arXiv subjects

Rachel A. Ward

Publications and source records attributed to Rachel A. Ward.

3 recordsLinked to original sources

arXiv2025

Modeling turbulent and self-gravitating fluids with Fourier neural operators

Neural Operators (NOs) are a leading method for surrogate modeling of partial differential equations. Unlike traditional neural networks, which approximate individual functions, NOs learn the mappings between function spaces. While NOs have been predominantly tested on simplified 1D and 2D problems, such as those explored in prior works, these studies fail to address the complexities of more realistic, high-dimensional, and high-dynamic range systems. Moreover, many real-world applications involve incomplete or noisy data, which has not been adequately explored in current NO literature. In this work, we present a novel application of NOs to astrophysical data, which involves high-dynamic range projections into an observational space. We train Fourier NO (FNO) models to predict the evolution of incomplete observational proxies with density variations spanning four orders of magnitude. We demonstrate that FNOs can predict the effects of unobserved dynamical variables. Our work lays the groundwork for future studies that forecast direct astronomical observables.

astro-ph.GA↗

arXiv2025

Generating synthetic data for neural operators

Recent advances in the literature show promising potential of deep learning methods, particularly neural operators, in obtaining numerical solutions to partial differential equations (PDEs) beyond the reach of current numerical solvers. However, existing data-driven approaches often rely on training data produced by numerical PDE solvers (e.g., finite difference or finite element methods). We introduce a "backward" data generation method that avoids solving the PDE numerically: by randomly sampling candidate solutions $u_j$ from the appropriate solution space (e.g., $H_0^1(Ω)$), we compute the corresponding right-hand side $f_j$ directly from the equation by differentiation. This produces training pairs ${(f_j, u_j)}$ by computing derivatives rather than solving a PDE numerically for each data point, enabling fast, large-scale data generation consisting of exact solutions. Experiments indicate that models trained on this synthetic data generalize well when tested on data produced by standard solvers. While the idea is simple, we hope this method will expand the potential of neural PDE solvers that do not rely on classical numerical solvers to generate their data.

cs.LG↗

arXiv2017

MC^2: A Two-Phase Algorithm for Leveraged Matrix Completion

Leverage scores, loosely speaking, reflect the importance of the rows and columns of a matrix. Ideally, given the leverage scores of a rank-$r$ matrix $M\in\mathbb{R}^{n\times n}$, that matrix can be reliably completed from just $O(rn\log^{2}n)$ samples if the samples are chosen randomly from a nonuniform distribution induced by the leverage scores. In practice, however, the leverage scores are often unknown a priori. As such, the sample complexity in uniform matrix completion---using uniform random sampling---increases to $O(η(M)\cdot rn\log^{2}n)$, where $η(M)$ is the largest leverage score of $M$. In this paper, we propose a two-phase algorithm called MC$^2$ for matrix completion: in the first phase, the leverage scores are estimated based on uniform random samples, and then in the second phase the matrix is resampled nonuniformly based on the estimated leverage scores and then completed. For well-conditioned matrices, the total sample complexity of MC$^2$ is no worse than uniform matrix completion, and for certain classes of well-conditioned matrices---namely, reasonably coherent matrices whose leverage scores exhibit mild decay---MC$^2$ requires substantially fewer samples. Numerical simulations suggest that the algorithm outperforms uniform matrix completion in a broad class of matrices, and in particular, is much less sensitive to the condition number than our theory currently requires.

cs.IT↗