Solving the Cosmological Vlasov-Poisson Equations with Physics-Informed Kolmogorov-Arnold Networks

TL;DR

This paper introduces a physics-informed neural network approach using Kolmogorov-Arnold networks to model cold dark matter evolution, achieving high accuracy without particle discretisation and offering potential for extension to higher dimensions.

Contribution

The paper presents a novel neural network method that directly models the continuous displacement field in cosmological simulations, bypassing traditional N-body particle discretisation.

Findings

Achieves sub-percent residual errors after multiple shell crossings

Matches N-body simulation results with resolution-free displacement fields

Displacement errors remain stable over time, unlike N-body methods

Abstract

Cold dark matter (CDM) evolves as a collisionless fluid under the Vlasov-Poisson equations, but N-body simulations approximate this evolution by discretising the distribution function into particles, introducing discreteness effects at small scales. We present a physics-informed neural network approach that evolves CDM fields without any use of N-body data or methods, using a Kolmogorov-Arnold network (KAN) to model the continuous displacement field for one-dimensional halo collapse. Physical constraints derived from the Vlasov-Poisson equations are embedded directly into the loss function, enabling accurate evolution beyond the first shell crossing. The trained model achieves sub-percent errors on the residuals even after seven shell crossings and matches N-body results while providing a resolution-free representation of the displacement field. In addition, displacement errors do not grow over time, a very interesting feature with respect to N-body methods. It can also reconstruct initial conditions through backward evolution when sufficient final-state information is available. Although current runtimes exceed those of N-body methods, this framework offers a new route to high-fidelity CDM evolution without particle discretisation, with prospects for extension to higher dimensions.