# Implicit Full Waveform Inversion with Deep Neural Representation

arXiv: https://arxiv.org/abs/2209.03525 · HTML: https://mlsanity.com/paper/2209.03525/

## Key facts

- Cited by 26 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 76 references that are in the map.
- First posted on arXiv 8 Sep 2022.

## Builds on (most cited first, 28 of 28)

- [PyTorch: An Imperative Style, High-Performance Deep Learning Library](https://mlsanity.com/paper/1912.01703/) (2019)
- [Dropout as a Bayesian Approximation: Representing Model Uncertainty in Deep Learning](https://mlsanity.com/paper/1506.02142/) (2015)
- [Simple and Scalable Predictive Uncertainty Estimation using Deep Ensembles](https://mlsanity.com/paper/1612.01474/) (2016)
- [DeepSDF: Learning Continuous Signed Distance Functions for Shape Representation](https://mlsanity.com/paper/1901.05103/) (2019)
- [Implicit Neural Representations with Periodic Activation Functions](https://mlsanity.com/paper/2006.09661/) (2020)
- [Fourier Features Let Networks Learn High Frequency Functions in Low Dimensional Domains](https://mlsanity.com/paper/2006.10739/) (2020)
- [Occupancy Networks: Learning 3D Reconstruction in Function Space](https://mlsanity.com/paper/1812.03828/) (2018)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [Learning Implicit Fields for Generative Shape Modeling](https://mlsanity.com/paper/1812.02822/) (2018)
- [Scene Representation Networks: Continuous 3D-Structure-Aware Neural Scene Representations](https://mlsanity.com/paper/1906.01618/) (2019)
- [Hands-on Bayesian Neural Networks -- a Tutorial for Deep Learning Users](https://mlsanity.com/paper/2007.06823/) (2020)
- [The Loss Surfaces of Multilayer Networks](https://mlsanity.com/paper/1412.0233/) (2014)
- [Escaping From Saddle Points --- Online Stochastic Gradient for Tensor Decomposition](https://mlsanity.com/paper/1503.02101/) (2015)
- [Learning Shape Templates with Structured Implicit Functions](https://mlsanity.com/paper/1904.06447/) (2019)
- [A Convergence Analysis of Gradient Descent for Deep Linear Neural Networks](https://mlsanity.com/paper/1810.02281/) (2018)
- [Pix2Vox: Context-aware 3D Reconstruction from Single and Multi-view Images](https://mlsanity.com/paper/1901.11153/) (2019)
- [Frequency Bias in Neural Networks for Input of Non-Uniform Density](https://mlsanity.com/paper/2003.04560/) (2020)
- [Convergence Analysis of Two-layer Neural Networks with ReLU Activation](https://mlsanity.com/paper/1705.09886/) (2017)
- [Matrix Completion has No Spurious Local Minimum](https://mlsanity.com/paper/1605.07272/) (2016)
- [Global Optimality of Local Search for Low Rank Matrix Recovery](https://mlsanity.com/paper/1605.07221/) (2016)
- [SGD Learns Over-parameterized Networks that Provably Generalize on Linearly Separable Data](https://mlsanity.com/paper/1710.10174/) (2017)
- [Hypernetwork functional image representation](https://mlsanity.com/paper/1902.10404/) (2019)
- [Globally Optimal Gradient Descent for a ConvNet with Gaussian Inputs](https://mlsanity.com/paper/1702.07966/) (2017)
- [An Analytical Formula of Population Gradient for two-layered ReLU network and its Applications in Convergence and Critical Point Analysis](https://mlsanity.com/paper/1703.00560/) (2017)
- [Deep-learning inversion: a next generation seismic velocity-model building method](https://mlsanity.com/paper/1902.06267/) (2019)
- [A Machine Learning Benchmark for Facies Classification](https://mlsanity.com/paper/1901.07659/) (2019)
- [PINNup: Robust neural network wavefield solutions using frequency upscaling and neuron splitting](https://mlsanity.com/paper/2109.14536/) (2021)
- [Total-variation regularization strategies in full-waveform inversion](https://mlsanity.com/paper/1608.06159/) (2016)

