# SPAMoE: Spectrum-Aware Hybrid Operator Framework for Full-Waveform Inversion

arXiv: https://arxiv.org/abs/2604.07421 · HTML: https://mlsanity.com/paper/2604.07421/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Placed in MoE theory and gating functions, one of 12 directions in Mixture-of-experts.
- Builds on 21 of its 30 references that are in the map.
- First posted on arXiv 8 Apr 2026.

## Placed in

- Mixture-of-experts / [MoE theory and gating functions](https://mlsanity.com/moe/moe-theory-and-gating-functions/)

## Builds on (most cited first, 21 of 21)

- [Outrageously Large Neural Networks: The Sparsely-Gated Mixture-of-Experts Layer](https://mlsanity.com/paper/1701.06538/) (2017)
- [Fourier Neural Operator for Parametric Partial Differential Equations](https://mlsanity.com/paper/2010.08895/) (2020)
- [DINOv3](https://mlsanity.com/paper/2508.10104/) (2025)
- [Fourier Neural Operator with Learned Deformations for PDEs on General Geometries](https://mlsanity.com/paper/2207.05209/) (2022)
- [U-FNO -- An enhanced Fourier neural operator-based deep-learning model for multiphase flow](https://mlsanity.com/paper/2109.03697/) (2021)
- [U-NO: U-shaped Neural Operators](https://mlsanity.com/paper/2204.11127/) (2022)
- [InversionNet: A Real-Time and Accurate Full Waveform Inversion with CNNs and continuous CRFs](https://mlsanity.com/paper/1811.07875/) (2018)
- [Fourier-DeepONet: Fourier-enhanced deep operator networks for full waveform inversion with improved accuracy, generalizability, and robustness](https://mlsanity.com/paper/2305.17289/) (2023)
- [OpenFWI: Large-Scale Multi-Structural Benchmark Datasets for Seismic Full Waveform Inversion](https://mlsanity.com/paper/2111.02926/) (2021)
- [Augmented Physics-Informed Neural Networks (APINNs): A gating network-based soft domain decomposition methodology](https://mlsanity.com/paper/2211.08939/) (2022)
- [Unsupervised Learning of Full-Waveform Inversion: Connecting CNN and Partial Differential Equation in a Loop](https://mlsanity.com/paper/2110.07584/) (2021)
- [Variational Autoencoding Neural Operators](https://mlsanity.com/paper/2302.10351/) (2023)
- [Multiscale Neural Operator: Learning Fast and Grid-independent PDE Solvers](https://mlsanity.com/paper/2207.11417/) (2022)
- [Mesh-Informed Neural Operator : A Transformer Generative Approach](https://mlsanity.com/paper/2506.16656/) (2025)
- [Latent Mamba Operator for Partial Differential Equations](https://mlsanity.com/paper/2505.19105/) (2025)
- [Local neural operator for solving transient partial differential equations on varied domains](https://mlsanity.com/paper/2203.08145/) (2022)
- [Ensemble and Mixture-of-Experts DeepONets For Operator Learning](https://mlsanity.com/paper/2405.11907/) (2024)
- [FreqMoE: Dynamic Frequency Enhancement for Neural PDE Solvers](https://mlsanity.com/paper/2505.06858/) (2025)
- [Ambient Noise Full Waveform Inversion with Neural Operators](https://mlsanity.com/paper/2503.15013/) (2025)
- [Mixture of neural operator experts for learning boundary conditions and model selection](https://mlsanity.com/paper/2502.04562/) (2025)
- [Differentiable Autoencoding Neural Operator for Interpretable and Integrable Latent Space Modeling](https://mlsanity.com/paper/2510.00233/) (2025)

## Abstract (arXiv)

Full-waveform inversion (FWI) is pivotal for reconstructing high-resolution subsurface velocity models but remains computationally intensive and ill-posed. While deep learning approaches promise efficiency, existing Convolutional Neural Networks (CNNs) and single-paradigm Neural Operators (NOs) struggle with one fundamental issue: frequency entanglement of multi-scale geological features. To address this challenge, we propose Spectral-Preserving Adaptive MoE (SPAMoE), a novel spectrum-aware framework for solving inverse problems with complex multi-scale structures. Our approach introduces a Spectral-Preserving DINO Encoder that enforces a lower bound on the high-to-low frequency energy ratio of the encoded representation, mitigating high-frequency collapse and stabilizing subsequent frequency-domain modeling. Furthermore, we design a novel Spectral Decomposition and Routing mechanism that dynamically assigns frequency bands to a Mixture-of-Experts (MoE) ensemble comprising FNO, MNO, and LNO. On the ten OpenFWI sub-datasets, experiments show that SPAMoE reduces the average MAE by 44.4% relative to the best officially reported OpenFWI baseline, thereby establishing a new architectural framework for learning-based full-waveform inversion. Our code and data are available at https://github.com/zhenyuwang12366/SPAMoE