ICLR 2025 Papers

Solving high wavenumber Helmholtz equations is notoriously challenging. Traditional solvers have yet to yield satisfactory results, and most neural network methods struggle to accurately solve cases with extremely high wavenumbers within heterogeneous media. This paper presents an advanced multigrid-hierarchical AI solver, tailored specifically for high wavenumber Helmholtz equations. We adapt the MGCNN architecture to align with the problem setting and incorporate a novel Fourier neural network (FNN) to match the characteristics of Helmholtz equations. FNN, mathematically akin to the convolutional neural network (CNN), enables faster propagation of source influence during the solve phase, making it particularly suitable for handling large size, high wavenumber problems. We conduct supervised learning tests against numerous neural operator learning methods to demonstrate the superior learning capabilities of our solvers. Additionally, we perform scalability tests using an unsupervised strategy to highlight our solvers' significant speedup over the most recent specialized AI solver and AI-enhanced traditional solver for high wavenumber Helmholtz equations. We also carry out an ablation study to underscore the effectiveness of the multigrid hierarchy and the benefits of introducing FNN. Notably, our solvers exhibit optimal convergence of $\mathcal{O}(k)$ up to $k \approx 2000$.

Partial label learning is a weakly supervised learning problem in which an instance is annotated with a set of candidate labels, among which only one is the correct label. However, in practice the correct label is not always in the candidate label set, leading to the noisy partial label learning (NPLL) problem. In this paper, we theoretically prove that the generalization error of the classifier constructed under NPLL paradigm is bounded by the noise rate and the average length of the candidate label set. Motivated by the theoretical guide, we propose a novel NPLL framework that can separate the noisy samples from the normal samples to reduce the noise rate and reconstruct the shorter candidate label sets for both of them. Extensive experiments on multiple benchmark datasets confirm the efficacy of the proposed method in addressing NPLL. For example, on CIFAR100 dataset with severe noise, our method improves the classification accuracy of the state-of-the-art one by 11.57%. The code is available at: https://github.com/pruirui/PLRC.

Bayesian optimisation (BO) using a Gaussian process (GP)-based surrogate model is a powerful tool for solving black-box optimisation problems but does not scale well to high-dimensional data. Previous works have proposed to use variational autoencoders (VAEs) to project high-dimensional data onto a low-dimensional latent space and to implement BO in the inferred latent space. In this work, we propose a conditional generative model for efficient high-dimensional BO that uses a GP surrogate model together with GP prior VAEs. A GP prior VAE extends the standard VAE by conditioning the generative and inference model on auxiliary covariates, capturing complex correlations across samples with a GP. Our model incorporates the observed target quantity values as auxiliary covariates learning a structured latent space that is better suited for the GP-based BO surrogate model. It handles partially observed auxiliary covariates using a unifying probabilistic framework and can also incorporate additional auxiliary covariates that may be available in real-world applications. We demonstrate that our method improves upon existing latent space BO methods on simulated datasets as well as on commonly used benchmarks.

Deep heteroscedastic regression models the mean and covariance of the target distribution through neural networks. The challenge arises from heteroscedasticity, which implies that the covariance is sample dependent and is often unknown. Consequently, recent methods learn the covariance through unsupervised frameworks, which unfortunately yield a trade-off between computational complexity and accuracy. While this trade-off could be alleviated through supervision, obtaining labels for the covariance is non-trivial. Here, we study self-supervised covariance estimation in deep heteroscedastic regression. We address two questions: (1) How should we supervise the covariance assuming ground truth is available? (2) How can we obtain pseudo labels in the absence of the ground-truth? We address (1) by analysing two popular measures: the KL Divergence and the 2-Wasserstein distance. Subsequently, we derive an upper bound on the 2-Wasserstein distance between normal distributions with non-commutative covariances that is stable to optimize. We address (2) through a simple neighborhood based heuristic algorithm which results in surprisingly effective pseudo labels for the covariance. Our experiments over a wide range of synthetic and real datasets demonstrate that the proposed 2-Wasserstein bound coupled with pseudo label annotations results in a computationally cheaper yet accurate deep heteroscedastic regression.

