ICML 2025 Papers

Fourier Neural Operators (FNOs) offer a principled approach for solving complex partial differential equations (PDEs). However, scaling them to handle more complex PDEs requires increasing the number of Fourier modes, which significantly expands the number of model parameters and makes hyperparameter tuning computationally impractical. To address this, we introduce $\mu$**Transfer-FNO**, a zero-shot hyperparameter transfer technique that enables optimal configurations, tuned on smaller FNOs, to be directly applied to billion-parameter FNOs _without_ additional tuning. Building on the Maximum Update Parametrization ($\mu$P) framework, we mathematically derive a parametrization scheme that facilitates the transfer of optimal hyperparameters across models with different numbers of Fourier modes in FNOs, which is validated through extensive experiments on various PDEs. Our empirical study shows that $\mu$Transfer-FNO reduces computational cost for tuning hyperparameters on large FNOs while maintaining or improving accuracy.

Automotive drag coefficient ($C_d$) is pivotal to energy efficiency, fuel consumption, and aerodynamic performance. However, costly computational fluid dynamics (CFD) simulations and wind tunnel tests struggle to meet the rapid-iteration demands of automotive design. We present DragSolver, a Transformer-based framework for rapid and accurate $C_d$ estimation from large-scale, diverse 3D vehicle models. DragSolver tackles four key real-world challenges: (1) multi-scale feature extraction to capture both global shape and fine local geometry; (2) heterogeneous scale normalization to handle meshes with varying sizes and densities; (3) surface-guided gating to suppress internal structures irrelevant to external aerodynamics; and (4) epistemic uncertainty estimation via Monte Carlo dropout for risk-aware design. Extensive evaluations on three industrial-scale datasets (DrivaerNet, DrivaerNet++, and DrivaerML) show that DragSolver outperforms existing approaches in accuracy and generalization, achieving an average reduction of relative $L_2$ error by 58.7% across real-world datasets. Crucially, DragSolver is the first to achieve reliable, real-time $C_d$ inference on production-level automotive geometries.

Protein backbone generation plays a central role in de novo protein design and is significant for many biological and medical applications. Although diffusion and flow-based generative models provide potential solutions to this challenging task, they often generate proteins with undesired designability and suffer computational inefficiency. In this study, we propose a novel rectified quaternion flow (ReQFlow) matching method for fast and high-quality protein backbone generation. In particular, our method generates a local translation and a 3D rotation from random noise for each residue in a protein chain, which represents each 3D rotation as a unit quaternion and constructs its flow by spherical linear interpolation (SLERP) in an exponential format. We train the model by quaternion flow (QFlow) matching with guaranteed numerical stability and rectify the QFlow model to accelerate its inference and improve the designability of generated protein backbones, leading to the proposed ReQFlow model. Experiments show that ReQFlow achieves on-par performance in protein backbone generation while requiring much fewer sampling steps and significantly less inference time (e.g., being 37$\times$ faster than RFDiffusion and 63$\times$ faster than Genie2 when generating a backbone of length 300), demonstrating its effectiveness and efficiency.

Molecular discovery has attracted significant attention in scientific fields for its ability to generate novel molecules with desirable properties. Although numerous methods have been developed to tackle this problem, most rely on an online setting that requires repeated online evaluation of candidate molecules using the oracle. However, in real-world molecular discovery, the oracle is often represented by wet-lab experiments, making this online setting impractical due to the significant time and resource demands. To fill this gap, we propose the Molecular Stitching (MolStitch) framework, which utilizes a fixed offline dataset to explore and optimize molecules without the need for repeated oracle evaluations. Specifically, MolStitch leverages existing molecules from the offline dataset to generate novel `stitched molecules' that combine their desirable properties. These stitched molecules are then used as training samples to fine-tune the generative model using preference optimization techniques. Experimental results on various offline multi-objective molecular optimization problems validate the effectiveness of MolStitch. The source code is available online.

Accurately solving partial differential equations (PDEs) is critical to understanding complex scientific and engineering phenomena, yet traditional numerical solvers are computationally expensive. Surrogate models offer a more efficient alternative, but their development is hindered by the cost of generating sufficient training data from numerical solvers. In this paper, we present a novel framework for active learning (AL) in PDE surrogate modeling that reduces this cost. Unlike the existing AL methods for PDEs that always acquire entire PDE trajectories, our approach strategically generates only the most important time steps with the numerical solver, while employing the surrogate model to approximate the remaining steps. This dramatically reduces the cost incurred by each trajectory and thus allows the active learning algorithm to try out a more diverse set of trajectories given the same budget. To accommodate this novel framework, we develop an acquisition function that estimates the utility of a set of time steps by approximating its resulting variance reduction. We demonstrate the effectiveness of our method on several benchmark PDEs, including the Burgers' equation, Korteweg–De Vries equation, Kuramoto–Sivashinsky equation, the incompressible Navier-Stokes equation, and the compressible Navier-Stokes equation. Experiments show that our approach improves performance by large margins over the best existing method. Our method not only reduces average error but also the 99\%, 95\%, and 50\% quantiles of error, which is rare for an AL algorithm. All in all, our approach offers a data-efficient solution to surrogate modeling for PDEs.

