Source-linked AI summary
Continually learning neural-operator surrogate for three-dimensional airborne electromagnetic Bayesian inversion
Jaehong Chung, Andrew Lockwood, Jef Caers
TL;DR
Three-dimensional Bayesian inversion of airborne electromagnetic data is limited by the enormous number of costly forward solves required. This paper develops a continually learning neural-operator surrogate with validity checking, reproduces the solver posterior, and inverts over two million soundings in seconds.
Problem
Three-dimensional Bayesian inversion requires many costly electromagnetic forward solves, limiting routine probabilistic inversion of large airborne surveys.
Method
A continually learning neural-operator surrogate learns the three-dimensional AEM forward operator across consecutive geological priors, using ensemble disagreement to route out-of-range soundings to the solver.
Results
The surrogate reproduces the solver posterior, with credible intervals covering the truth within 2.6 percentage points of nominal, while two million soundings cost seconds instead of about 26,300 years for the solver.
Takeaways & Limitations
Probabilistic inversion with calibrated uncertainty becomes available at airborne-survey scale, while geological priors can be tested and revised against entire surveys.
Takeaways & Limitations
The ensemble-disagreement validity score degrades as ensemble members become more accurate, with partial correlation falling from +0.485 to −0.014.
Abstract
from arXiv · showhide
Three-dimensional probabilistic inversion of time-domain airborne electromagnetic (AEM) data is limited by the cost of the forward solve. Even though one simulation takes only tens of seconds, a Bayesian inversion of a survey of millions of soundings requires of order $10^{10}$ forward evaluations. To address this, we develop a continually learning neural-operator surrogate of the three-dimensional AEM forward operator that replaces the solver inside the Bayesian inversion. We start from the point of view that regardless of what geological prior is specified, Maxwell's laws remain invariant. Secondly, we avoid the limitation of learning on a single prior by continual learning on consecutive priors, which means our surrogate becomes richer as it is applied in future case studies, either by the authors, or by the scientific community. We use a validity check built on ensemble disagreement to divert cases with measurements outside the training range to the solver. Driven by the surrogate, the identical Markov chain Monte Carlo sampler reproduces the full-solver posterior, and its credible intervals cover the truth within 2.6 percentage points. Applied to the 2013 Capricorn TEMPEST survey in Western Australia, the surrogate inverts over two million soundings in seconds, a computation infeasible for the solver. Testing the geological prior against the entire survey costs minutes. The framework delivers uncertainty-quantified conductivity imaging at survey scale, which we believe is essential to perform near real-time mineral-systems targeting with geophysics.
1 Introduction
AEM Bayesian inversion can quantify non-uniqueness but is computationally prohibitive at survey scale. The paper addresses this with a continually learning surrogate that spans geological priors, detects out-of-range soundings, and reproduces solver-based inversion.
- AEM data support conductivity imaging for mineral exploration, groundwater assessment, and structural mapping, but deterministic inversion returns only one model despite non-uniqueness.Multiple conductivity models can fit the same noisy data, leaving follow-up decisions exposed to uncertainty and cost risks.
- Three-dimensional Bayesian inversion is costly because each sounding requires thousands of forward solves, while surveys contain hundreds of thousands to millions of soundings.A three-dimensional electromagnetic solve takes tens of seconds, producing an impractical aggregate computational burden.
- Static forward surrogates are accurate only within the support of the single geostatistical prior used for training, creating risks when geology or survey coverage shifts.The limitation is especially severe for compact, strong conductors of economic interest.
- The proposed operator continually learns across consecutive priors by retaining earlier models and constraining updates through experience replay and function-space distillation.Maxwell’s equations remain invariant across priors, so the operator accumulates prior coverage rather than replacing one learned mapping with another.
- A validity check flags soundings outside the training support and routes them to the full solver, while the surrogate reproduces solver posteriors and enables full-survey inversion.These capabilities are demonstrated on real Western Australian data at a scale infeasible for the traditional solver.
2 AEM survey and geological priors
The study tests continual learning across a Western Australian AEM survey and four additional geological priors. These priors represent contrasting conductivity structures constructed from geological information rather than survey data.
