Mitigating Boundary Artefacts in Fourier Neural Operators: An Investigation of Symmetric Boundary Conditions using the Sivashinsky Equation
(2025) MVKM01 20251Department of Energy Sciences
- Abstract
- Although Fourier Neural Operators (FNOs) are computationally efficient and theoretically mesh-independent, their standard formulation assumes periodic boundary conditions and rectangular domains. When applied to non-periodic conditions, FNOs often exhibit oscillations, edge effects, and reduced accuracy. To mitigate these issues, this study investigates modifications to the baseline FNO, including padding with smoothstep polynomial (FNOpad), cosine padding (FNOcosPad) and FNO utilising the cosine transform (FNOcos). The methodology involved generating symmetric data for various parameter combinations of (β, ρ) for the Sivashinsky equation and employing machine learning approach.
The results demonstrate that all FNO variants consistently... (More) - Although Fourier Neural Operators (FNOs) are computationally efficient and theoretically mesh-independent, their standard formulation assumes periodic boundary conditions and rectangular domains. When applied to non-periodic conditions, FNOs often exhibit oscillations, edge effects, and reduced accuracy. To mitigate these issues, this study investigates modifications to the baseline FNO, including padding with smoothstep polynomial (FNOpad), cosine padding (FNOcosPad) and FNO utilising the cosine transform (FNOcos). The methodology involved generating symmetric data for various parameter combinations of (β, ρ) for the Sivashinsky equation and employing machine learning approach.
The results demonstrate that all FNO variants consistently outperformed the baseline FNO for short-
term forecasting (up to approximately 600 time steps), significantly reducing edge artefacts and oscillations that are evident in the baseline model. FNOcos generally showed the best performance for short-term predictions, achieving the lowest mean L2 loss by effectively exploiting even function properties using the Discrete Cosine Transform type II (DCT-II). For intermediate- and long-term forecasting (1000 and 4000 time steps), all operators exhibited a tendency to diverge, particularly for
ρ = 1, which is the regime for cusp formation. Despite this, the padding-based variants, FNOpad and FNOcosPad, generally showed reduced average errors compared to the baseline FNO for lower β values. FNOcosPad emerged as the most reliable option for higher β, balancing stability and accuracy, while FNOcos showed mixed long-term results, being effective for lower (β, ρ) but unstable for chaotic dynamics and larger β. The choice of model effectiveness depends on the prediction regime, with
FNOcos best for short- and intermediate-term forecasting, and FNOcosPad offering the best trade-off for accuracy and long-term stability, especially with higher β. The instability in long-term predictions for FNOcos is attributed to the mirroring of data introducing new modes which are truncated, unlike FNOcosPad where extended padding is based on low cosine modes.
Future studies should extend these adaptations to more general boundary conditions, such as Robin boundaries, and focus on improving long-term stability. (Less) - Popular Abstract
- Machine learning methods are now being used to solve complex physics simulations quickly, but often struggles at domain boundaries, producing oscillations and other artefacts. This project improves how Fourier neural operators deal with edges, making them more reliable beyond their original periodic constraints.
Please use this url to cite or link to this publication:
http://lup.lub.lu.se/student-papers/record/9213930
- author
- Kadhem, Ali LU
- supervisor
- organization
- course
- MVKM01 20251
- year
- 2025
- type
- H2 - Master's Degree (Two Years)
- subject
- report number
- ISRN LUTMDN/TMPH-25/5664-SE
- ISSN
- 0282-1990
- language
- English
- id
- 9213930
- date added to LUP
- 2025-10-15 08:19:53
- date last changed
- 2025-10-15 08:19:53
@misc{9213930,
abstract = {{Although Fourier Neural Operators (FNOs) are computationally efficient and theoretically mesh-independent, their standard formulation assumes periodic boundary conditions and rectangular domains. When applied to non-periodic conditions, FNOs often exhibit oscillations, edge effects, and reduced accuracy. To mitigate these issues, this study investigates modifications to the baseline FNO, including padding with smoothstep polynomial (FNOpad), cosine padding (FNOcosPad) and FNO utilising the cosine transform (FNOcos). The methodology involved generating symmetric data for various parameter combinations of (β, ρ) for the Sivashinsky equation and employing machine learning approach.
The results demonstrate that all FNO variants consistently outperformed the baseline FNO for short-
term forecasting (up to approximately 600 time steps), significantly reducing edge artefacts and oscillations that are evident in the baseline model. FNOcos generally showed the best performance for short-term predictions, achieving the lowest mean L2 loss by effectively exploiting even function properties using the Discrete Cosine Transform type II (DCT-II). For intermediate- and long-term forecasting (1000 and 4000 time steps), all operators exhibited a tendency to diverge, particularly for
ρ = 1, which is the regime for cusp formation. Despite this, the padding-based variants, FNOpad and FNOcosPad, generally showed reduced average errors compared to the baseline FNO for lower β values. FNOcosPad emerged as the most reliable option for higher β, balancing stability and accuracy, while FNOcos showed mixed long-term results, being effective for lower (β, ρ) but unstable for chaotic dynamics and larger β. The choice of model effectiveness depends on the prediction regime, with
FNOcos best for short- and intermediate-term forecasting, and FNOcosPad offering the best trade-off for accuracy and long-term stability, especially with higher β. The instability in long-term predictions for FNOcos is attributed to the mirroring of data introducing new modes which are truncated, unlike FNOcosPad where extended padding is based on low cosine modes.
Future studies should extend these adaptations to more general boundary conditions, such as Robin boundaries, and focus on improving long-term stability.}},
author = {{Kadhem, Ali}},
issn = {{0282-1990}},
language = {{eng}},
note = {{Student Paper}},
title = {{Mitigating Boundary Artefacts in Fourier Neural Operators: An Investigation of Symmetric Boundary Conditions using the Sivashinsky Equation}},
year = {{2025}},
}