Summary
The paper explores the extension of Fourier Neural Operators (FNOs) to irregular geometries and topology changes. To leverage the computational speed benefits of the fast Fourier transform (FFT) employed by FNOs, they enclose the physical domain with a period box. They then adapt the FNO kernel by multiplying the integrand with a smoothed characteristic function that encodes domain information. This encoding enables their method to generalize to new geometries while preserving the FFT computational efficiency. They outperform several FNO baselines on two benchmark datasets, and showcase the properties of their method on a real-world crack propagation dataset, which contains evolving topology.
Strengths
The paper is well written and clear. The different choices of architecture are well justified.
- It tackles an interesting and useful problem for real-world application of FNO.
- The idea of incorporating the domain information in the kernel is simple, but effective and grounded, and the smoothing of the domain characteristic function helps avoiding dicontinituty problems.
- Experimental results on three datasets are strong.
- The crack propagation dataset is well-considered and helps highlighting the advantages of DAFNO, especially its ability to adapt to evolving topologies.
Weaknesses
- The comparison with several baselines is interesting, but given the recent improvements seen over FNO and Geo-FNO, having F-FNO [1] as a baseline seems important. In addition to neural operators, other grid-independent methods that can be used on irregular domains have been developed, for instance [2] implemented a continuous model using Implicit Neural Representations (INRs). It would have been interesting to see a comparison or at least a discussion about the advantage of DAFNO in the light of these recent improvements in the community (for e.g. computational cost?).
- As explained in the limitations and clear from the methodology itself, the architecture struggles to handle non-uniformly meshed grids due to the inability to use FFT on such grids. This limitation significantly restricts the range of problems that can be efficiently tackled using this architecture. For example, if applied to real-world airfoil surrogate modeling problems like [3], it seems that DAFNO would not offer significant improvements over the original FNO.
- While the idea behind DAFNO is interesting and novel, the architectural improvement upon the regular FNO architecture seems limited. The authors address this issue in a ‘remark’ paragraph, but I am not totally convinced that it is a sufficient improvement, especially since the architecture does not appear to efficiently handle non-uniformly meshed grids.
- This is more a remark than a weakness. In Figure 3, the zero-shot super-resolution prediction from eDAFNO does not effectively highlights the resolution invariant properties, since there is no ground truth to compare it to. While some degree of "deblurring" is noticeable, it remains unclear if the network successfully captures higher frequency phenomena that may occur at higher resolutions. The appendix, specifically Figure 10, sheds light on this property for the crack propagation dataset. However, Figure 3 does not truly demonstrates this resolution invariant property.
[1]: Tran, Alasdair, et al. "Factorized Fourier Neural Operators." The Eleventh International Conference on Learning Representations. 2022.
[2]: Yin, Yuan, et al. "Continuous PDE Dynamics Forecasting with Implicit Neural Representations." The Eleventh International Conference on Learning Representations. 2023.
[3]: Bonnet, Florent, et al. "AirfRANS: High Fidelity Computational Fluid Dynamics Dataset for Approximating Reynolds-Averaged Navier–Stokes Solutions." Advances in Neural Information Processing Systems 35 (2022): 23463-23478.
Questions
1. I wonder why Geo-FNO is not in the baselines for the Crack propagation dataset. It would have been an interesting baseline, especially to see if it could adapt to evolving geometries better than a masked FNO. It is claimed that Geo-FNO cannot handle topology change, but once a continuous mapping is learned, it can be applied. I agree that it should not adapt well, but is there a specific reason for its absence in these experimental results?
2. Just to confirm my understanding, for the FNO baseline, the grid is interpolated as in Geo-FNO, and then a mask is given as an additional input?
3. I don’t quite understand the link from equation 5 to the equation before 6. Why is $\chi(x)$ in factor of $W^l h(x) + c^l$ ? Is it to ensure that DAFNO will not produce results for points outside of the domain?
4. Given the link of the smoothing function to a signed distance function (SDF) with the $\\tanh(\\beta \\text{dist}(x, \partial \Omega))$ formulation, is there a reason for not directly using a SDF instead of the smoothing function ? It may have given more (or a different) information to the network, as in different previous works (for e.g. [4]).
5. Out of curiosity, have you experimented with using other smoothing functions besides tanh (for e.g. a logistic function)?
[4]: Guo, Xiaoxiao, Wei Li, and Francesco Iorio. "Convolutional neural networks for steady flow approximation." Proceedings of the 22nd ACM SIGKDD international conference on knowledge discovery and data mining. 2016.
Rating
6: Weak Accept: Technically solid, moderate-to-high impact paper, with no major concerns with respect to evaluation, resources, reproducibility, ethical considerations.
Confidence
3: You are fairly confident in your assessment. It is possible that you did not understand some parts of the submission or that you are unfamiliar with some pieces of related work. Math/other details were not carefully checked.
Limitations
The main limitations are already presented in the article.