# Towards Understanding the Benefits of Neural Network Parameterizations in Geophysical Inversions: A Study With Neural Fields

arXiv: https://arxiv.org/abs/2503.17503 · HTML: https://mlsanity.com/paper/2503.17503/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- Builds on 30 of its 66 references that are in the map.
- First posted on arXiv 21 Mar 2025.

## Builds on (most cited first, 30 of 30)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Scaling Laws for Neural Language Models](https://mlsanity.com/paper/2001.08361/) (2020)
- [Delving Deep into Rectifiers: Surpassing Human-Level Performance on ImageNet Classification](https://mlsanity.com/paper/1502.01852/) (2015)
- [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)
- [Plenoxels: Radiance Fields without Neural Networks](https://mlsanity.com/paper/2112.05131/) (2021)
- [On the Spectral Bias of Neural Networks](https://mlsanity.com/paper/1806.08734/) (2018)
- [NeRF in the Wild: Neural Radiance Fields for Unconstrained Photo Collections](https://mlsanity.com/paper/2008.02268/) (2020)
- [Deep Image Prior](https://mlsanity.com/paper/1711.10925/) (2017)
- [A Convergence Theory for Deep Learning via Over-Parameterization](https://mlsanity.com/paper/1811.03962/) (2018)
- [Neural Fields in Visual Computing and Beyond](https://mlsanity.com/paper/2111.11426/) (2021)
- [Bayesian Deep Learning and a Probabilistic Perspective of Generalization](https://mlsanity.com/paper/2002.08791/) (2020)
- [On the Optimization of Deep Networks: Implicit Acceleration by Overparameterization](https://mlsanity.com/paper/1802.06509/) (2018)
- [Generalization in diffusion models arises from geometry-adaptive harmonic representations](https://mlsanity.com/paper/2310.02557/) (2023)
- [NeRP: Implicit Neural Representation Learning with Prior Embedding for Sparsely Sampled Image Reconstruction](https://mlsanity.com/paper/2108.10991/) (2021)
- [On Translation Invariance in CNNs: Convolutional Layers can Exploit Absolute Spatial Location](https://mlsanity.com/paper/2003.07064/) (2020)
- [Implicit Neural Representation in Medical Imaging: A Comparative Survey](https://mlsanity.com/paper/2307.16142/) (2023)
- [On the Implicit Bias in Deep-Learning Algorithms](https://mlsanity.com/paper/2208.12591/) (2022)
- [Dynamic CT Reconstruction from Limited Views with Implicit Neural Representations and Parametric Motion Fields](https://mlsanity.com/paper/2104.11745/) (2021)
- [Recovery of Continuous 3D Refractive Index Maps from Discrete Intensity-Only Measurements using Neural Fields](https://mlsanity.com/paper/2112.00002/) (2021)
- [Implicit Bias in Leaky ReLU Networks Trained on High-Dimensional Data](https://mlsanity.com/paper/2210.07082/) (2022)
- [Rethinking Positional Encoding](https://mlsanity.com/paper/2107.02561/) (2021)
- [Implicit Full Waveform Inversion with Deep Neural Representation](https://mlsanity.com/paper/2209.03525/) (2022)
- [Inversion using a new low-dimensional representation of complex binary geological media based on a deep neural network](https://mlsanity.com/paper/1710.09196/) (2017)
- [Bayes in the age of intelligent machines](https://mlsanity.com/paper/2311.10206/) (2023)
- [Deep-Learning based Inverse Modeling Approaches: A Subsurface Flow Example](https://mlsanity.com/paper/2007.15580/) (2020)
- [Deep Learning Inversion of Electrical Resistivity Data](https://mlsanity.com/paper/1904.05265/) (2019)
- [Automatic Differentiation-based Full Waveform Inversion with Flexible Workflows](https://mlsanity.com/paper/2412.00486/) (2024)
- [A Test-Time Learning Approach to Reparameterize the Geophysical Inverse Problem with a Convolutional Neural Network](https://mlsanity.com/paper/2312.04752/) (2023)
- [Enhancing Dynamic CT Image Reconstruction with Neural Fields and Optical Flow](https://mlsanity.com/paper/2406.01299/) (2024)

## Built on by (most cited first, 2 of 2)

- [Three-dimensional inversion of gravity data using implicit neural representations and scientific machine learning](https://mlsanity.com/paper/2510.17876/) (2025)
- [An automatic-differentiation framework for time-lapse electrical resistivity tomography inversion of hydrologic dynamics](https://mlsanity.com/paper/2608.14661/) (2026)

## Abstract (arXiv)

In this work, we employ neural fields, which use neural networks to map a coordinate to the corresponding physical property value at that coordinate, in a test-time learning manner. For a test-time learning method, the weights are learned during the inversion, as compared to traditional approaches which require a network to be trained using a training dataset. Results for synthetic examples in seismic tomography and direct current resistivity inversions are shown first. We then perform a singular value decomposition analysis on the Jacobian of the weights of the neural network (SVD analysis) for both cases to explore the effects of neural networks on the recovered model. The results show that the test-time learning approach can eliminate unwanted artifacts in the recovered subsurface physical property model caused by the sensitivity of the survey and physics. Therefore, NFs-Inv improves the inversion results compared to the conventional inversion in some cases such as the recovery of the dip angle or the prediction of the boundaries of the main target. In the SVD analysis, we observe similar patterns in the left-singular vectors as were observed in some diffusion models, trained in a supervised manner, for generative tasks in computer vision. This observation provides evidence that there is an implicit bias, which is inherent in neural network structures, that is useful in supervised learning and test-time learning models. This implicit bias has the potential to be useful for recovering models in geophysical inversions.