-level uniform convexity. For uniformly convex lower-level functions with exponent $p\geq 2$, we establish a novel implicit differentiation theorem characterizing the hyperobjective's smoothness property. Building on this, we design a new stochastic algorithm, termed UniBiO, with provable convergence guarantees, based on an oracle that provides stochastic gradient and Hessian-vector product information for the bilevel problems. Our algorithm achieves $\widetilde{O}(\epsilon^{-5p+6})$ oracle complexity bound for finding $\epsilon$-stationary points. Notably, our complexity bounds match the optimal rates in terms of the $\epsilon$ dependency for strongly convex lower-level functions ($p=2$), up to logarithmic factors. Our theoretical findings are validated through experiments on synthetic tasks and data hyper-cleaning, demonstrating the effectiveness of our proposed algorithm.

Enzyme Commission (EC) number prediction is vital for elucidating enzyme functions and advancing biotechnology applications. However, current methods struggle to capture the hierarchical relationships among enzymes and often overlook critical structural and active site features. To bridge this gap, we introduce PoinnCARE, a novel framework that jointly encodes and aligns multi-modal data from enzyme sequences, structures, and active sites in hyperbolic space. By integrating graph diffusion and alignment techniques, PoinnCARE mitigates data sparsity and enriches functional representations, while hyperbolic embedding preserves the intrinsic hierarchy of the EC system with theoretical guarantees in low-dimensional spaces. Extensive experiments on four datasets from the CARE benchmark demonstrate that PoinnCARE consistently and significantly outperforms state-of-the-art methods in EC number prediction.

We consider the problem of modeling high-speed flows using machine learning methods. While most prior studies focus on low-speed fluid flows in which uniform time-stepping is practical, flows approaching and exceeding the speed of sound exhibit sudden changes such as shock waves. In such cases, it is essential to use adaptive time-stepping methods to allow a temporal resolution sufficient to resolve these phenomena while simultaneously balancing computational costs. Here, we propose a two-phase machine learning method, known as ShockCast, to model high-speed flows with adaptive time-stepping. In the first phase, we propose to employ a machine learning model to predict the timestep size. In the second phase, the predicted timestep is used as an input along with the current fluid fields to advance the system state by the predicted timestep. We explore several physically-motivated components for timestep prediction and introduce timestep conditioning strategies inspired by neural ODE and Mixture of Experts. We evaluate our methods by generating three supersonic flow datasets.

Solving high-dimensional partial differential equations (PDEs) is a critical challenge where modern data-driven solvers often lack reliability and rigorous error guarantees. We introduce Simulation-Calibrated Scientific Machine Learning (SCaSML), a framework that systematically improves pre-trained PDE solvers at inference time without any retraining. Our core idea is to derive a new PDE, which we term the Law of Defect, that precisely governs the error of a given surrogate model. Because this defect PDE retains the structure of the original problem, we can solve it efficiently with traditional stochastic simulators, yielding a targeted correction to the initial machine-learned solution. We prove that SCaSML achieves a faster convergence rate, with a final error bounded by the product of the surrogate and simulation errors. On challenging PDEs up to 160 dimensions, SCaSML reduces the error of various surrogate models, including PINNs and Gaussian Processes, by 20-80%. SCaSML provides a principled method to fuse the speed of machine learning with the rigor of numerical simulation, enhancing the trustworthiness of Al for scientific discovery.

Hypergraphs provide a natural way to represent higher-order interactions among multiple entities. While undirected hypergraphs have been extensively studied, the case of directed hypergraphs, which can model oriented group interactions, remains largely under-explored despite its relevance for many applications. Recent approaches in this direction often exhibit an implicit bias toward homophily, which limits their effectiveness in heterophilic settings. Rooted in the algebraic topology notion of Cellular Sheaves, Sheaf Neural Networks (SNNs) were introduced as an effective solution to circumvent such a drawback. While a generalization to hypergraphs is known, it is only suitable for undirected hypergraphs, failing to tackle the directed case. In this work, we introduce Directional Sheaf Hypergraph Networks (DSHN), a framework integrating sheaf theory with a principled treatment of asymmetric relations within a hypergraph. From it, we construct the Directed Sheaf Hypergraph Laplacian, a complex-valued operator by which we unify and generalize many existing Laplacian matrices proposed in the graph-and hypergraph-learning literature. Across 7 real-world datasets and against 13 baselines, DSHN achieves relative accuracy gains from 2% up to 20%, showing how a principled treatment of directionality in hypergraphs, combined with the expressive power of sheaves, can substantially improve performance.

