arXiv subjects

Emmanuel E. Oguadimma

Publications and source records attributed to Emmanuel E. Oguadimma.

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

Operator-Split Bayesian Learning for Elliptic PDEs with Unequal Interior and Boundary Data

We propose an operator-split Bayesian learning framework for second-order uniformly elliptic Dirichlet problems with unequal numbers of interior and boundary observations. The data consist of noisy measurements of the source in the domain and noisy measurements of the boundary values. Independent Bayesian neural-network (BNN) priors are assigned to these two quantities, and the resulting product posterior is pushed forward through the elliptic solution operator. We prove that the posterior induced by this construction contracts around the true solution. The contraction radius separates a domain contribution, governed by the second-order elliptic operator, from a boundary contribution, governed by the intrinsic dimension of the boundary. Together with the minimax lower bound of \cite{ZhaoLu2026}, this yields a near-minimax upper bound up to logarithmic factors. Our numerical experiments illustrate the propagation of source and boundary uncertainty and the effects of unequal sampling budgets on the posterior reconstruction.

math.NA↗

arXiv2026

Analysis of Nonlinear Random Polarization in Dispersive Dielectrics

We present a study on the time-domain propagation of electromagnetic waves in dielectric materials modeled by a nonlinear Debye medium with random perturbations. Polynomial Chaos Expansions are employed to transform the random nonlinear Debye polarization model into a deterministic framework. We extend the Yee discretization to the resulting coupled system, establish second order accuracy, and verify convergence numerically. We investigate the sensitivity of nonlinear properties to uncertainty, particularly when the amplitude of the input signal is large. Given the challenges in manufacturing where uncertainties can cause optimal parameters to vary and potentially disrupt nonlinear effects, our approach incorporates these uncertainties within the simulation. This can enable the model-based design identification of realizable materials that maintain their desired effects despite variations. The findings from this study contribute to a deeper understanding of wave propagation in complex media, with potential implications for applications in optical communications, material science, and electromagnetic wave control.

physics.optics↗