We consider layerwise function-space learning rates, which measure the magnitude of the change in a neural network's output function in response to an update to a parameter tensor. This contrasts with traditional learning rates, which describe the magnitude of changes in parameter space. We develop efficient methods to measure and set function-space learning rates in arbitrary neural networks, requiring only minimal computational overhead through a few additional backward passes that can be performed at the start of, or periodically during, training. We demonstrate two key applications: (1) analysing the dynamics of standard neural network optimisers in function space, rather than parameter space, and (2) introducing FLeRM (Function-space Learning Rate Matching), a novel approach to hyperparameter transfer across model scales. FLeRM records function-space learning rates while training a small, cheap base model, then automatically adjusts parameter-space layerwise learning rates when training larger models to maintain consistent function-space updates. FLeRM gives hyperparameter transfer across model width, depth, initialisation scale, and LoRA rank in various architectures including MLPs with residual connections and transformers with different layer normalisation schemes.

Graph Transformers (GTs) have emerged as a powerful alternative to message-passing neural networks, yet their performance heavily depends on effectively embedding structural inductive biases. In this work, we introduce novel structural encodings (SEs) grounded in a rigorous analysis of random walks (RWs), leveraging Green and Martin kernels that we have carefully redefined for AI applications while preserving their mathematical essence.These kernels capture the long-term behavior of RWs on graphs and allow for enhanced representation of complex topologies, including non-aperiodic and directed acyclic substructures.Empirical evaluations across eight benchmark datasets demonstrate strong performance across diverse tasks, notably in molecular and circuit domains.We attribute this performance boost to the improved ability of our kernel-based SEs to encode intricate structural information, thereby strengthening the global attention and inductive bias within GTs.This work highlights the effectiveness of theoretically grounded kernel methods in advancing Transformer-based models for graph learning.

Crystal symmetry plays a fundamental role in determining its physical, chemical, and electronic properties such as electrical and thermal conductivity, optical and polarization behavior, and mechanical strength. Almost all known crystalline materials have internal symmetry. However, this is often inadequately addressed by existing generative models, making the consistent generation of stable and symmetrically valid crystal structures a significant challenge. We introduce WyFormer, a generative model that directly tackles this by formally conditioning on space group symmetry. It achieves this by using Wyckoff positions as the basis for an elegant, compressed, and discrete structure representation. To model the distribution, we develop a permutation-invariant autoregressive model based on the Transformer encoder and an absence of positional encoding. Extensive experimentation demonstrates WyFormer's compelling combination of attributes: it achieves best-in-class symmetry-conditioned generation, incorporates a physics-motivated inductive bias, produces structures with competitive stability, predicts material properties with competitive accuracy even without atomic coordinates, and exhibits unparalleled inference speed.

This paper addresses weak features learning (WFL), focusing on learning scenarios characterized by low-quality input features (weak features; WFs) that arise due to missingness, measurement errors, or ambiguous observations. We present a theoretical formalization and error analysis of WFL for continuous WFs (continuous WFL), which has been insufficiently explored in existing literature. A previous study established formalization and error analysis for WFL with discrete WFs (discrete WFL); however, this analysis does not extend to continuous WFs due to the inherent constraints of discreteness. To address this, we propose a theoretical framework specifically designed for continuous WFL, systematically capturing the interactions between feature estimation models for WFs and label prediction models for downstream tasks. Furthermore, we derive the theoretical conditions necessary for both sequential and iterative learning methods to achieve consistency. By integrating the findings of this study on continuous WFL with the existing theory of discrete WFL, we demonstrate that the WFL framework is universally applicable, providing a robust theoretical foundation for learning with low-quality features across diverse application domains.

