Three-dimensional inversion of gravity data using implicit neural representations and scientific machine learning

Anand Singh; Jochen Kamm; Pankaj K Mishra; Sanni Laaksonen

arxiv: 2510.17876 · v2 · submitted 2025-10-17 · ⚛️ physics.geo-ph · cs.LG

Three-dimensional inversion of gravity data using implicit neural representations and scientific machine learning

Pankaj K Mishra , Sanni Laaksonen , Jochen Kamm , Anand Singh This is my paper

Pith reviewed 2026-05-18 06:40 UTC · model grok-4.3

classification ⚛️ physics.geo-ph cs.LG

keywords gravity inversionimplicit neural representationsscientific machine learning3D density modelingmesh-free inversionsubsurface structure recoverygeophysical forward modeling

0 comments

The pith

Implicit neural representations let gravity inversion recover detailed density structures without regularization or depth weighting.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper introduces a method that represents the subsurface density distribution as the output of a neural network rather than a fixed grid of values. The network is trained by directly minimizing the difference between measured gravity data and the gravity field predicted by the forward physics operator applied to the network's continuous density output. Spatial encoding is added to the input coordinates so the network can represent sharp boundaries instead of oversmoothing them. Tests on synthetic models with smooth features and a dipping block show that geologically plausible structures emerge at multiple depths. The approach requires no explicit regularization term and no depth weighting, and the total number of trainable parameters does not grow with finer discretization.

Core claim

An implicit neural representation parametrizes the three-dimensional density field as a continuous function of spatial coordinates. A deep network is optimized solely by back-propagating the misfit between observed gravity anomalies and those computed from the forward gravity operator applied to the network output. This yields reconstructions of smooth and blocky synthetic structures with plausible boundaries, without any added regularization or depth-weighting terms, while the number of free parameters decreases relative to the size of the equivalent discretized problem.

What carries the argument

Implicit neural representation of density: a coordinate-based neural network that maps any spatial point to a density value and is trained end-to-end through the gravity forward-model loss.

If this is right

Synthetic smooth and dipping-block models are recovered with detailed structure and geologically plausible boundaries.
No explicit regularization or depth weighting is required to obtain stable solutions.
The number of inversion parameters decreases as the spatial extent or resolution of the problem increases.
The same continuous representation can be extended to other geophysical forward operators and to joint multiphysics inversions.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The continuous field could support locally adaptive resolution without a global increase in parameter count.
Real-field tests with noise and incomplete coverage would be needed to confirm whether solutions remain unique and interpretable.
The same INR parametrization might be reused across multiple data types to enforce consistency in multiphysics settings.

Load-bearing premise

A neural network trained only on the physics forward-model loss will converge to a unique, geologically meaningful density field rather than an arbitrary or non-unique fit to the data.

What would settle it

Applying the trained network to noisy real gravity measurements over a site with independent borehole or seismic constraints and finding large mismatches between the recovered density and those constraints would show the method does not reliably produce meaningful models.

read the original abstract

Inversion of gravity data is an important method for investigating subsurface density variations relevant to mineral exploration, geothermal assessment, carbon storage, natural hydrogen, groundwater resources, and tectonic evolution. Here we present a scientific machine-learning approach for three-dimensional gravity inversion that represents subsurface density as a continuous field using an implicit neural representation (INR). The method trains a deep neural network directly through a physics-based forward-model loss, mapping spatial coordinates to a continuous density field without predefined meshes or discretisation. Spatial encoding enhances the network's capacity to capture sharp contrasts and short-wavelength features that conventional coordinate-based networks tend to oversmooth due to spectral bias. We demonstrate the approach on synthetic examples including smooth models, representing realistic geological complexity, and a dipping block model to assess recovery of structures at different depths. The INR framework reconstructs detailed structure and geologically plausible boundaries without explicit regularisation or depth weighting, while reducing the number of inversion parameters as the problem size grows bigger. These results highlight the potential of implicit representations to enable scalable, flexible, and interpretable large-scale geophysical inversion. This framework could generalise to other geophysical methods and for joint/multiphysics inversion.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper puts implicit neural reps to work on 3D gravity inversion via a direct physics loss, which is a clean new framing but rests on thin synthetic evidence.

