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

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.