arXiv subjects

Usama Nadeem

Publications and source records attributed to Usama Nadeem.

4 recordsLinked to original sources

arXiv2024

Quasi-generalised KPZ equation

We derive the renormalised equation for the quasi-generalised KPZ equation with space-time white noise. We complement the program initiated by Gerencsér and Hairer for solving quasi-linear equations using regularity structures by an algebraic machinery that gives a systematic tool to remove non-local counterterms and provide a precise expression of the renormalised equation that is consistent with the semilinear case. The solution theory satisfies the chain rule and a natural notion of Itô isometry, which can be combined to obtain global in time solution.

math.PR↗

arXiv2023

Convergence of space-discretised gKPZ via Regularity Structures

In this work, we show a convergence result for the discrete formulation of the generalised KPZ equation $\partial_t u = (Δu) + g(u)(\nabla u)^2 + k(\nabla u) + h(u) + f(u)ξ_t(x)$, where the $ξ$ is a real-valued random field, $Δ$ is the discrete Laplacian, and $\nabla$ is a discrete gradient, without fixing the spatial dimension. Our convergence result is established within the discrete regularity structures introduced by Hairer and Erhard [arXiv:1705.02836]. We extend with new ideas the convergence result found in [arXiv:2103.13479] that deals with a discrete form of the Parabolic Anderson model driven by a (rescaled) symmetric simple exclusion process. This is the first time that a discrete generalised KPZ equation is treated and it is a major step toward a general convergence result that will cover a large family of discrete models.

math.PR↗

arXiv2023

Diagram-free approach for convergence of tree-based models in Regularity Structures

In this work, we translate at the level of decorated trees some of the crucial arguments which have been used in arXiv:2112.10739 for proposing a diagram-free approach for the convergence of the model in Regularity Structures. This allows us to broaden the perspective and enlarge the scope of singular SPDEs covered by this approach. It also sheds new light on algebraic structures introduced in the foundational paper of Martin Hairer on Regularity structures which was used later for recursively described renormalised models.

math.PR↗

arXiv2022

CodeDSI: Differentiable Code Search

Reimplementing solutions to previously solved software engineering problems is not only inefficient but also introduces inadequate and error-prone code. Many existing methods achieve impressive performance on this issue by using autoregressive text-generation models trained on code. However, these methods are not without their flaws. The generated code from these models can be buggy, lack documentation, and introduce vulnerabilities that may go unnoticed by developers. An alternative to code generation -- neural code search -- is a field of machine learning where a model takes natural language queries as input and, in turn, relevant code samples from a database are returned. Due to the nature of this pre-existing database, code samples can be documented, tested, licensed, and checked for vulnerabilities before being used by developers in production. In this work, we present CodeDSI, an end-to-end unified approach to code search. CodeDSI is trained to directly map natural language queries to their respective code samples, which can be retrieved later. In an effort to improve the performance of code search, we have investigated docid representation strategies, impact of tokenization on docid structure, and dataset sizes on overall code search performance. Our results demonstrate CodeDSI strong performance, exceeding conventional robust baselines by 2-6% across varying dataset sizes.

cs.SE↗