David W. Hogg

Soledad Villar, Weichi Yao, David W. Hogg et al.

Units equivariance (or units covariance) is the exact symmetry that follows from the requirement that relationships among measured quantities of physics relevance must obey self-consistent dimensional scalings. Here, we express this symmetry in terms of a (non-compact) group action, and we employ dimensional analysis and ideas from equivariant machine learning to provide a methodology for exactly units-equivariant machine learning: For any given learning task, we first construct a dimensionless version of its inputs using classic results from dimensional analysis, and then perform inference in the dimensionless space. Our approach can be used to impose units equivariance across a broad range of machine learning methods which are equivariant to rotations and other groups. We discuss the in-sample and out-of-sample prediction accuracy gains one can obtain in contexts like symbolic regression and emulation, where symmetry is important. We illustrate our approach with simple numerical examples involving dynamical systems in physics and ecology.

Soledad Villar, David W. Hogg, Weichi Yao et al.

Any representation of data involves arbitrary investigator choices. Because those choices are external to the data-generating process, each choice leads to an exact symmetry, corresponding to the group of transformations that takes one possible representation to another. These are the passive symmetries; they include coordinate freedom, gauge symmetry, and units covariance, all of which have led to important results in physics. In machine learning, the most visible passive symmetry is the relabeling or permutation symmetry of graphs. Our goal is to understand the implications for machine learning of the many passive symmetries in play. We discuss dos and don'ts for machine learning practice if passive symmetries are to be respected. We discuss links to causal modeling, and argue that the implementation of passive symmetries is particularly valuable when the goal of the learning problem is to generalize out of sample. This paper is conceptual: It translates among the languages of physics, mathematics, and machine-learning. We believe that consideration and implementation of passive symmetries might help machine learning in the same ways that it transformed physics in the twentieth century.

Pablo Mercader-Perez, Carolina Cuesta-Lazaro, Daniel Muthukrishna et al.

Data collected from the physical world is always a combination of multiple sources: an underlying signal from the physical process of interest and a signal from measurement-dependent artifacts from the sensor or instrument. This secondary signal acts as a confounding factor, limiting our ability to extract information about the physics underlying the phenomena we observe. Furthermore, it complicates the combination of observations in heterogeneous or multi-instrument settings. We propose a deep learning framework that leverages overlapping observations, a dual-encoder architecture, and a counterfactual generation objective to disentangle these factors of variation. The resulting representations explicitly separate intrinsic signals from sensor-specific distortions and noise, and can be used for counterfactual view generation, parameter inference unconfounded by measurement distortions, and instrument-independent similarity search. We demonstrate the effectiveness of our approach on astrophysical galaxy images from the DESI Legacy Imaging Survey (Legacy) and the Hyper Suprime-Cam (HSC) Survey as a representative multi-instrument setting. This framework provides a general recipe for scientific and multi-modal self-supervised pretraining: construct training pairs from overlapping observations of the same physical system, treat sensor- or modality-specific effects as augmentations, and learn invariant representations through counterfactual generation.

Valentino F. Foit, David W. Hogg, Soledad Villar

Many machine learning tasks in the natural sciences are precisely equivariant to particular symmetries. Nonetheless, equivariant methods are often not employed, perhaps because training is perceived to be challenging, or the symmetry is expected to be learned, or equivariant implementations are seen as hard to build. Group averaging is an available technique for these situations. It happens at test time; it can make any trained model precisely equivariant at a (often small) cost proportional to the size of the group; it places no requirements on model structure or training. It is known that, under mild conditions, the group-averaged model will have a provably better prediction accuracy than the original model. Here we show that an inexpensive group averaging can improve accuracy in practice. We take well-established benchmark machine learning models of differential equations in which certain symmetries ought to be obeyed. At evaluation time, we average the models over a small group of symmetries. Our experiments show that this procedure always decreases the average evaluation loss, with improvements of up to 37\% in terms of the VRMSE. The averaging produces visually better predictions for continuous dynamics. This short paper shows that, under certain common circumstances, there are no disadvantages to imposing exact symmetries; the ML4PS community should consider group averaging as a cheap and simple way to improve model accuracy.

Weichi Yao, Kate Storey-Fisher, David W. Hogg et al.

Physical systems obey strict symmetry principles. We expect that machine learning methods that intrinsically respect these symmetries should have higher prediction accuracy and better generalization in prediction of physical dynamics. In this work we implement a principled model based on invariant scalars, and release open-source code. We apply this Scalars method to a simple chaotic dynamical system, the springy double pendulum. We show that the Scalars method outperforms state-of-the-art approaches for learning the properties of physical systems with symmetries, both in terms of accuracy and speed. Because the method incorporates the fundamental symmetries, we expect it to generalize to different settings, such as changes in the force laws in the system.

Jeroen Audenaert, Daniel Muthukrishna, Paul F. Gregory et al.

Foundation models for structured time series data must contend with a fundamental challenge: observations often conflate the true underlying physical phenomena with systematic distortions introduced by measurement instruments. This entanglement limits model generalization, especially in heterogeneous or multi-instrument settings. We present a causally-motivated foundation model that explicitly disentangles physical and instrumental factors using a dual-encoder architecture trained with structured contrastive learning. Leveraging naturally occurring observational triplets (i.e., where the same target is measured under varying conditions, and distinct targets are measured under shared conditions) our model learns separate latent representations for the underlying physical signal and instrument effects. Evaluated on simulated astronomical time series designed to resemble the complexity of variable stars observed by missions like NASA's Transiting Exoplanet Survey Satellite (TESS), our method significantly outperforms traditional single-latent space foundation models on downstream prediction tasks, particularly in low-data regimes. These results demonstrate that our model supports key capabilities of foundation models, including few-shot generalization and efficient adaptation, and highlight the importance of encoding causal structure into representation learning for structured data.

