OptiXDE: A fast optical-inspired solver for differential equations
Abstract
OptiXDE is a matrix-free spectral operator framework for differential equations on uniform grids and embedded domains. Inspired by angular-spectrum propagation in Fourier optics, it maps transform-diagonal spatial operators to analytical modal multipliers and composes them with physical-space operators for nonlinearities, geometry and boundary enforcement. A common transform--operator--inverse-transform backbone is demonstrated across transient diffusion, periodic and embedded-domain Poisson problems, the cubic nonlinear Schr"odinger equation, viscous Burgers dynamics, the two-dimensional Allen--Cahn equation and incompressible flows from the Taylor--Green vortex to embedded-cylinder vortex shedding. Transform-compatible linear problems are recovered near the floating-point limit, whereas errors on the singular L-shaped domain remain localized near the re-entrant corner and regularized interface. Nonlinear benchmarks recover second-order temporal convergence and the expected conservative or dissipative behavior, while incompressibility remains near round-off level during long-time vortex shedding. The matrix-free updates require πͺ(Nlog N) work and πͺ(N) memory. Device-resident transform workloads reach 94.9Γ GPU acceleration, and the complete embedded-cylinder solver achieves a 42.1Γ CPU--GPU speedup under matched numerical settings. These results establish OptiXDE as a deterministic and extensible operator-centric framework for structured and embedded-domain differential equations.
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BibTeX
@misc{yang2026optixde,
title = {{OptiXDE: A fast optical-inspired solver for differential equations}},
author = {Yang Yang and Mingjiao Yan and Zongliang Zhang},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.01009},
eprint = {2609.01009},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.01009},
}
RIS
TY - UNPB
TI - OptiXDE: A fast optical-inspired solver for differential equations
AU - Yang Yang
AU - Mingjiao Yan
AU - Zongliang Zhang
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.01009
UR - https://arxiv.org/abs/2609.01009
AB - OptiXDE is a matrix-free spectral operator framework for differential equations on uniform grids and embedded domains. Inspired by angular-spectrum propagation in Fourier optics, it maps transform-diagonal spatial operators to analytical modal multipliers and composes them with physical-space operators for nonlinearities, geometry and boundary enforcement. A common transform--operator--inverse-transform backbone is demonstrated across transient diffusion, periodic and embedded-domain Poisson problems, the cubic nonlinear Schr"odinger equation, viscous Burgers dynamics, the two-dimensional Allen--Cahn equation and incompressible flows from the Taylor--Green vortex to embedded-cylinder vortex shedding. Transform-compatible linear problems are recovered near the floating-point limit, whereas errors on the singular L-shaped domain remain localized near the re-entrant corner and regularized interface. Nonlinear benchmarks recover second-order temporal convergence and the expected conservative or dissipative behavior, while incompressibility remains near round-off level during long-time vortex shedding. The matrix-free updates require πͺ(Nlog N) work and πͺ(N) memory. Device-resident transform workloads reach 94.9Γ GPU acceleration, and the complete embedded-cylinder solver achieves a 42.1Γ CPU--GPU speedup under matched numerical settings. These results establish OptiXDE as a deterministic and extensible operator-centric framework for structured and embedded-domain differential equations.
ER -