Self-supervised learning (SSL) approaches have brought tremendous success across many tasks and domains. It has been argued that these successes can be attributed to a link between SSL and identifiable representation learning: Temporal structure and auxiliary variables ensure that latent representations are related to the true underlying generative factors of the data. Here, we deepen this connection and show that SSL can perform system identification in latent space. We propose dynamics contrastive learning, a framework to uncover linear, switching linear and non-linear dynamics under a non-linear observation model, give theoretical guarantees and validate them empirically.

Recent advancements in operator-type neural networks have shown promising results in approximating the solutions of spatiotemporal Partial Differential Equations (PDEs). However, these neural networks often entail considerable training expenses, and may not always achieve the desired accuracy required in many scientific and engineering disciplines. In this paper, we propose a new learning framework to address these issues. A new spatiotemporal adaptation is proposed to generalize any Fourier Neural Operator (FNO) variant to learn maps between Bochner spaces, which can perform an arbitrary-length temporal super-resolution for the first time. To better exploit this capacity, a new paradigm is proposed to refine the commonly adopted end-to-end neural operator training and evaluations with the help from the wisdom from traditional numerical PDE theory and techniques. Specifically, in the learning problems for the turbulent flow modeled by the Navier-Stokes Equations (NSE), the proposed paradigm trains an FNO only for a few epochs. Then, only the newly proposed spatiotemporal spectral convolution layer is fine-tuned without the frequency truncation. The spectral fine-tuning loss function uses a negative Sobolev norm for the first time in operator learning, defined through a reliable functional-type a posteriori error estimator whose evaluation is exact thanks to the Parseval identity. Moreover, unlike the difficult nonconvex optimization problems in the end-to-end training, this fine-tuning loss is convex. Numerical experiments on commonly used NSE benchmarks demonstrate significant improvements in both computational efficiency and accuracy, compared to end-to-end evaluation and traditional numerical PDE solvers under certain conditions. The source code is publicly available at https://github.com/scaomath/torch-cfd.

Learning long-term behaviors in chaotic dynamical systems, such as turbulent flows and climate modelling, is challenging due to their inherent instability and unpredictability. These systems exhibit positive Lyapunov exponents, which significantly hinder accurate long-term forecasting. As a result, understanding long-term statistical behavior is far more valuable than focusing on short-term accuracy. While autoregressive deep sequence models have been applied to capture long-term behavior, they often lead to exponentially increasing errors in learned dynamics. To address this, we shift the focus from simple prediction errors to preserving an invariant measure in dissipative chaotic systems. These systems have attractors, where trajectories settle, and the invariant measure is the probability distribution on attractors that remains unchanged under dynamics. Existing methods generate long trajectories of dissipative chaotic systems by aligning invariant measures, but it is not always possible to obtain invariant measures for arbitrary datasets. We propose the Poincaré Flow Neural Network (PFNN), a novel operator learning framework designed to capture behaviors of chaotic systems without any explicit knowledge of the invariant measure. PFNN employs an auto-encoder to map the chaotic system to a finite-dimensional feature space, effectively linearizing the chaotic evolution. It then learns the linear evolution operators to match the physical dynamics by addressing two critical properties in dissipative chaotic systems: (1) contraction, the system’s convergence toward its attractors, and (2) measure invariance, trajectories on the attractors following a probability distribution invariant to the dynamics. Our experiments on a variety of chaotic systems, including Lorenz systems, Kuramoto-Sivashinsky equation and Navier–Stokes equation, demonstrate that PFNN has more accurate predictions and physical statistics compared to competitive baselines including the Fourier Neural Operator and the Markov Neural Operator.

