Systematically improved potential energy surfaces via sinNN models and sparse grid sampling

Abstract

Accurate, global Potential Energy Surfaces (PES) expressed in sum-of-products (SOP) form are a prerequisite for efficient high-dimensional quantum dynamics simulations using the MCTDH method. This work introduces a methodology for constructing such surfaces by combining hierarchical sparse grid sampling with a single-layer neural network using sinusoidal activation functions (sinNN). The sparse grid strategy provides a rigorous, unbiased discretization of the configuration space, enabling systematic improvability of the PES fidelity, where accuracy is strictly controlled by the refinement level, while successfully mitigating the curse of dimensionality. The sinNN fitting approach leverages a trigonometric factorization identity to maintain a compact SOP form, offering superior numerical stability compared to standard exponential-based networks (expNN) for the systems investigated. The flexibility of the sparse grid methodology is demonstrated through a dual-reference strategy, where grids centered on distinct isomers are merged to eliminate topological bias. This optimized sampling yields a global PES that reproduces fundamental vibrational transition energies for both trans- and cis-HONO with spectroscopic precision (< 2.5 cm-1) and high data efficiency. Finally, the methodology is applied to fit potential energies computed via the AI-enhanced quantum mechanical method AIQM2. The resulting AIQM2-based PES for HONO reproduces experimental vibrational frequencies with a root mean square deviation of about 16 cm-1, a performance comparable to high-level ab initio methods. The robustness of the approach is further confirmed on larger molecules, formic acid (HCOOH) and carbamic acid (H2NCOOH), establishing the combination of sparse grid sampling and sinNN fitting as a powerful, automated tool for generating topologically sound, spectroscopic-quality potential energy surfaces.