Predicting the intermediate trajectories between an initial and target distribution is a central problem in generative modeling. Existing approaches, such as flow matching and Schrödinger Bridge Matching, effectively learn mappings between two distributions by modeling a single stochastic path. However, these methods are inherently limited to unimodal transitions and cannot capture *branched* or *divergent* evolution from a common origin to multiple distinct outcomes. To address this, we introduce **Branched Schrödinger Bridge Matching (BranchSBM)**, a novel framework that learns branched Schrödinger bridges. BranchSBM parameterizes multiple time-dependent velocity fields and growth processes, enabling the representation of population-level divergence into multiple terminal distributions. We show that BranchSBM is not only more expressive but also essential for tasks involving multi-path surface navigation, modeling cell fate bifurcations from homogeneous progenitor states, and simulating diverging cellular responses to perturbations.

Simulation-free training frameworks have been at the forefront of the generative modelling revolution in continuous spaces, leading to large-scale diffusion and flow matching models. However, such modern generative models suffer from expensive inference, inhibiting their use in numerous scientific applications like Boltzmann Generators (BGs) for molecular conformations that require fast likelihood evaluation. In this paper, we revisit classical normalizing flows in the context of BGs that offer efficient sampling and likelihoods, but whose training via maximum likelihood is often unstable and computationally challenging. We propose Regression Training of Normalizing Flows (RegFlow), a novel and scalable regression-based training objective that bypasses the numerical instability and computational challenge of conventional maximum likelihood training in favour of a simple $\ell_2$-regression objective. Specifically, RegFlow maps prior samples under our flow to targets computed using optimal transport couplings or a pre-trained continuous normalizing flow (CNF). To enhance numerical stability, RegFlow employs effective regularization strategies such as a new forward-backward self-consistency loss that enjoys painless implementation. Empirically, we demonstrate that RegFlow unlocks a broader class of architectures that were previously intractable to train for BGs with maximum likelihood. We also show RegFlow exceeds the performance, computational cost, and stability of maximum likelihood training in equilibrium sampling in Cartesian coordinates of alanine dipeptide, tripeptide, and tetrapeptide, showcasing its potential in molecular systems.

Accurately predicting molecular properties requires effective integration of structural information from both 2D molecular graphs and their corresponding equilibrium conformer ensembles. In this work, we propose FACET, a scalable Structure-Aware Graph Transformer that efficiently aggregates features from multiple 3D conformers while incorporating fragment-level information from 2D graphs. Unlike prior methods that rely on static geometric solvers or rigid fusion strategies, our approach utilizes a differentiable graph transformer to theoretically approximate the computationally expensive Fused Gromov–Wasserstein (FGW), enabling dynamic and scalable fusion of 2D and 3D structural information. We further enhance this mechanism by injecting fragment-specific structural priors into the attention layers, enabling the model to capture fine-grained molecular details. This unified design scales to large datasets, handling up to 75,000 molecules and hundreds of thousands of conformers, and provides over a 6× speedup compared to geometry-aware FGW-based baselines. Our method also achieves state-of-the-art results in molecular property prediction, Boltzmann-weighted ensemble modeling, and reaction-level tasks, and is particularly effective on chemically diverse compounds, including organocatalysts and transition-metal complexes.

With rapid urbanization, predicting urban microclimates has become critical, as it affects building energy demand and public health risks. However, existing generative and homogeneous graph approaches fall short in capturing physical consistency, spatial dependencies, and temporal variability. \revise{To address this, we introduce UrbanGraph, a framework founded on a novel structure-based inductive bias. Unlike implicit graph learning, UrbanGraph transforms physical first principles into a dynamic causal topology, explicitly encoding time-varying causalities (e.g., shading and convection) directly into the graph structure to ensure physical consistency and data efficiency. Results show that UrbanGraph achieves state-of-the-art performance across all baselines. Specifically, the use of explicit causal pruning significantly reduces the model's floating-point operations (FLOPs) by 73.8\% and increases training speed by 21\% compared to implicit graphs. Our contribution includes the first high-resolution benchmark for spatio-temporal microclimate modeling, and a generalizable explicit topological encoding paradigm applicable to urban spatio-temporal dynamics governed by known physical equations.

Parameter shift rules (PSRs) are key techniques for efficient gradient estimation in variational quantum eigensolvers (VQEs). In this paper, we propose their Bayesian variant, where Gaussian processes with appropriate kernels are used to estimate the gradient of the VQE objective. Our Bayesian PSR offers flexible gradient estimation from observations at arbitrary locations with uncertainty information, and reduces to the generalized PSR in special cases. In stochastic gradient descent (SGD), the flexibility of Bayesian PSR allows reuse of observations in previous steps, which accelerates the optimization process. Furthermore, the accessibility to the posterior uncertainty, along with our proposed notion of gradient confident region (GradCoRe), enables us to minimize the observation costs in each SGD step. Our numerical experiments show that the VQE optimization with Bayesian PSR and GradCoRe significantly accelerates SGD, and outperforms the state-of-the-art methods, including sequential minimal optimization.

