# Physics-guided Full Waveform Inversion using Encoder-Solver Convolutional Neural Networks

arXiv: https://arxiv.org/abs/2405.17696 · HTML: https://mlsanity.com/paper/2405.17696/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 21 of its 69 references that are in the map.
- First posted on arXiv 27 May 2024.

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

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Going Deeper with Convolutions](https://mlsanity.com/paper/1409.4842/) (2014)
- [Perceptual Losses for Real-Time Style Transfer and Super-Resolution](https://mlsanity.com/paper/1603.08155/) (2016)
- [Automatic differentiation in machine learning: a survey](https://mlsanity.com/paper/1502.05767/) (2015)
- [JFB: Jacobian-Free Backpropagation for Implicit Networks](https://mlsanity.com/paper/2103.12803/) (2021)
- [InversionNet: A Real-Time and Accurate Full Waveform Inversion with CNNs and continuous CRFs](https://mlsanity.com/paper/1811.07875/) (2018)
- [Deep-Learning Inversion of Seismic Data](https://mlsanity.com/paper/1901.07733/) (2019)
- [Unsupervised Learning of Full-Waveform Inversion: Connecting CNN and Partial Differential Equation in a Loop](https://mlsanity.com/paper/2110.07584/) (2021)
- [Implicit Full Waveform Inversion with Deep Neural Representation](https://mlsanity.com/paper/2209.03525/) (2022)
- [Multigrid-augmented deep learning preconditioners for the Helmholtz equation](https://mlsanity.com/paper/2203.11025/) (2022)
- [A Helmholtz equation solver using unsupervised learning: Application to transcranial ultrasound](https://mlsanity.com/paper/2010.15761/) (2020)
- [Multigrid-Augmented Deep Learning Preconditioners for the Helmholtz Equation using Compact Implicit Layers](https://mlsanity.com/paper/2306.17486/) (2023)
- [A penalty method for PDE-constrained optimization in inverse problems](https://mlsanity.com/paper/1504.02249/) (2015)
- [Deep Recurrent Architectures for Seismic Tomography](https://mlsanity.com/paper/1908.07824/) (2019)
- [Efficient Inversion of Multiple-Scattering Model for Optical Diffraction Tomography](https://mlsanity.com/paper/1708.00745/) (2017)
- [Encoder-Decoder Architecture for 3D Seismic Inversion](https://mlsanity.com/paper/2207.14789/) (2022)
- [Mesh-free Eulerian Physics-Informed Neural Networks](https://mlsanity.com/paper/2206.01545/) (2022)
- [Full waveform inversion guided by travel time tomography](https://mlsanity.com/paper/1607.00968/) (2016)
- [Improving full waveform inversion by wavefield reconstruction with the alternating direction method of multipliers](https://mlsanity.com/paper/1809.00891/) (2018)
- [Deep Compressed Learning for 3D Seismic Inversion](https://mlsanity.com/paper/2311.00107/) (2023)
- [Full waveform inversion using extended and simultaneous sources](https://mlsanity.com/paper/2102.06014/) (2021)

## Abstract (arXiv)

Full Waveform Inversion (FWI) is an inverse problem for estimating the wave velocity distribution in a given domain, based on observed data on the boundaries. The inversion is computationally demanding because we are required to solve multiple forward problems, either in time or frequency domains, to simulate data that are then iteratively fitted to the observed data. We consider FWI in the frequency domain, where the Helmholtz equation is used as a forward model, and its repeated solution is the main computational bottleneck of the inversion process. To ease this cost, we integrate a learning process of an encoder-solver preconditioner that is based on convolutional neural networks (CNNs). The encoder-solver is trained to effectively precondition the discretized Helmholtz operator given velocity medium parameters. Then, by re-training the CNN between the iterations of the optimization process, the encoder-solver is adapted to the iteratively evolving velocity medium as part of the inversion. Without retraining, the performance of the solver deteriorates as the medium changes. Using our light retraining procedures, we obtain the forward simulations effectively throughout the process. We demonstrate our approach to solving FWI problems using 2D geophysical models with high-frequency data.