A Hybrid FNO-Diffusion Framework for Uncertainty-Aware Source Energy Estimation in Atmospheric Waveguides

Estimating the source of acoustic waves propagating in a vertically stratified medium poses significant challenges due to the high variability of the acoustic fields at long-range distances caused by heterogeneous vertical sound speed profiles. This renders the problem an inverse and ill-posed one. To address this challenge, we propose a three-step approach utilizing a Bayesian framework. First, we show that using only the low-frequency components (up to 1.5 Hz) of the acoustic fields is sufficient to capture the source parameters. Second, we develop a fast surrogate forward model based on a Fourier Neural Operator (FNO) [1] to bypass the computational burden of traditional numerical solvers. Finally, we trained diffusion models to represent the complex prior [2] of the atmospheric profiles and to accurately estimate the posterior distribution [3] in the context of our inference problem. The models are trained on a database comprising over 20,000 simulations generated using a normal mode code [4]. Our results show that our FNO model achieves a relative least squares error of approximately 8%. The combined FNO and diffusion model framework [5] is demonstrated to yield more reliable energy estimates when compared to the utilization of the FNO framework alone.

[1] N. Perrone, F. Lehmann, H. Gabrielidis, S. Fresca, and F. Gatti, “Integrating Fourier Neural Operators with Diffusion Models to improve Spectral Representation of Synthetic Earthquake Ground Motion Response,” arXiv preprint arXiv:2504.00757, 2025. doi: 10.48550/arXiv.2504.00757

[2] F. Lehmann, F. Gatti, M. Bertin, and D. Clouteau, “3D elastic wave propagation with a Factorized Fourier Neural Operator (F-FNO),” Computer Methods in Applied Mechanics and Engineering, vol. 417, art. no. 116718, 2023. doi: 10.1016/j.cma.2023.116718

[3] F. Bergamin, C. Diaconu, A. Shysheya, P. Perdikaris, J. M. Hernández-Lobato, R. E. Turner, and E. Mathieu, “Guided Autoregressive Diffusion Models with Applications to PDE Simulation,” in ICLR 2024 Workshop on AI4DifferentialEquations In Science, 2024. 

[4] T. Karras, M. Aittala, S. Laine, and T. Aila, “Elucidating the design space of diffusion-based generative models,” in Proceedings of the 36th International Conference on Neural Information Processing Systems (NeurIPS), 2022, pp. 26565–26577. doi: 10.5555/3600270.3602196

[5] M. Bertin, C. Millet, and D. Bouche, “A low-order reduced model for the long range propagation of infrasound in the atmosphere,” The Journal of the Acoustical Society of America, vol. 136, no. 5, pp. 2693–2705, 2014. doi: 10.1121/1.4896776