arXiv subjects

Victory C. Obieke

Publications and source records attributed to Victory C. Obieke.

4 recordsLinked to original sources

arXiv2026

Geometry-Conditioned Fourier Neural Operators for Cubic Nonlinear Schrodinger Dynamics on Periodic Domains

We consider the cubic nonlinear Schrödinger (NLS) equation on two-dimensional flat tori with varying aspect ratios. In this formulation, the choice of aspect ratio governs the Fourier resonance structure, so rational and irrational geometries can exhibit different high-frequency cascade behaviors. We present a geometry-conditioned Fourier neural operator (FNO) for the cubic defocusing NLS equation, where the input consists of the real and imaginary parts of the solution together with the aspect-ratio parameter $ω^2$. The model is trained to approximate the one-step solution operator and is evaluated on unseen trajectories generated from random-phase initial data using Fourier pseudospectral method. Our numerical experiments show that the learned operator captures the main solution dynamics on both tori and reproduces the distinct Sobolev norm behavior of the two geometries, with stronger $H^2$-growth on the rational torus and more constrained behavior on the irrational torus, consistent with the findings of \cite{hrabski2021energy}. We perform ablation studies to examine the roles of retained Fourier modes, activation functions, Fourier-layer depth, and explicit geometry conditioning. The results indicate that including $ω^2$ improves long-time predictive accuracy, especially for the rational geometry, and supports the use of geometry-aware neural operators for learning spectral-transfer phenomena in nonlinear dispersive partial differential equations.

cs.LG↗

arXiv2026

Structure-Informed Neural Operators for Long-Time Prediction of Parametric Hamiltonian PDEs

Hamiltonian partial differential equations (PDEs) often exhibit long-time dynamics governed by conserved quantities such as mass, momentum, and Hamiltonian energy. Standard Fourier neural operators (FNOs) provide efficient data-driven approximations of solution operators, but may not preserve these invariants during autoregressive rollout, and can develop drift in conserved quantities, phase error, and loss of qualitative accuracy. We propose an energy-projection Fourier neural operator (EP-FNO), a structure-informed operator learning architecture that combines a residual FNO time-stepping update with an invariant projection for long-time prediction of parametric Hamiltonian PDEs. We also provide a theoretical analysis showing that EP-FNO can approximate operators associated with PDEs efficiently, we also suggest a stability estimate. We evaluate the approach on the Zakharov--Kuznetsov, Kadomtsev--Petviashvili, and sine--Gordon equations. Numerical experiments show that the projected model improves long-time stability, and gives more accurate propagation of soliton and coherent wave structures compared with a standard FNO baseline. Our results demonstrate that invariant projection improves the reliability of learned surrogates for long-time Hamiltonian PDE simulation.

math-ph↗

arXiv2026

Communication Strategy Selection for Multi-GPU 3D FDTD with Convolutional Perfectly Matched Boundary Layers

In this paper we describe a communication-strategy study for multi-GPU three-dimensional finite-difference time-domain computation with convolutional perfectly matched layer boundary conditions using CUDA. The metrics used to determine the most effective implementation include runtime, throughput in millions of output points per second, strong-scaling efficiency, CPML overhead, host-staged versus direct GPU-to-GPU exchange speedup, and enlarged-ghost speedup. On a single NVIDIA Quadro RTX 6000 GPU, the CPML implementation sustains 2,889--3,290 million output points per second with less than 1\% boundary-layer overhead, providing the single-GPU baseline for the multi-GPU study. The results show that direct GPU-to-GPU peer exchange is the dominant optimization with a 2.46--2.76$\times$ speedup over host-staged exchange, while enlarged ghost regions give only modest benefits because the reduced communication frequency is partly offset by redundant computation and additional memory traffic. On NVIDIA Quadro RTX 8000 GPUs, the implementation gives up to a 1.51$\times$ speedup on two GPUs for the tested strong-scaling cases, while four GPUs enable larger grids that approach or exceed single-GPU memory capacity.

cs.DC↗

arXiv2026

An Energy Stable Approach for Learning Derivative Operators from Noisy Data for Maxwells Equations

We develop a structure-preserving ADMM method, denoted SP-ADMM, for learning energy-stable spatial derivative stencils for Maxwell equations from noisy data. Starting from the source-free Maxwell system, we focus on a one-dimensional reduction whose energy conservation depends on the skew-adjointness of the spatial derivative operator. The learned derivative is represented by a compact periodic convolution stencil. Unlike standard constrained ADMM, which learns the full stencil and imposes skew-adjointness through equality constraints, SP-ADMM enforces skew-adjointness by construction through a reduced parameterization using only the independent positive-side stencil coefficients. Numerical experiments show that SP-ADMM is especially effective in hidden operator and noisy-data regimes. Across clean data, noisy derivative data, multiple initial conditions, different hidden skew-adjoint operators, training-set sizes, regularization parameters, constraint ablations, and long-time simulations, SP-ADMM achieves the smallest final-time electric-field error while preserving energy to roundoff accuracy. A layered-medium Maxwell propagation test further shows that the learned structure-preserving stencil remains competitive with classical finite differences in a physical reflection/transmission setting. Overall, SP-ADMM provides a data-driven way to learn accurate Maxwell stencils while retaining the energy-conserving structure of the underlying equations.

math.NA↗