The Well: a Large-Scale Collection of Diverse Physics Simulations for Machine Learning

Machine learning based surrogate models offer researchers powerful tools for accelerating simulation-based workflows. However, as standard datasets in this space often cover small classes of physical behavior, it can be difficult to evaluate the efficacy of new approaches. To address this gap, we introduce the Well: a large-scale collection of datasets containing numerical simulations of a wide variety of spatiotemporal physical systems. The Well draws from domain experts and numerical software developers to provide 15TB of data across 16 datasets covering diverse domains such as biological systems, fluid dynamics, acoustic scattering, as well as magneto-hydrodynamic simulations of extra-galactic fluids or supernova explosions. These datasets can be used individually or as part of a broader benchmark suite. To facilitate usage of the Well, we provide a unified PyTorch interface for training and evaluating models. We demonstrate the function of this library by introducing example baselines that highlight the new challenges posed by the complex dynamics of the Well. The code and data is available at https://github.com/PolymathicAI/the_well.

Contextual Counting: A Mechanistic Study of Transformers on a Quantitative Task

Transformers have revolutionized machine learning across diverse domains, yet understanding their behavior remains crucial, particularly in high-stakes applications. This paper introduces the contextual counting task, a novel toy problem aimed at enhancing our understanding of Transformers in quantitative and scientific contexts. This task requires precise localization and computation within datasets, akin to object detection or region-based scientific analysis. We present theoretical and empirical analysis using both causal and non-causal Transformer architectures, investigating the influence of various positional encodings on performance and interpretability. In particular, we find that causal attention is much better suited for the task, and that no positional embeddings lead to the best accuracy, though rotary embeddings are competitive and easier to train. We also show that out of distribution performance is tightly linked to which tokens it uses as a bias term.

Listening to the Noise: Blind Denoising with Gibbs Diffusion

In recent years, denoising problems have become intertwined with the development of deep generative models. In particular, diffusion models are trained like denoisers, and the distribution they model coincide with denoising priors in the Bayesian picture. However, denoising through diffusion-based posterior sampling requires the noise level and covariance to be known, preventing blind denoising. We overcome this limitation by introducing Gibbs Diffusion (GDiff), a general methodology addressing posterior sampling of both the signal and the noise parameters. Assuming arbitrary parametric Gaussian noise, we develop a Gibbs algorithm that alternates sampling steps from a conditional diffusion model trained to map the signal prior to the family of noise distributions, and a Monte Carlo sampler to infer the noise parameters. Our theoretical analysis highlights potential pitfalls, guides diagnostic usage, and quantifies errors in the Gibbs stationary distribution caused by the diffusion model. We showcase our method for 1) blind denoising of natural images involving colored noises with unknown amplitude and spectral index, and 2) a cosmology problem, namely the analysis of cosmic microwave background data, where Bayesian inference of "noise" parameters means constraining models of the evolution of the Universe.

Removing Dust from CMB Observations with Diffusion Models

In cosmology, the quest for primordial $B$-modes in cosmic microwave background (CMB) observations has highlighted the critical need for a refined model of the Galactic dust foreground. We investigate diffusion-based modeling of the dust foreground and its interest for component separation. Under the assumption of a Gaussian CMB with known cosmology (or covariance matrix), we show that diffusion models can be trained on examples of dust emission maps such that their sampling process directly coincides with posterior sampling in the context of component separation. We illustrate this on simulated mixtures of dust emission and CMB. We show that common summary statistics (power spectrum, Minkowski functionals) of the components are well recovered by this process. We also introduce a model conditioned by the CMB cosmology that outperforms models trained using a single cosmology on component separation. Such a model will be used in future work for diffusion-based cosmological inference.

xVal: A Continuous Number Encoding for Large Language Models

Large Language Models have not yet been broadly adapted for the analysis of scientific datasets due in part to the unique difficulties of tokenizing numbers. We propose xVal, a numerical encoding scheme that represents any real number using just a single token. xVal represents a given real number by scaling a dedicated embedding vector by the number value. Combined with a modified number-inference approach, this strategy renders the model end-to-end continuous when considered as a map from the numbers of the input string to those of the output string. This leads to an inductive bias that is generally more suitable for applications in scientific domains. We empirically evaluate our proposal on a number of synthetic and real-world datasets. Compared with existing number encoding schemes, we find that xVal is more token-efficient and demonstrates improved generalization.

