Numerical Methods

3 items

RESEARCHarXiv CS.LG·4/20/2026

Python library supporting Discrete Variational Formulations and training solutions with Collocation-based Robust Variational Physics Informed Neural Networks (DVF-CRVPINN)

This paper explores solving Partial Differential Equations (PDEs) using discrete weak formulations and a discrete neural network representation. It proposes a Python environment and a DVF-CRVPINN approach for training solutions, applying discrete automatic differentiation for equations like 2D Stokes.

variational methods Numerical Methods Physics-Informed Neural Networks partial differential equations

RESEARCHarXiv CS.LG·29d ago

On the Role of Strain and Vorticity in Numerical Integration Error for Flow Matching

This paper analyzes the role of strain and vorticity in numerical integration error for flow matching, proving that strain controls exponential error amplification while vorticity contributes linearly. It further shows that optimal transport velocity fields are irrotational, leading to second-order Euler accuracy.

Numerical Methods machine learning AI Flow Matching

RESEARCHarXiv CS.LG·4/30/2026

A Randomized PDE Energy driven Iterative Framework for Efficient and Stable PDE Solutions

This work introduces a PDE energy-driven iterative framework for solving partial differential equations efficiently and stably, without relying on traditional matrix-based discretizations or costly data-driven neural network training. It evolves random initial fields through physically constrained diffusion iterations and Gaussian smoothing, strictly enforcing boundary conditions, and demonstrates stable convergence on Poisson, Heat, and viscous Burgers equations.

Numerical Methods machine learning Scientific Computing Algorithms