## Built on by (most cited first, 26 of 26)

- [Controllable seismic velocity synthesis using generative diffusion models](https://mlsanity.com/paper/2402.06277/) (2024)
- [Microseismic source imaging using physics-informed neural networks with hard constraints](https://mlsanity.com/paper/2304.04315/) (2023)
- [Automatic Differentiation-based Full Waveform Inversion with Flexible Workflows](https://mlsanity.com/paper/2412.00486/) (2024)
- [Continuous Representation Methods, Theories, and Applications: An Overview and Perspectives](https://mlsanity.com/paper/2505.15222/) (2025)
- [TAEN: A Model-Constrained Tikhonov Autoencoder Network for Forward and Inverse Problems](https://mlsanity.com/paper/2412.07010/) (2024)
- [ODE-DPS: ODE-based Diffusion Posterior Sampling for Inverse Problems in Partial Differential Equation](https://mlsanity.com/paper/2404.13496/) (2024)
- [Towards Understanding the Benefits of Neural Network Parameterizations in Geophysical Inversions: A Study With Neural Fields](https://mlsanity.com/paper/2503.17503/) (2025)
- [Initial Model Incorporation for Deep Learning FWI: Pretraining or Denormalization?](https://mlsanity.com/paper/2506.05484/) (2025)
- [Bayesian full waveform inversion with sequential surrogate model refinement](https://mlsanity.com/paper/2505.03246/) (2025)
- [A Differentiable Framework for Full and Phaseless Data Inversion Using Neural Implicit Contrast-Source Representation](https://mlsanity.com/paper/2508.10555/) (2025)
- [Unveiling the Mechanism of Continuous Representation Full-Waveform Inversion: A Wave Based Neural Tangent Kernel Framework](https://mlsanity.com/paper/2603.22362/) (2026)
- [Overview of Physics-Informed Machine Learning Inversion of Geophysical Data](https://mlsanity.com/paper/2310.08109/) (2023)
- [IntraSeismic: a coordinate-based learning approach to seismic inversion](https://mlsanity.com/paper/2312.10568/) (2023)
- [Physics-guided Full Waveform Inversion using Encoder-Solver Convolutional Neural Networks](https://mlsanity.com/paper/2405.17696/) (2024)
- [Survey of Deep Learning and Physics-Based Approaches in Computational Wave Imaging](https://mlsanity.com/paper/2410.08329/) (2024)
- [A new practical and effective source-independent full-waveform inversion with a velocity-distribution supported deep image prior: Applications to two real datasets](https://mlsanity.com/paper/2503.00658/) (2025)
- [Full waveform inversion with CNN-based velocity representation extension](https://mlsanity.com/paper/2504.15826/) (2025)
- [Inspired by machine learning optimization: can gradient-based optimizers solve cycle skipping in full waveform inversion given sufficient iterations?](https://mlsanity.com/paper/2509.14919/) (2025)
- [Three-dimensional inversion of gravity data using implicit neural representations and scientific machine learning](https://mlsanity.com/paper/2510.17876/) (2025)
- [Self-Flow-Matching assisted Full Waveform Inversion](https://mlsanity.com/paper/2603.13425/) (2026)
- [Total Generalized Variation regularization closes the gap between neural-eld and classical methods in seismic travel-time tomography](https://mlsanity.com/paper/2605.09960/) (2026)
- [Deciphering Neural Reparameterized Full-Waveform Inversion with Neural Sensitivity Kernel and Wave Tangent Kernel](https://mlsanity.com/paper/2605.14370/) (2026)
- [Bayesian three-dimensional seismic travel-time tomography for active- and passive-source seismic data using physics-informed neural network](https://mlsanity.com/paper/2606.21789/) (2026)
- [Tensor Train Decomposition-based 3D Implicit Full Waveform Inversion with Multi-scale Structural Similarity](https://mlsanity.com/paper/2606.22867/) (2026)
- [An automatic-differentiation framework for time-lapse electrical resistivity tomography inversion of hydrologic dynamics](https://mlsanity.com/paper/2608.14661/) (2026)
- [Image Fidelity is Not Field Fidelity: Joint Thermodynamic Reconstruction and Error Localization in Neural Tomography](https://mlsanity.com/paper/2609.28868/) (2026)

## Abstract (arXiv)

Full waveform inversion (FWI) commonly stands for the state-of-the-art approach for imaging subsurface structures and physical parameters, however, its implementation usually faces great challenges, such as building a good initial model to escape from local minima, and evaluating the uncertainty of inversion results. In this paper, we propose the implicit full waveform inversion (IFWI) algorithm using continuously and implicitly defined deep neural representations. Compared to FWI, which is sensitive to the initial model, IFWI benefits from the increased degrees of freedom with deep learning optimization, thus allowing to start from a random initialization, which greatly reduces the risk of non-uniqueness and being trapped in local minima. Both theoretical and experimental analyses indicates that, given a random initial model, IFWI is able to converge to the global minimum and produce a high-resolution image of subsurface with fine structures. In addition, uncertainty analysis of IFWI can be easily performed by approximating Bayesian inference with various deep learning approaches, which is analyzed in this paper by adding dropout neurons. Furthermore, IFWI has a certain degree of robustness and strong generalization ability that are exemplified in the experiments of various 2D geological models. With proper setup, IFWI can also be well suited for multi-scale joint geophysical inversion.