Wilson G. Gregory, David W. Hogg, Ben Blum-Smith et al.

Machine learning methods are increasingly being employed as surrogate models in place of computationally expensive and slow numerical integrators for a bevy of applications in the natural sciences. However, while the laws of physics are relationships between scalars, vectors, and tensors that hold regardless of the frame of reference or chosen coordinate system, surrogate machine learning models are not coordinate-free by default. We enforce coordinate freedom by using geometric convolutions in three model architectures: a ResNet, a Dilated ResNet, and a UNet. In numerical experiments emulating 2D compressible Navier-Stokes, we see better accuracy and improved stability compared to baseline surrogate models in almost all cases. The ease of enforcing coordinate freedom without making major changes to the model architecture provides an exciting recipe for any CNN-based method applied to an appropriate class of problems

Soledad Villar, David W. Hogg, Kate Storey-Fisher et al.

There has been enormous progress in the last few years in designing neural networks that respect the fundamental symmetries and coordinate freedoms of physical law. Some of these frameworks make use of irreducible representations, some make use of high-order tensor objects, and some apply symmetry-enforcing constraints. Different physical laws obey different combinations of fundamental symmetries, but a large fraction (possibly all) of classical physics is equivariant to translation, rotation, reflection (parity), boost (relativity), and permutations. Here we show that it is simple to parameterize universally approximating polynomial functions that are equivariant under these symmetries, or under the Euclidean, Lorentz, and Poincaré groups, at any dimensionality $d$. The key observation is that nonlinear O($d$)-equivariant (and related-group-equivariant) functions can be universally expressed in terms of a lightweight collection of scalars -- scalar products and scalar contractions of the scalar, vector, and tensor inputs. We complement our theory with numerical examples that show that the scalar-based method is simple, efficient, and scalable.

David W. Hogg, Soledad Villar

There are many uses for linear fitting; the context here is interpolation and denoising of data, as when you have calibration data and you want to fit a smooth, flexible function to those data. Or you want to fit a flexible function to de-trend a time series or normalize a spectrum. In these contexts, investigators often choose a polynomial basis, or a Fourier basis, or wavelets, or something equally general. They also choose an order, or number of basis functions to fit, and (often) some kind of regularization. We discuss how this basis-function fitting is done, with ordinary least squares and extensions thereof. We emphasize that it is often valuable to choose far more parameters than data points, despite folk rules to the contrary: Suitably regularized models with enormous numbers of parameters generalize well and make good predictions for held-out data; over-fitting is not (mainly) a problem of having too many parameters. It is even possible to take the limit of infinite parameters, at which, if the basis and regularization are chosen correctly, the least-squares fit becomes the mean of a Gaussian process. We recommend cross-validation as a good empirical method for model selection (for example, setting the number of parameters and the form of the regularization), and jackknife resampling as a good empirical method for estimating the uncertainties of the predictions made by the model. We also give advice for building stable computational implementations.

Ningyuan Huang, David W. Hogg, Soledad Villar

Overparameterization in deep learning is powerful: Very large models fit the training data perfectly and yet often generalize well. This realization brought back the study of linear models for regression, including ordinary least squares (OLS), which, like deep learning, shows a "double-descent" behavior: (1) The risk (expected out-of-sample prediction error) can grow arbitrarily when the number of parameters $p$ approaches the number of samples $n$, and (2) the risk decreases with $p$ for $p>n$, sometimes achieving a lower value than the lowest risk for $p<n$. The divergence of the risk for OLS can be avoided with regularization. In this work, we show that for some data models it can also be avoided with a PCA-based dimensionality reduction (PCA-OLS, also known as principal component regression). We provide non-asymptotic bounds for the risk of PCA-OLS by considering the alignments of the population and empirical principal components. We show that dimensionality reduction improves robustness while OLS is arbitrarily susceptible to adversarial attacks, particularly in the overparameterized regime. We compare PCA-OLS theoretically and empirically with a wide range of projection-based methods, including random projections, partial least squares (PLS), and certain classes of linear two-layer neural networks. These comparisons are made for different data generation models to assess the sensitivity to signal-to-noise and the alignment of regression coefficients with the features. We find that methods in which the projection depends on the training data can outperform methods where the projections are chosen independently of the training data, even those with oracle knowledge of population quantities, another seemingly paradoxical phenomenon that has been identified previously. This suggests that overparameterization may not be necessary for good generalization.

Bernhard Schölkopf, David W. Hogg, Dun Wang et al.

We describe a method for removing the effect of confounders in order to reconstruct a latent quantity of interest. The method, referred to as half-sibling regression, is inspired by recent work in causal inference using additive noise models. We provide a theoretical justification and illustrate the potential of the method in a challenging astronomy application.

Dustin Lang, David W. Hogg, Bernhard Scholkopf

We describe a system that builds a high dynamic-range and wide-angle image of the night sky by combining a large set of input images. The method makes use of pixel-rank information in the individual input images to improve a "consensus" pixel rank in the combined image. Because it only makes use of ranks and the complexity of the algorithm is linear in the number of images, the method is useful for large sets of uncalibrated images that might have undergone unknown non-linear tone mapping transformations for visualization or aesthetic reasons. We apply the method to images of the night sky (of unknown provenance) discovered on the Web. The method permits discovery of astronomical objects or features that are not visible in any of the input images taken individually. More importantly, however, it permits scientific exploitation of a huge source of astronomical images that would not be available to astronomical research without our automatic system.