Multiple Physics Pretraining for Physical Surrogate Models

We introduce multiple physics pretraining (MPP), an autoregressive task-agnostic pretraining approach for physical surrogate modeling. MPP involves training large surrogate models to predict the dynamics of multiple heterogeneous physical systems simultaneously by learning features that are broadly useful across diverse physical tasks. In order to learn effectively in this setting, we introduce a shared embedding and normalization strategy that projects the fields of multiple systems into a single shared embedding space. We validate the efficacy of our approach on both pretraining and downstream tasks over a broad fluid mechanics-oriented benchmark. We show that a single MPP-pretrained transformer is able to match or outperform task-specific baselines on all pretraining sub-tasks without the need for finetuning. For downstream tasks, we demonstrate that finetuning MPP-trained models results in more accurate predictions across multiple time-steps on new physics compared to training from scratch or finetuning pretrained video foundation models. We open-source our code and model weights trained at multiple scales for reproducibility and community experimentation.

AstroCLIP: Cross-Modal Pre-Training for Astronomical Foundation Models

We present AstroCLIP, a strategy to facilitate the construction of astronomical foundation models that bridge the gap between diverse observational modalities. We demonstrate that a cross-modal contrastive learning approach between images and optical spectra of galaxies yields highly informative embeddings of both modalities. In particular, we apply our method on multi-band images and optical spectra from the Dark Energy Spectroscopic Instrument (DESI), and show that: (1) these embeddings are well-aligned between modalities and can be used for accurate cross-modal searches, and (2) these embeddings encode valuable physical information about the galaxies -- in particular redshift and stellar mass -- that can be used to achieve competitive zero- and few- shot predictions without further finetuning. Additionally, in the process of developing our approach, we also construct a novel, transformer-based model and pretraining approach for processing galaxy spectra.

MoMo: Momentum Models for Adaptive Learning Rates

We present new adaptive learning rates that can be used with any momentum method. To showcase our new learning rates we develop MoMo and MoMo-Adam, which are SGD with momentum (SGDM) and Adam together with our new adaptive learning rates. Our MoMo methods are motivated through model-based stochastic optimization, wherein we use momentum estimates of the batch losses and gradients sampled at each iteration to build a model of the loss function. Our model also makes use of any known lower bound of the loss function by using truncation. Indeed most losses are bounded below by zero. We then approximately minimize this model at each iteration to compute the next step. For losses with unknown lower bounds, we develop new on-the-fly estimates of the lower bound that we use in our model. Numerical experiments show that our MoMo methods improve over SGDM and Adam in terms of accuracy and robustness to hyperparameter tuning for training image classifiers on MNIST, CIFAR10, CIFAR100, Imagenet32, DLRM on the Criteo dataset, and a transformer model on the translation task IWSLT14.

Shedding a PAC-Bayesian Light on Adaptive Sliced-Wasserstein Distances

The Sliced-Wasserstein distance (SW) is a computationally efficient and theoretically grounded alternative to the Wasserstein distance. Yet, the literature on its statistical properties with respect to the distribution of slices, beyond the uniform measure, is scarce. To bring new contributions to this line of research, we leverage the PAC-Bayesian theory and the central observation that SW actually hinges on a slice-distribution-dependent Gibbs risk, the kind of quantity PAC-Bayesian bounds have been designed to characterize. We provide four types of results: i) PAC-Bayesian generalization bounds that hold on what we refer as adaptive Sliced-Wasserstein distances, i.e. distances defined with respect to any distribution of slices, ii) a procedure to learn the distribution of slices that yields a maximally discriminative SW, by optimizing our PAC-Bayesian bounds, iii) an insight on how the performance of the so-called distributional Sliced-Wasserstein distance may be explained through our theory, and iv) empirical illustrations of our findings.

Complex-to-Real Random Features for Polynomial Kernels

Kernel methods are ubiquitous in statistical modeling due to their theoretical guarantees as well as their competitive empirical performance. Polynomial kernels are of particular importance as their feature maps model the interactions between the dimensions of the input data. However, the construction time of explicit feature maps scales exponentially with the polynomial degree and a naive application of the kernel trick does not scale to large datasets. In this work, we propose Complex-to-Real (CtR) random features for polynomial kernels that leverage intermediate complex random projections and can yield kernel estimates with much lower variances than their real-valued analogs. The resulting features are real-valued, simple to construct and have the following advantages over the state-of-the-art: 1) shorter construction times, 2) lower kernel approximation errors for commonly used degrees, 3) they enable us to obtain a closed-form expression for their variance.