We develop a framework to quantify predictive uncertainty in survival analysis, providing a reliable lower predictive bound (LPB) for the true, unknown patient survival time. Recently, conformal prediction has been used to construct such valid LPBs for *type-I right-censored data*, with the guarantee that the bound holds with high probability. Crucially, under the type-I setting, the censoring time is observed for all data points. As such, informative LPBs can be constructed by framing the calibration as an estimation task with covariate shift, relying on the conditionally independent censoring assumption. This paper expands the conformal toolbox for survival analysis, with the goal of handling the ubiquitous *general right-censored setting*, in which either the censoring or survival time is observed, but not both. The key challenge here is that the calibration cannot be directly formulated as a covariate shift problem anymore. Yet, we show how to construct LPBs with distribution-free finite-sample guarantees, under the same assumptions as conformal approaches for type-I censored data. Experiments demonstrate the informativeness and validity of our methods in simulated settings and showcase their practical utility using several real-world datasets.

In recent years, neural operators have emerged as a prominent approach for learning mappings between function spaces, such as the solution operators of parametric PDEs. A notable example is the Fourier Neural Operator (FNO), which models the integral kernel as a convolution operator and uses the Convolution Theorem to learn the kernel directly in the frequency domain. The parameters are decoupled from the resolution of the data, allowing the FNO to take inputs of different resolutions. However, training at a lower resolution and inferring at a finer resolution does not guarantee consistent performance, nor can fine details, present only in fine-scale data, be learned solely from coarse data. In this work, we address this misconception by defining and examining the discretization mismatch error: the discrepancy between the outputs of the neural operator when using different discretizations of the input data. We demonstrate that neural operators may suffer from discretization mismatch errors that hinder their effectiveness when inferred on data with resolutions different from that of the training data or when trained on data with varying resolutions. As neural operators underpin many critical cross-resolution scientific tasks, such as climate modeling and fluid dynamics, understanding discretization mismatch errors is essential. Based on our findings, we propose a Cross-Resolution Operator-learning Pipeline that is free of aliasing and discretization mismatch errors, enabling efficient cross-resolution and multi-spatial-scale learning, and resulting in superior performance.

In this paper, we address conditional testing problems through the conformal inference framework. We define the localized conformal $p$-values by inverting prediction intervals and prove their theoretical properties. These defined $p$-values are then applied to several conditional testing problems to illustrate their practicality. Firstly, we propose a conditional outlier detection procedure to test for outliers in the conditional distribution with finite-sample false discovery rate (FDR) control. We also introduce a novel conditional label screening problem with the goal of screening multivariate response variables and propose a screening procedure to control the family-wise error rate (FWER). Finally, we consider the two-sample conditional distribution test and define a weighted U-statistic through the aggregation of localized $p$-values. Numerical simulations and real-data examples validate the superior performance of our proposed strategies.

In this paper, we use the Fire Dynamics Simulator (FDS) combined with the {\fontfamily{lmtt}\selectfont \textit{supercomputer}} support to create a \textbf{C}ombustion \textbf{K}inetics (CK) dataset for machine learning and scientific research. This dataset captures the development of fires in industrial parks with high-precision Computational Fluid Dynamics (CFD) simulations. It includes various physical fields such as temperature and pressure, and covers multiple environmental combinations for exploring \underline{multi-physics} field coupling phenomena. Additionally, we evaluate several advanced machine learning architectures across our {\fontfamily{lmtt}\selectfont {Open-CK}} benchmark using a substantial computational setup of 64 NVIDIA A100 GPUs: \ding{182} vision backbone; \ding{183} spatio-temporal predictive models; \ding{184} operator learning frameworks. These architectures uniquely excel at handling complex physical field data. We also introduce three benchmarks to demonstrate their potential in enhancing the exploration of downstream tasks: (a) capturing continuous changes in combustion kinetics; (b) a neural partial differential equation solver for learning temperature fields and turbulence; (c) reconstruction of sparse physical observations. The Open-CK dataset and benchmarks aim to advance research in combustion kinetics driven by machine learning, providing a reliable baseline for developing and comparing cutting-edge technologies and models. We hope to further promote the application of deep learning in earth sciences. Our project is available at \url{https://github.com/whscience/Open-CK}.

