Asma Jamali

Asma Jamali, Tin Sum Cheng, Rodrigo A. Vargas-Hernández

Understanding the spectral properties of kernels offers a principled perspective on generalization and representation quality. While deep models achieve state-of-the-art accuracy in molecular property prediction, kernel methods remain widely used for their robustness in low-data regimes and transparent theoretical grounding. Despite extensive studies of kernel spectra in machine learning, systematic spectral analyses of molecular kernels are scarce. In this work, we provide the first comprehensive spectral analysis of kernel ridge regression on the QM9 dataset, molecular fingerprint, pretrained transformer-based, global and local 3D representations across seven molecular properties. Surprisingly, richer spectral features, measured by four different spectral metrics, do not consistently improve accuracy. Pearson correlation tests further reveal that for transformer-based and local 3D representations, spectral richness can even have a negative correlation with performance. We also implement truncated kernels to probe the relationship between spectrum and predictive performance: in many kernels, retaining only the top 2% of eigenvalues recovers nearly all performance, indicating that the leading eigenvalues capture the most informative features. Our results challenge the common heuristic that "richer spectra yield better generalization" and highlight nuanced relationships between representation, kernel features, and predictive performance. Beyond molecular property prediction, these findings inform how kernel and self-supervised learning methods are evaluated in data-limited scientific and real-world tasks.

Hamidreza Moazzami, Asma Jamali, Nicholas Kevlahan et al.

Data assimilation (DA) is crucial for enhancing solutions to partial differential equations (PDEs), such as those in numerical weather prediction, by optimizing initial conditions using observational data. Variational DA methods are widely used in oceanic and atmospheric forecasting, but become computationally expensive, especially when Hessian information is involved. To address this challenge, we propose a meta-learning framework that employs the Fourier Neural Operator (FNO) to approximate the inverse Hessian operator across a family of DA problems, thereby providing an effective initialization for the conjugate gradient (CG) method. Numerical experiments on a linear advection equation demonstrate that the resulting FNO-CG approach reduces the average relative error by $62\%$ and the number of iterations by $17\%$ compared to the standard CG. These improvements are most pronounced in ill-conditioned scenarios, highlighting the robustness and efficiency of FNO-CG for challenging DA problems.