arXiv subjects

Álvaro Fernández Corral

Publications and source records attributed to Álvaro Fernández Corral.

5 recordsLinked to original sources

arXiv2025

Enhancing polynomial approximation of continuous functions by composition with homeomorphisms

We enhance the approximation capabilities of algebraic polynomials by composing them with homeomorphisms. This composition yields families of functions that remain dense in the space of continuous functions, while enabling more accurate approximations. For univariate continuous functions exhibiting a finite number of local extrema, we prove that there exist a polynomial of finite degree and a homeomorphism whose composition approximates the target function to arbitrary accuracy. The construction is especially relevant for multivariate approximation problems, where standard numerical methods often suffer from the curse of dimensionality. To support our theoretical results, we investigate both regression tasks and the construction of molecular potential-energy surfaces, parametrizing the underlying homeomorphism using invertible neural networks. The numerical experiments show strong agreement with our theoretical analysis.

math.NA↗

arXiv2025

Transferability and interpretability of vibrational normalizing-flow coordinates

The choice of vibrational coordinates is crucial for the accuracy, efficiency, and interpretability of molecular vibrational dynamics and spectra calculations. We explore the recently proposed normalizing-flow vibrational coordinates, which are learned molecule-specific coordinate transformations optimized for a given basis set. Much like how spherical coordinates naturally simplify the hydrogen atom by embedding physical insight into the coordinate system, normalizing-flow coordinates offload complexity from the basis functions into the coordinate transformation itself. This shift not only improves basis-set convergence, but also enhances interpretability of vibrational motions. We provide an analysis of the utility, interpretation and associated constraints of normalizing-flow vibrational coordinates. Moreover, we demonstrate that these coordinates can be generalized across different isotopologues, and even structurally related molecules, achieved with minimal fine-tuning of selected output parameters.

physics.chem-ph↗

arXiv2025

Taylor-mode automatic differentiation for constructing molecular rovibrational Hamiltonian operators

We present an automated framework for constructing Taylor series expansions of rovibrational kinetic and potential energy operators for arbitrary molecules, internal coordinate systems, and molecular frame embedding conditions. Expressing operators in a sum-of-products form allows for computationally efficient evaluations of matrix elements in product basis sets. Our approach uses automatic differentiation tools from the Python machine learning ecosystem, particularly the JAX library, to efficiently and accurately generate high-order Taylor expansions of rovibrational operators. The implementation is available at https://github.com/robochimps/vibrojet.

physics.atm-clus↗

arXiv2025

Computing excited states of molecules using normalizing flows

Calculations of highly excited and delocalized molecular vibrational states are computationally challenging tasks, which strongly depends on the choice of coordinates for describing vibrational motions. We introduce a new method that leverages normalizing flows -- parametrized invertible functions -- to learn optimal vibrational coordinates that satisfy the variational principle. This approach produces coordinates tailored to the vibrational problem at hand, significantly increasing the accuracy and enhancing basis-set convergence of the calculated energy spectrum. The efficiency of the method is demonstrated in calculations of the 100 lowest excited vibrational states of H$_2$S, H$_2$CO, and HCN/HNC. The method effectively captures the essential vibrational behavior of molecules by enhancing the separability of the Hamiltonian and hence allows for an effective assignment of approximate quantum numbers. We demonstrate that the optimized coordinates are transferable across different levels of basis-set truncation, enabling a cost-efficient protocol for computing vibrational spectra of high-dimensional systems.

physics.chem-ph↗

arXiv2024

Learning phase-space flows using time-discrete implicit Runge-Kutta PINNs

We present a computational framework for obtaining multidimensional phase-space solutions of systems of non-linear coupled differential equations, using high-order implicit Runge-Kutta Physics- Informed Neural Networks (IRK-PINNs) schemes. Building upon foundational work originally solving differential equations for fields depending on coordinates [J. Comput. Phys. 378, 686 (2019)], we adapt the scheme to a context where the coordinates are treated as functions. This modification enables us to efficiently solve equations of motion for a particle in an external field. Our scheme is particularly useful for explicitly time-independent and periodic fields. We apply this approach to successfully solve the equations of motion for a mass particle placed in a central force field and a charged particle in a periodic electric field.

cs.LG↗