In this study, we examine the representation learning abilities of Denoising Diffusion Models (DDM) that were originally purposed for image generation. Our philosophy is to deconstruct a DDM, gradually transforming it into a classical Denoising Autoencoder (DAE). This deconstructive process allows us to explore how various components of modern DDMs influence self-supervised representation learning. We observe that only a very few modern components are critical for learning good representations, while many others are nonessential. Our study ultimately arrives at an approach that is highly simplified and to a large extent resembles a classical DAE. We hope our study will rekindle interest in a family of classical methods within the realm of modern self-supervised learning.

We address the challenge of zeroth-order online convex optimization where the objective function's gradient exhibits sparsity, indicating that only a small number of dimensions possess non-zero gradients. Our aim is to leverage this sparsity to obtain useful estimates of the objective function's gradient even when the only information available is a limited number of function samples. Our motivation stems from the optimization of large-scale queueing networks that process time-sensitive jobs. Here, a job must be processed by potentially many queues in sequence to produce an output, and the service time at any queue is a function of the resources allocated to that queue. Since resources are costly, the end-to-end latency for jobs must be balanced with the overall cost of the resources used. While the number of queues is substantial, the latency function primarily reacts to resource changes in only a few, rendering the gradient sparse. We tackle this problem by introducing the Compressive Online Gradient Optimization framework which allows compressive sensing methods previously applied to stochastic optimization to achieve regret bounds with an optimal dependence on the time horizon without the full problem dimension appearing in the bound. For specific algorithms, we reduce the samples required per gradient estimate to scale with the gradient's sparsity factor rather than its full dimensionality. Numerical simulations and real-world microservices benchmarks demonstrate CONGO's superiority over gradient descent approaches that do not account for sparsity.

Modeling the evolution of physical systems is critical to many applications in science and engineering. As the evolution of these systems is governed by partial differential equations (PDEs), there are a number of computational simulations which resolve these systems with high accuracy. However, as these simulations incur high computational costs, they are infeasible to be employed for large-scale analysis. A popular alternative to simulators are neural network surrogates which are trained in a data-driven manner and are much more computationally efficient. However, these surrogate models suffer from high rollout error when used autoregressively, especially when confronted with training data paucity. Existing work proposes to improve surrogate rollout error by either including physical loss terms directly in the optimization of the model or incorporating computational simulators as `differentiable layers' in the neural network. Both of these approaches have their challenges, with physical loss functions suffering from slow convergence for stiff PDEs and simulator layers requiring gradients which are not always available, especially in legacy simulators. We propose the Hybrid PDE Predictor with Reinforcement Learning (HyPER) model: a model-agnostic, RL based, cost-aware model which combines a neural surrogate, RL decision model, and a physics simulator (with or without gradients) to reduce surrogate rollout error significantly. In addition to reducing in-distribution rollout error by **47%-78%**, HyPER learns an intelligent policy that is adaptable to changing physical conditions and resistant to noise corruption. Code available at https://github.com/scailab/HyPER.

Semi-supervised learning (SSL), exemplified by FixMatch (Sohn et al., 2020), has shown significant generalization advantages over supervised learning (SL), particularly in the context of deep neural networks (DNNs). However, it is still unclear, from a theoretical standpoint, why FixMatch-like SSL algorithms generalize better than SL on DNNs. In this work, we present the first theoretical justification for the enhanced test accuracy observed in FixMatch-like SSL applied to DNNs by taking convolutional neural networks (CNNs) on classification tasks as an example. Our theoretical analysis reveals that the semantic feature learning processes in FixMatch and SL are rather different. In particular, FixMatch learns all the discriminative features of each semantic class, while SL only randomly captures a subset of features due to the well-known lottery ticket hypothesis. Furthermore, we show that our analysis framework can be applied to other FixMatch-like SSL methods, e.g., FlexMatch, FreeMatch, Dash, and SoftMatch. Inspired by our theoretical analysis, we develop an improved variant of FixMatch, termed Semantic-Aware FixMatch (SA-FixMatch). Experimental results corroborate our theoretical findings and the enhanced generalization capability of SA-FixMatch.