Modern Earth observation (EO) increasingly leverages deep learning to harness the scale and diversity of satellite imagery across sensors and regions. While recent foundation models have demonstrated promising generalization across EO tasks, many remain limited by the scale, geographical coverage, and spectral diversity of their training data, factors critical for learning globally transferable representations. In this work, we introduce TerraFM, a scalable self-supervised learning model that leverages globally distributed Sentinel-1 and Sentinel-2 imagery, combined with large spatial tiles and land-cover aware sampling to enrich spatial and semantic coverage. By treating sensing modalities as natural augmentations in our self-supervised approach, we unify radar and optical inputs via modality-specific patch embeddings and adaptive cross-attention fusion. Our training strategy integrates local-global contrastive learning and introduces a dual-centering mechanism that incorporates class-frequency-aware regularization to address long-tailed distributions in land cover. TerraFM achieves strong generalization on both classification and segmentation tasks, outperforming prior models on GEO-Bench and Copernicus-Bench. Our code and pretrained models will be publicly released.

Accurate tumor analysis is central to clinical radiology and precision oncology, where early detection, reliable lesion characterization, and pathology-level risk assessment directly guide diagnosis, staging, and treatment planning. Chain-of-Thought (CoT) reasoning is particularly critical in this setting, as it enables stepwise interpretation from imaging findings to clinical impressions and pathology-level conclusions, ensuring traceability and reducing diagnostic errors. Here, we target the clinical tumor analysis task and build a large-scale benchmark that operationalizes a multimodal reasoning pipeline, spanning findings, impressions, and pathology predictions. We curate TumorCoT, a large-scale dataset of 1.5M CoT-labeled VQA instructions paired with 3D CT scans, with step-aligned rationales and cross-modal alignments along the “findings → impression → pathology” trajectory, enabling standardized evaluation of both final accuracy and reasoning consistency. We further propose TumorChain, a multimodal interleaved reasoning framework that tightly couples 3D imaging encoders, clinical text understanding, and organ-level vision-language alignment. Through cross-modal alignment and iterative interleaved causal reasoning, TumorChain grounds visual evidence, aggregates conclusions, and issues pathology predictions after multiple rounds of self-refinement, improving traceability and reducing hallucination risk. TumorChain demonstrates consistent gains over strong unimodal and pipeline baselines in lesion detection, impression quality, and pathology classification, and successfully generalizes to the public DeepTumorVQA benchmark. Ablations validate the key contributions of interleaved reasoning and clinical CoT. Clinically, these advances lay the groundwork for reliable, interpretable tumor assessment to support real-world decision-making. We release the task, benchmark, and evaluation protocol to advance safe, explainable, and reproducible multimodal reasoning for high-stakes tumor analysis. Our project is available at https://anonymous.4open.science/r/TumorChain-D6E6.

Full-waveform inversion (FWI) estimates physical parameters in the wave equation from limited measurements and has been widely applied in geophysical exploration, medical imaging, and non-destructive testing. Conventional FWI methods are limited by their notorious sensitivity to the accuracy of the initial models. Recent progress in continuous representation FWI (CR-FWI) demonstrates that representing parameter models with a coordinate-based neural network, such as implicit neural representation (INR), can mitigate the dependence on initial models. However, its underlying mechanism remains unclear, and INR-based FWI shows slower high-frequency convergence. In this work, we investigate the general CR-FWI framework and develop a unified theoretical understanding by extending the neural tangent kernel (NTK) for FWI to establish a wave-based NTK framework. Unlike standard NTK, our analysis reveals that wave-based NTK is not constant, both at initialization and during training, due to the inherent nonlinearity of FWI. We further show that the eigenvalue decay behavior of the wave-based NTK can explain why CR-FWI alleviates the dependency on initial models and shows slower high-frequency convergence. Building on these insights, we propose several CR-FWI methods with tailored eigenvalue decay properties for FWI, including a novel hybrid representation combining INR and multi-resolution grid (termed IG-FWI) that achieves a more balanced trade-off between robustness and high-frequency convergence rate. Applications in geophysical exploration on Marmousi, 2D SEG/EAGE Salt and Overthrust, 2004 BP model, and the more realistic 2014 Chevron models show the superior performance of our proposed methods compared to conventional FWI and existing INR-based FWI methods.

