# Multilevel Graph Wavelet Compressed Sensing with Scale-Aware Neural Recovery

arXiv: https://arxiv.org/abs/2607.20857 · HTML: https://mlsanity.com/paper/2607.20857/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 40 references that are in the map.
- First posted on arXiv 23 Jul 2026.

## Builds on (most cited first, 20 of 20)

- [Attention Is All You Need](https://mlsanity.com/paper/1706.03762/) (2017)
- [Semi-Supervised Classification with Graph Convolutional Networks](https://mlsanity.com/paper/1609.02907/) (2016)
- [Inductive Representation Learning on Large Graphs](https://mlsanity.com/paper/1706.02216/) (2017)
- [DeepONet: Learning nonlinear operators for identifying differential equations based on the universal approximation theorem of operators](https://mlsanity.com/paper/1910.03193/) (2019)
- [Variational Graph Auto-Encoders](https://mlsanity.com/paper/1611.07308/) (2016)
- [Relational inductive biases, deep learning, and graph networks](https://mlsanity.com/paper/1806.01261/) (2018)
- [Learning to Simulate Complex Physics with Graph Networks](https://mlsanity.com/paper/2002.09405/) (2020)
- [Hierarchical Graph Representation Learning with Differentiable Pooling](https://mlsanity.com/paper/1806.08804/) (2018)
- [Graph U-Nets](https://mlsanity.com/paper/1905.05178/) (2019)
- [The Well: a Large-Scale Collection of Diverse Physics Simulations for Machine Learning](https://mlsanity.com/paper/2412.00568/) (2024)
- [Discrete Signal Processing on Graphs: Sampling Theory](https://mlsanity.com/paper/1503.05432/) (2015)
- [Graph Unrolling Networks: Interpretable Neural Networks for Graph Signal Denoising](https://mlsanity.com/paper/2006.01301/) (2020)
- [Physics vs Distributions: Pareto Optimal Flow Matching with Physics Constraints](https://mlsanity.com/paper/2506.08604/) (2025)
- [Sampling of graph signals with successive local aggregations](https://mlsanity.com/paper/1504.04687/) (2015)
- [Random sampling of bandlimited signals on graphs](https://mlsanity.com/paper/1511.05118/) (2015)
- [Maximum Entropy Weighted Independent Set Pooling for Graph Neural Networks](https://mlsanity.com/paper/2107.01410/) (2021)
- [Eigendecomposition-Free Sampling Set Selection for Graph Signals](https://mlsanity.com/paper/1809.01827/) (2018)
- [Fast Graph Sampling Set Selection Using Gershgorin Disc Alignment](https://mlsanity.com/paper/1907.06179/) (2019)
- [Graph sampling with determinantal processes](https://mlsanity.com/paper/1703.01594/) (2017)
- [Sampling and Recovery of Graph Signals based on Graph Neural Networks](https://mlsanity.com/paper/2011.01412/) (2020)

## Abstract (arXiv)

Scientific machine learning methods such as neural operators and physics-informed neural networks have advanced engineering applications and inverse problems, but their training typically requires large volumes of simulated data. This makes data preparation and model training expensive. We propose Graph Wavelet Compressed Sensing (GWCS), a learning-based framework for offline compression of graph signals by representing them as sparse, interpretable wavelet-domain representations using the spectral graph wavelet transform. The framework combines a nonparametric multilevel importance sampler, which retains high-energy wavelet coefficients within each scale for a given compression ratio, with a scale-aware graph neural network that reconstructs the signal from the sparse coefficients. We evaluate the proposed framework on synthetic approximately band-limited graph signals over random graphs and four PDE simulation datasets over meshes, which include Turbulent Radiative Layer, Viscoelastic Instability, Kolmogorov Flow, and Dynamic Stall. We compare against graph signal sampling methods and graph autoencoder baselines. Results demonstrate that the framework achieves high reconstruction fidelity and substantial data compression compared to existing benchmarks.