The coordinate is right - Augmenting basis sets via normalizing flows

Approximating functions using a basis set of the functions’ space guarantees convergence of the approximation. However, high-accuracy calculations require memory costs that scale exponentially with the dimensionality, formally known as the curse of dimensionality. In this talk, I will introduce augmented basis sets, generated by pushing forward standard basis sets through normalizing flows, i.e., invertible neural networks, which is equivalent to a transformation of the basis’ coordinates. Multidimensional basis sets are often built from direct-products of univariate functions. These constructions struggle to capture complex structures involving different coordinates. Normalizing-flows coordinates reduce the coupling between dimensions, mitigating the curse of dimensionality.I will demonstrate the efficacy of augmented basis sets to approximate eigenpairs of the vibrational Schrödinger equation. Unlike standard neural-network-based methods that directly model the eigenfunctions, our method preserves the basis properties, ensuring robustness in the approximation of many highly excited states. Additionally, optimal normalizing-flows coordinates encode physical information of the molecular motion, which allows for the interpretability of the method, and enables transferability to different basis set truncations sizes and even to structurally similar molecular systems.