read the letter

The main point is that they represent density as a continuous field inside a neural network and train it end-to-end by minimizing the mismatch with the gravity forward operator. No mesh is needed, the number of parameters stays fixed as the domain grows, and spatial encoding is added to help capture sharper features that plain coordinate networks tend to blur. On the two synthetic cases shown—a smooth model and a dipping block—the outputs look plausible and recover boundaries without any hand-tuned regularization or depth weighting. That is a genuine difference from the usual discretized, regularized inversions in the literature they cite. The parameter-efficiency claim is straightforward and holds by construction. The soft spots are the missing pieces that matter for this problem. Gravity inversion is badly non-unique, and the paper offers only qualitative pictures from noise-free synthetics. There are no error norms, no recovery statistics, no architecture or training details, and no real-data example. It is not clear whether the network’s implicit bias will reliably pick a geologically sensible model once noise or model mismatch enters, or whether the same setup will still work when the true structure sits outside the network’s preferred function class. The abstract is honest about the scope, but the central claim about “geologically plausible boundaries” without extra constraints needs more than visuals to stand up. This is for geophysicists who already follow scientific machine learning and want to test mesh-free ideas on potential-field data. A reader who needs a working tool for mineral exploration or carbon storage today will not get it here, but someone looking for a starting point to combine INR with physics losses could. I would send it to peer review. The idea is distinct enough that referees can usefully check the implementation and press on the non-uniqueness question.

Referee Report

3 major / 2 minor

Summary. The manuscript presents a scientific machine learning method for three-dimensional gravity inversion that represents the subsurface density as a continuous field using an implicit neural representation (INR). A neural network is trained directly by minimizing a loss derived from the physics forward model, without meshes, discretization, explicit regularization, or depth weighting. The approach is demonstrated on synthetic examples including smooth models with realistic geological complexity and a dipping block model, with claims of recovering detailed structures and geologically plausible boundaries while reducing the number of parameters as problem size increases.

Significance. If the central claims hold under more rigorous testing, the work could advance scalable geophysical inversion by providing a mesh-free framework that leverages network architecture for implicit regularization and offers parameter efficiency for large problems. The physics-based loss and focus on boundary recovery in synthetics are strengths that align with trends in scientific machine learning. Credit is given for the parameter-reduction aspect and the attempt to handle sharp contrasts via spatial encoding.

major comments (3)

Abstract: The central claim that the INR framework 'reconstructs detailed structure and geologically plausible boundaries without explicit regularisation or depth weighting' is load-bearing for the paper's novelty but is supported only by qualitative descriptions of success on synthetic models; no quantitative metrics (e.g., RMS misfit, density error norms, or resolution tests) or comparisons to traditional regularized inversions are reported, leaving the handling of gravity's non-uniqueness unverified.
Results (synthetic examples): For the dipping block model, recovery of structures at different depths is asserted without depth weighting, yet the physics-only loss does not inherently resolve the null-space ambiguity that grows with depth; tests with added noise or multiple random initializations are needed to show the solution is not an arbitrary member of the equivalent model family.
Method: The abstract references spatial encoding to mitigate spectral bias and enable sharp features, but provides no details on the encoding scheme, network architecture (depth, width, activation), training procedure, or exact form of the embedded forward-model loss, all of which are required to evaluate reproducibility and the claimed parameter reduction.

minor comments (2)

Define acronyms such as INR on first use and ensure consistent notation for the density field throughout.
Clarify whether the forward operator is analytic or numerical and how it is differentiated through the network during training.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their constructive and detailed comments, which have identified key areas where the manuscript can be strengthened in terms of quantitative support, robustness testing, and methodological transparency. We address each major comment below and will incorporate revisions to improve the paper.

read point-by-point responses

Referee: Abstract: The central claim that the INR framework 'reconstructs detailed structure and geologically plausible boundaries without explicit regularisation or depth weighting' is load-bearing for the paper's novelty but is supported only by qualitative descriptions of success on synthetic models; no quantitative metrics (e.g., RMS misfit, density error norms, or resolution tests) or comparisons to traditional regularized inversions are reported, leaving the handling of gravity's non-uniqueness unverified.