Sound source localization (SSL) is a fundamental task in spatial audio understanding, yet most deep neural network-based methods are constrained by fixed array geometries and predefined directional grids, limiting generalizability and scalability. To address these issues, we propose _audio-geometry-grid representation learning_ (AGG-RL), a novel framework that jointly learns audio-geometry and grid representations in a shared latent space, enabling both geometry-invariant and grid-flexible SSL. Moreover, to enhance generalizability and interpretability, we introduce two physics-informed components: a _learnable non-uniform discrete Fourier transform_ (LNuDFT), which optimizes the dense allocation of frequency bins in a non-uniform manner to emphasize informative phase regions, and a _relative microphone positional encoding_ (rMPE), which encodes relative microphone coordinates in accordance with the nature of inter-channel time differences. Experiments on synthetic and real datasets demonstrate that AGG-RL achieves superior performance, particularly under unseen conditions. The results highlight the potential of representation learning with physics-informed design towards a universal solution for spatial acoustic scene understanding across diverse scenarios.

Large language models (LLMs) have recently shown strong progress on scientific reasoning, yet two major bottlenecks remain. First, explicit retrieval fragments reasoning, imposing a hidden tool tax of extra tokens and steps. Second, multi-agent pipelines often dilute strong solutions by averaging across all candidates. We address these challenges with a unified framework that combines implicit retrieval and structured collaboration. At its foundation, a Monitor-based retrieval module operates at the token level, integrating external knowledge with minimal disruption to reasoning. On top of this substrate, Hierarchical Solution Refinement (HSR) iteratively designates each candidate as an anchor to be repaired by its peers, while Quality-Aware Iterative Reasoning (QAIR) adapts refinement to solution quality. On Humanity’s Last Exam (HLE) Bio/Chem Gold, our framework achieves 48.3% accuracy—the highest reported to date, surpassing the strongest agent baseline by 13.4 points and leading frontier LLMs by up to 18.1 points, while simultaneously reducing token usage by 53.5% and agent steps by 43.7%. Results on SuperGPQA and TRQA confirm robustness across domains. Error analysis shows that reasoning failures and knowledge gaps co-occur in over 85% of cases, while diversity analysis reveals a clear dichotomy: retrieval tasks benefit from solution variety, whereas reasoning tasks favor consensus. Together, these findings demonstrate how implicit augmentation and structured refinement overcome the inefficiencies of explicit tool use and uniform aggregation.

Dual-arm robots play a crucial role in improving efficiency and flexibility in complex multitasking scenarios. While existing methods have achieved promising results in task planning, they often fail to fully optimize task parallelism, limiting the potential of dual-arm collaboration. To address this issue, we propose RoboPARA, a novel large language model (LLM)-driven framework for dual-arm task parallelism planning. RoboPARA employs a two-stage process: (1) Dependency Graph-based Planning Candidates Generation, which constructs directed acyclic graphs (DAGs) to model task dependencies and eliminate redundancy, and (2) Graph Re-Traversal-based Dual-Arm Parallel Planning, which optimizes DAG traversal to maximize parallelism while maintaining task coherence. In addition, we introduce the Cross-Scenario Dual-Arm Parallel Task dataset (X-DAPT dataset), the first dataset specifically designed to evaluate dual-arm task parallelism across diverse scenarios and difficulty levels. Extensive experiments demonstrate that RoboPARA significantly outperforms existing planning methods, achieving higher efficiency and reliability, particularly in complex task combinations.Our code is publicly available at https://github.com/AiDuanshiying/RoboPARA.

Modern spatio-temporal learning techniques usually exploit sampled discrete observations to foresee the future. Actually, spatio-temporal dynamics are continuous and evolve everytime and everywhere, thus modeling spatio-temporal dynamics in a continuous space can be long-standing challenge. Existing deep learning architectures often fail to generalize to unseen regions or graph topologies, while many physics-driven approaches are confined to Euclidean grids and scale poorly to complex graph structures. To address this gap, we propose PhySTA, a physics-inspired spatio-temporal learning framework designed for efficient and scalable arbitrary inference over graph-structured data. PhySTA integrates two key modules, i.e., (1) Continuous Operator-based Spectrum-Temporal Learning (CoSTL), which leverages a Graph-Time Fourier Neural Operator combined with Time-Gated Spectral Segmentation Perception to model continuous dynamics in operator space, (2) Adaptive Multi-scale Interaction (AMI) that constructs multi-scale subgraphs and introduces node-edge coupled convolution to capture discrete interaction patterns and refine continuous predictions. By bridging operator learning with node-edge-graph interaction, PhySTA achieves both continuity-aware dynamic modeling and hierarchical interactive refinement. Extensive experiments across large-scale benchmarks demonstrate that PhySTA attains state-of-the-art accuracy while reducing computation cost and lowering parameter overhead.