- The experiment uses the 2013 Capricorn TEMPEST survey in Western Australia and sequentially assimilates four additional regional geological priors.The design tests whether one operator can work across different geological settings.
- The Capricorn survey contains 2,155,272 soundings on 191 flight lines, with fifteen time gates spanning 0.013 to 16.2 ms.The fixed-wing system has a nominal transmitter clearance of 120 m, with the receiver trailing 117 m behind and 41.5 m below.
- The Capricorn prior encodes a three-layer stratigraphy in which conductive lateritic regolith overlies siliciclastics and resistive fresh banded iron formation.This conductivity contrast is derived from the regional geology and determines the prior’s vertical structure.
- The comparison priors cover resistive crystalline terrain with compact conductors, layered glacial sediments, a coastal fresh-over-saline interface, and conductive cover over resistive bedrock.They are built from geological information alone, without using survey data from those regions.
3 Methodology
The methodology combines a physics-based three-dimensional forward model with a continually adaptable neural-operator surrogate and Bayesian sampler. It represents geological priors, predicts gate responses, and evaluates posterior conductivity fields while retaining solver-based validation.
- 3.5 Surrogate accelerated Bayesian inversion: The surrogate replaces the solver inside an identical Metropolis-based probability-perturbation sampler, whose likelihood evaluates predicted-data misfit.Better-fitting proposals are always accepted, while worse-fitting proposals are accepted probabilistically according to the tempered Metropolis rule.
- 3.1 Governing equations for forward simulation: AEM forward simulation models diffusive electromagnetic responses governed by Maxwell’s equations in the quasi-static limit.The conductivity field determines the electric-field diffusion, from which the magnetic field is sampled at receiver locations and gate times.
- 3.1 Governing equations for forward simulation: The survey response is represented as an alternating-sign superposition of step-off responses for successive square-wave polarity reversals.The sum converges within a few half-cycles and converts the step-off solution into the 100% duty-cycle response used for the data.
- 3.1 Governing equations for forward simulation: Each sounding is simulated on a local 32 × 32 × 24 voxel crop using a three-dimensional finite-volume OcTree solve.The local footprint is approximately one kilometer for this system and depth range.
- 3.2 Neural operator surrogate for forward simulation: The neural operator maps conductivity fields to airborne decay responses at query gate times and geometries within the training range.A three-dimensional Fourier Neural Operator encodes the conductivity field, while a query network produces responses from the encoded representation.
- 3.2 Neural operator surrogate for forward simulation: A permutation-invariant prior embedding conditions the surrogate on geological-prior statistics while preserving the shared governing physics.The embedding uses conductivity statistics including mean, standard deviation, correlation lengths, conductive fraction, and vertical contrast.
- 3.2 Neural operator surrogate for forward simulation: The deployed surrogate predicts log10 |d̂| directly, whereas a controlled checkpoint can add a one-dimensional layered-earth baseline and a learned three-dimensional correction.The baseline supplies positivity and late-time behavior; production and field checkpoints set it to zero.
4 Results
The surrogate remains accurate across geological priors, avoids catastrophic forgetting, reproduces solver-based posterior behavior, and enables survey-scale inversion with prior updating.
- Surrogate accuracy and validity: R2 = 0.992 on log10 |Bz| and a median gate error of 4.7% demonstrate close agreement with the forward solver on Capricorn test data.Across ten initializations, the median gate error is 5.0 ± 0.3%.
- Surrogate accuracy and validity: ε ∝n−0.46 as training size increases, with error still declining at 1,000 training soundings.The results indicate that training-data quantity, rather than network capacity, limits accuracy in this experiment.
- Surrogate accuracy and validity: 67.7% error on Seward contrasts with 5.4% on Wisconsin, 6.6% on Denmark, and 11.8% on Zeeland when trained only on Capricorn.The validity check flags 32.8% of Seward and 16.8% of Zeeland, versus 0.8% of Denmark and Wisconsin and 4.0% of Capricorn test data.