Authors: We agree that quantitative metrics and comparisons would provide stronger support for the claims regarding handling of non-uniqueness. In the revised manuscript we will add RMS data misfit values, L2 density error norms against the true models, and a side-by-side comparison with a conventional regularized inversion (smoothness plus depth weighting) on the same synthetic datasets. These additions will quantify performance and clarify the role of implicit regularization from the network and encoding. revision: yes
Referee: Results (synthetic examples): For the dipping block model, recovery of structures at different depths is asserted without depth weighting, yet the physics-only loss does not inherently resolve the null-space ambiguity that grows with depth; tests with added noise or multiple random initializations are needed to show the solution is not an arbitrary member of the equivalent model family.

Authors: We acknowledge that the physics loss alone does not remove the depth-dependent null space inherent to gravity inversion. The spatial encoding and network architecture are meant to supply implicit regularization that favors geologically reasonable solutions. In the revision we will add experiments with 5% Gaussian noise and results from multiple random initializations for the dipping-block case, reporting consistency of recovered structures and misfit statistics to demonstrate that the solutions are not arbitrary members of the equivalence class. revision: yes
Referee: Method: The abstract references spatial encoding to mitigate spectral bias and enable sharp features, but provides no details on the encoding scheme, network architecture (depth, width, activation), training procedure, or exact form of the embedded forward-model loss, all of which are required to evaluate reproducibility and the claimed parameter reduction.

Authors: We appreciate the referee highlighting the need for complete methodological details. Although a methods section exists in the manuscript, we will expand it substantially in the revision to specify the exact spatial encoding (Fourier feature mapping with chosen frequency scales), network architecture (layer count, width, activation functions), training hyperparameters and procedure, and the precise mathematical form of the physics-informed loss that incorporates the gravity forward operator. We will also include a quantitative comparison of parameter counts versus discretized approaches to support the scalability claim. revision: yes

Circularity Check

0 steps flagged

No significant circularity in the INR-based gravity inversion derivation

full rationale

The paper's central procedure trains a coordinate-based neural network to represent a continuous density field by minimizing a physics forward-model loss, without any self-definitional loops, fitted parameters renamed as predictions, or load-bearing self-citations that reduce the result to its own inputs. The claimed advantages (detailed structure recovery without explicit regularization, parameter count reduction with problem size) follow directly from the INR architecture and empirical tests on synthetics; these do not collapse by construction to the training data or prior author results. The derivation remains self-contained against external benchmarks such as the embedded forward operator and standard neural network properties.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The method rests on standard geophysical forward modeling and neural network training assumptions rather than introducing new physical entities or ad-hoc postulates beyond the choice of INR representation.

free parameters (1)

Neural network architecture and training hyperparameters
Depth, width, learning rate, and encoding parameters are selected to enable convergence on the synthetic test cases.

axioms (1)

domain assumption The Newtonian gravity forward operator accurately maps a given density distribution to surface gravity measurements.
Invoked implicitly as the physics-based loss; this is a standard assumption in all gravity inversion work.

pith-pipeline@v0.9.0 · 5741 in / 1263 out tokens · 56048 ms · 2026-05-18T06:40:46.484477+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

The INR framework reconstructs detailed structure and geologically plausible boundaries without explicit regularisation or depth weighting, while reducing the number of inversion parameters as the problem size grows bigger.
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

Positional encoding enhances the network’s capacity to capture sharp contrasts and short-wavelength features that conventional coordinate-based networks tend to oversmooth due to spectral bias.

What do these tags mean?

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supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.