Machine learning is playing an increasingly important role in computational chemistry and materials science, complementing expensive ab initio and first-principles methods. However, machine-learned interatomic potentials (MLIPs) often struggle with generalization and robustness, leading to unphysical energy and force predictions in atomistic simulations. To address this, we propose a physics-informed, weakly supervised training framework for MLIPs. Our method introduces two novel loss functions: one based on Taylor expansions of the potential energy and another enforcing conservative force constraints. This approach enhances accuracy, particularly in low-data regimes, and reduces the reliance on large, expensive training datasets. Extensive experiments across benchmark datasets show up to 2× reductions in energy and force errors for multiple baseline models. Additionally, our method improves the stability of molecular dynamics simulations and facilitates effective fine-tuning of ML foundation models on sparse, high-accuracy ab initio data. An implementation of our method and scripts for executing experiments are available at \url{https://github.com/nec-research/PICPS-ML4Sci}.

Watermarking for large language models (LLMs) offers a promising approach to identifying AI-generated text. Existing approaches, however, either compromise the distribution of original generated text by LLMs or are limited to embedding zero-bit information that only allows for watermark detection but ignores identification. We present StealthInk, a stealthy multi-bit watermarking scheme that preserves the original text distribution while enabling the embedding of provenance data, such as userID, TimeStamp, and modelID, within LLM-generated text. This enhances fast traceability without requiring access to the language model's API or prompts. We derive a lower bound on the number of tokens necessary for watermark detection at a fixed equal error rate, which provides insights on how to enhance the capacity. Comprehensive empirical evaluations across diverse tasks highlight the stealthiness, detectability, and resilience of StealthInk, establishing it as an effective solution for LLM watermarking applications.

Auction plays a crucial role in many modern trading environments, including online advertising and public resource allocation. As the number of competing bidders increases, learning Bayesian Nash Equilibrium (BNE) in auctions faces significant scalability challenges. Existing methods often experience slow convergence in large-scale auctions. For example, in a classic symmetric auction setting, the convergence rate depends on the number of bidders quadratically. To address this issue, we propose the *Approximate Best Response Gradient* method, a new approach for learning BNE efficiently in auction games. We leverage an analytic solution for gradient estimation to enable efficient gradient computation during optimization. Moreover, we introduce the *Best Response Distance* objective, which serves as an upper bound of approximation quality to BNE. By optimizing the new objective, our method is proven to achieve a local convergence rate independent of bidder numbers and circumvent the traditional quadratic complexity in the classic symmetric setting. Extensive experiments across various auction formats demonstrate that our approach accelerates convergence and enhances learning efficiency in complex auction settings.

Symbolic models play a key role in cognitive science, expressing computationally precise hypotheses about how the brain implements a cognitive process. Identifying an appropriate model typically requires a great deal of effort and ingenuity on the part of a human scientist. Here, we adapt FunSearch (Romera-Paredes et al. 2024), a recently developed tool that uses Large Language Models (LLMs) in an evolutionary algorithm, to automatically discover symbolic cognitive models that accurately capture human and animal behavior. We consider datasets from three species performing a classic reward-learning task that has been the focus of substantial modeling effort, and find that the discovered programs outperform state-of-the-art cognitive models for each. The discovered programs can readily be interpreted as hypotheses about human and animal cognition, instantiating interpretable symbolic learning and decision-making algorithms. Broadly, these results demonstrate the viability of using LLM-powered program synthesis to propose novel scientific hypotheses regarding mechanisms of human and animal cognition.

Machine learning algorithms often struggle to control complex real-world systems. In the case of nuclear fusion, these challenges are exacerbated, as the dynamics are notoriously complex, data is poor, hardware is subject to failures, and experiments often affect dynamics beyond the experiment's duration. Existing tools like reinforcement learning, supervised learning, and Bayesian optimization address some of these challenges but fail to provide a comprehensive solution. To overcome these limitations, we present a multi-scale Bayesian optimization approach that integrates a high-frequency data-driven dynamics model with a low-frequency Gaussian process. By updating the Gaussian process between experiments, the method rapidly adapts to new data, refining the predictions of the less reliable dynamical model. We validate our approach by controlling tearing instabilities in the DIII-D nuclear fusion plant. Offline testing on historical data shows that our method significantly outperforms several baselines. Results on live experiments on the DIII-D tokamak, conducted under high-performance plasma scenarios prone to instabilities, shows a 50\% success rate — marking a 117\% improvement over historical outcomes.