- Continual learning across priors: −0.6 percentage points of average forgetting under continual learning contrasts with 7.5 percentage points under fine-tuning.The continual surrogate improves each region by the end of the sequence, whereas fine-tuning errors rise after later updates.
- Continual learning across priors: 4.4%, 4.9%, 4.4% and 4.5% final errors across four arrival orders show order-robust continual learning, compared with 11.6%–25.9% for fine-tuning.Retraining all five regions simultaneously reaches 4.6%, versus 4.4% for the sequence.
- Equivalence of the surrogate and solver posteriors: The surrogate posterior reaches 96% of the solver uncertainty width, with 85% overlap and acceptance 0.30 versus 0.28.At the stated sampling budget, solver halves differ by 12.2 times the pooled-state split, while surrogate-versus-solver differs by 10.3 times.
- Equivalence of the surrogate and solver posteriors: Credible intervals cover the truth within 2.6 percentage points of nominal across 50%, 80%, 90% and 95% levels.Observed coverages are 50.6%, 79.3%, 88.7% and 92.4%, respectively.
- Survey-scale inversion and prior updating: 2,131,667 soundings are inverted in 24.9 s on one GPU, versus about 26,300 years for the forward solver.The validity check flags no sounding within the trained 90–180 m clearance band, so no solver fallback is required.
5 Discussion
The framework addresses distribution shift by continually learning the invariant electromagnetic forward operator, while diagnostics identify unsupported inputs and control prior updates. Its cost structure enables survey-scale probabilistic inversion and makes geological priors testable against whole surveys.
- 5 Discussion: Replay and function-space distillation preserve earlier knowledge as the surrogate learns consecutive geological priors governed by unchanged Maxwell physics.The method treats each new prior as a covariate shift rather than a conflicting task.
- 5 Discussion: Two million soundings, infeasible for the solver at any budget, cost seconds, while testing a prior against an entire survey costs minutes.The framework changes probabilistic inversion from solver-limited computation to a survey-scale workflow.
- 5 Discussion: The ensemble disagreement score detects departures from training support, but its correlation with error falls from +0.485 to −0.014 as ensemble accuracy improves.This degradation means the validity diagnostic becomes less informative as ensemble members become more accurate.
- 5 Discussion: Coverage gains diagnose a better prior only against a control widened by the same amount without re-centering.The widened control occupied the largest model space yet explained fewer lines than the correlated mixture and lost 19 previously explained lines.
6 Conclusion
The study develops a continually learning neural-operator surrogate with validity checking for three-dimensional AEM inversion. It learns successive priors without forgetting, reproduces solver posteriors with calibrated uncertainty, and makes prior revision and full-survey inversion computationally practical.
- 6 Conclusion: The surrogate learns the three-dimensional AEM forward map across sequential geostatistical priors, flags out-of-support soundings, and routes them to the full solver.These are the framework’s core operator-learning, validity-check, and fallback components.
- 6 Conclusion: Every prior ends the sequence more accurate than when first learned because new conductivity models expand the operator’s learned domain.The reported accuracy gain survives every tested arrival order.
- 6 Conclusion: The surrogate departs from the solver by less than two independent solver-driven chains, while credible intervals cover the truth within 2.6 percentage points of nominal.The comparison uses the identical sampler at matched settings.
- 6 Conclusion: A second prior built from one failing line raises the explained lines from 127 to 184 of 190 and falsifies none previously explained.The result indicates that the survey revises prior structure rather than merely requiring greater prior width.
- 6 Conclusion: 11.7 µs per sounding reduces inversion of the survey’s two million soundings to seconds versus about 26,300 years for the solver.The framework provides probabilistic inversion with calibrated uncertainty at airborne-survey scale.
Data and code availability
The Capricorn TEMPEST survey data are publicly available from Geoscience Australia, and the code is planned for public release on GitHub upon publication acceptance.
- Data and code availability: The Capricorn TEMPEST survey data are publicly available from Geoscience Australia at dataset record 81642.The release includes acquisition and processing information.
- Data and code availability: The code will be public on GitHub upon acceptance for publication.