A growing number of machine learning scenarios rely on knowledge distillation where one uses the output of a surrogate model as labels to supervise the training of a target model. In this work, we provide a sharp characterization of this process for ridgeless, high-dimensional regression, under two settings: *(i)* model shift, where the surrogate model is arbitrary, and *(ii)* distribution shift, where the surrogate model is the solution of empirical risk minimization with out-of-distribution data. In both cases, we characterize the precise risk of the target model through non-asymptotic bounds in terms of sample size and data distribution under mild conditions. As a consequence, we identify the form of the optimal surrogate model, which reveals the benefits and limitations of discarding weak features in a data-dependent fashion. In the context of weak-to-strong (W2S) generalization, this has the interpretation that *(i)* W2S training, with the surrogate as the weak model, can provably outperform training with strong labels under the same data budget, but *(ii)* it is unable to improve the data scaling law. We validate our results on numerical experiments both on ridgeless regression and on neural network architectures.

Model merging has emerged as an effective approach to combining multiple single-task models into a multitask model. This process typically involves computing a weighted average of the model parameters without additional training. Existing model-merging methods focus on improving average task accuracy. However, interference and conflicts between the objectives of different tasks can lead to trade-offs during the merging process. In real-world applications, a set of solutions with various trade-offs can be more informative, helping practitioners make decisions based on diverse preferences. In this paper, we introduce a novel and low-compute algorithm, Model Merging with Amortized Pareto Front (MAP). MAP efficiently identifies a Pareto set of scaling coefficients for merging multiple models, reflecting the trade-offs involved. It amortizes the substantial computational cost of evaluations needed to estimate the Pareto front by using quadratic approximation surrogate models derived from a preselected set of scaling coefficients. Experimental results on vision and natural language processing tasks demonstrate that MAP can accurately identify the Pareto front, providing practitioners with flexible solutions to balance competing task objectives. We also introduce Bayesian MAP for scenarios with a relatively low number of tasks and Nested MAP for situations with a high number of tasks, further reducing the computational cost of evaluation.

Gaussian Processes (GPs) are widely seen as the state-of-the-art surrogate models for Bayesian optimization (BO) due to their ability to model uncertainty and their performance on tasks where correlations are easily captured (such as those defined by Euclidean metrics) and their ability to be efficiently updated online. However, the performance of GPs depends on the choice of kernel, and kernel selection for complex correlation structures is often difficult or must be made bespoke. While Bayesian neural networks (BNNs) are a promising direction for higher capacity surrogate models, they have so far seen limited use due to poor performance on some problem types. In this paper, we propose an approach which shows competitive performance on many problem types, including some that BNNs typically struggle with. We build on variational Bayesian last layers (VBLLs), and connect training of these models to exact conditioning in GPs. We exploit this connection to develop an efficient online training algorithm that interleaves conditioning and optimization. Our findings suggest that VBLL networks significantly outperform GPs and other BNN architectures on tasks with complex input correlations, and match the performance of well-tuned GPs on established benchmark tasks.

Surrogate models are used to predict the behavior of complex energy systems that are too expensive to simulate with traditional numerical methods. Our work introduces the use of language descriptions, which we call "system captions" or SysCaps, to interface with such surrogates. We argue that interacting with surrogates through text, particularly natural language, makes these models more accessible for both experts and non-experts. We introduce a lightweight multimodal text and timeseries regression model and a training pipeline that uses large language models (LLMs) to synthesize high-quality captions from simulation metadata. Our experiments on two real-world simulators of buildings and wind farms show that our SysCaps-augmented surrogates have better accuracy on held-out systems than traditional methods while enjoying new generalization abilities, such as handling semantically related descriptions of the same test system. Additional experiments also highlight the potential of SysCaps to unlock language-driven design space exploration and to regularize training through prompt augmentation.