arXiv subjects

Kamran Arora

Publications and source records attributed to Kamran Arora.

3 recordsLinked to original sources

arXiv2026

On reduced SPDE models for stochastic Cucker-Smale flocking dynamics

Flocking refers to a phenomenon in collective motion where particles align their velocities with each other, commonly modelled using Cucker-Smale dynamics. In this work, we introduce and analyse two general reduced SPDE models for stochastic Cucker-Smale particle dynamics that encompass environmental and i.i.d.-type noises. Our models are computationally cheaper alternatives to the underlying particle system and preserve the flocking behaviour. To derive our reduced models, we consider suitable empirical particle densities which depend on position and time, but feature velocity dependency in a multiplicative fashion: this setup crucially allows for a very intuitive and convenient notion of flocking at the SPDE level. In $d=1$ we prove flocking of the reduced SPDE models, and, for the environmental noise, we also quantify the discrepancy between the reduced SPDE and mollified particle system. We also discuss the technical modifications required to extend the results to higher dimensions. We then provide extensive numerical experiments to verify our theoretical results in $d=1$ and to explore the behaviour of our models in $d=2$.

math.AP↗

arXiv2026

A discontinuous Galerkin approximation of the Dean--Kawasaki equation

We introduce and analyse an arbitrary order spatial discontinuous Galerkin (dG) method for the Dean--Kawasaki equation, a highly singular SPDE modelling density fluctuations of $N$ diffusing particles in the regime of large particle number $N \gg 1$. Our starting point is a general procedure for discretising multiplicative, divergence-form noise on finite element spaces whilst preserving its cross-variation structure at the discrete level; the construction is explicit, elementwise, applies to continuous and discontinuous spaces alike, and extends to general mobilities. Using it, we prove weak error estimates of order $O(h^p)$ between fluctuations of the semi-discrete scheme and those of the underlying particle system, together with a correction that is exponentially small in the scaling regime $Nh^d \gg 1$ and arises because the scheme does not preserve positivity. The resulting method is locally and globally conservative and applies on unstructured simplicial meshes. Quantitative numerical experiments support the analysis, and further experiments illustrate the method beyond the scope of the theory: indicator function observables, external and interaction potentials, convection-dominated regimes and reflecting boundary conditions.

math.NA↗

arXiv2025

Reversible Deep Equilibrium Models

Deep Equilibrium Models (DEQs) are an interesting class of implicit model where the model output is implicitly defined as the fixed point of a learned function. These models have been shown to outperform explicit (fixed-depth) models in large-scale tasks by trading many deep layers for a single layer that is iterated many times. However, gradient calculation through DEQs is approximate. This often leads to unstable training dynamics and requires regularisation or many function evaluations to fix. Here, we introduce Reversible Deep Equilibrium Models (RevDEQs) that allow for exact gradient calculation, no regularisation and far fewer function evaluations than DEQs. We show that RevDEQs significantly improve performance on language modelling and image classification tasks against comparable implicit and explicit models.

cs.LG↗