Mixture of Experts (MoE) have shown remarkable success in leveraging specialized expert networks for complex machine learning tasks. However, their susceptibility to adversarial attacks presents a critical challenge for deployment in robust applications. This paper addresses the critical question of how to incorporate robustness into MoEs while maintaining high natural accuracy. We begin by analyzing the vulnerability of MoE components, finding that expert networks are notably more susceptible to adversarial attacks than the router. Based on this insight, we propose a targeted robust training technique that integrates a novel loss function to enhance the adversarial robustness of MoE, requiring only the robustification of one additional expert without compromising training or inference efficiency. Building on this, we introduce a dual-model strategy that linearly combines a standard MoE model with our robustified MoE model using a smoothing parameter. This approach allows for flexible control over the robustness-accuracy trade-off. We further provide theoretical foundations by deriving certified robustness bounds for both the single MoE and the dual-model. To push the boundaries of robustness and accuracy, we propose a novel joint training strategy JTDMoE for the dual-model. This joint training enhances both robustness and accuracy beyond what is achievable with separate models. Experimental results on CIFAR-10 and TinyImageNet datasets using ResNet18 and Vision Transformer (ViT) architectures demonstrate the effectiveness of our proposed methods.

Subgraph GNNs have emerged as promising architectures that overcome the expressiveness limitations of Graph Neural Networks (GNNs) by processing bags of subgraphs. Despite their compelling empirical performance, these methods are afflicted by a high computational complexity: they process bags whose size grows linearly in the number of nodes, hindering their applicability to larger graphs. In this work, we propose an effective and easy-to-implement approach to dramatically alleviate the computational cost of Subgraph GNNs and unleash broader applications thereof. Our method, dubbed HyMN, leverages walk-based centrality measures to sample a small number of relevant subgraphs and drastically reduce the bag size. By drawing a connection to perturbation analysis, we highlight the strength of the proposed centrality-based subgraph sampling, and further prove that these walk-based centralities can be additionally used as Structural Encodings for improved discriminative power. A comprehensive set of experimental results demonstrates that HyMN provides an effective synthesis of expressiveness, efficiency, and downstream performance, unlocking the application of Subgraph GNNs to dramatically larger graphs. Not only does our method outperform more sophisticated subgraph sampling approaches, it is also competitive, and sometimes better, than other state-of-the-art approaches for a fraction of their runtime.

Marginalization -- summing a function over all assignments to a subset of its inputs -- is a fundamental computational problem with applications from probabilistic inference to formal verification. Despite its computational hardness in general, there exist many classes of functions (e.g., probabilistic models) for which marginalization remains tractable, and they can all be commonly expressed by arithmetic circuits computing multilinear polynomials. This raises the question, can *all* functions with polynomial time marginalization algorithms be succinctly expressed by such circuits? We give a negative answer, exhibiting simple functions with tractable marginalization yet no efficient representation by known models, assuming $\\mathsf{FP} \\neq \\#\\mathsf{P}$ (an assumption implied by $\\mathsf{P} \\neq \\mathsf{NP}$). To this end, we identify a hierarchy of complexity classes corresponding to stronger forms of marginalization, all of which are efficiently computable on the known circuit models. We conclude with a completeness result, showing that whenever there is an efficient real RAM performing virtual evidence marginalization for a function, then there are small arithmetic circuits for that function's multilinear representation.

Divergence-free symmetric tensors (DFSTs) are fundamental in continuum mechanics, encoding conservation laws such as mass and momentum conservation. We introduce Riemann Tensor Neural Networks (RTNNs), a novel neural architecture that inherently satisfies the DFST condition to machine precision, providing a strong inductive bias for enforcing these conservation laws. We prove that RTNNs can approximate any sufficiently smooth DFST with arbitrary precision and demonstrate their effectiveness as surrogates for conservative PDEs, achieving improved accuracy across benchmarks. This work is the first to use DFSTs as an inductive bias in neural PDE surrogates and to explicitly enforce the conservation of both mass and momentum within a physics-constrained neural architecture.

Event cameras are innovative sensors that capture brightness changes as asynchronous events rather than traditional intensity frames. These cameras offer substantial advantages over conventional cameras, including high temporal resolution, high dynamic range, and the elimination of motion blur. However, defocus blur, a common image quality degradation resulting from out-of-focus lenses, complicates the challenge of event-based imaging. Due to the unique imaging mechanism of event cameras, existing focusing algorithms struggle to operate efficiently on sparse event data. In this work, we propose EvFocus, a novel architecture designed to reconstruct sharp images from defocus event streams for the first time. Our work includes the development of an event-based out-of-focus camera model and a simulator to generate realistic defocus event streams for robust training and testing. EvDefous integrates a temporal information encoder, a blur-aware two-branch decoder, and a reconstruction and re-defocus module to effectively learn and correct defocus blur. Extensive experiments on both simulated and real-world datasets demonstrate that EvFocus outperforms existing methods across varying lighting conditions and blur sizes, proving its robustness and practical applicability in event-based defocus imaging.