Continually learning neural-operator surrogate for three-dimensional airborne electromagnetic Bayesian inversion

Jaehong Chung; Andrew Lockwood; Jef Caers

physics.geo-phAug 26, 2026

Continually learning neural-operator surrogate for three-dimensional airborne electromagnetic Bayesian inversion

Authors: Jaehong Chung, Andrew Lockwood, Jef Caers

Organizations: Mineral-X, Department of Earth and Planetary Sciences, Stanford University, USA · Xcalibur Smart Mapping, Perth, Australia

Abstract

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.

Figures & tables

Figure 1: The Capricorn TEMPEST survey. (a) The 191 flight lines, colored by the gate time at which each sounding’s decay reaches the noise level. Line 1005801 is marked in black, and the inset locates the survey in Australia. (b) Decay section of line 1005801, $\log_{10}|B_{z}|$ (fT) over the fifteen gates along the 178 km line. Dashed vertical lines mark the three soundings of panel (c). (c) Decays of those three soundings, spanning slow (conductive), median, and fast (resistive) responses. The dashed gray curve is the noise model, and gates below it are shaded and drawn open.

Figure 2: Conceptual geology of the five regions, on one shared $\log_{10}\sigma$ scale. (a) Capricorn, (b) Seward Peninsula, (c) Denmark, (d) Zeeland, and (e) northeastern Wisconsin.

Figure 3: Two realizations of each geological prior, sampled from a Gaussian process. Each row is one prior, and each realization is shown as the three-dimensional volume and a cut-away along a horizontal and a vertical mid-plane.

Prior	Type	Variable	$U[a,b]$	$\ell_{h}$ (m)	$\ell_{v}$ (m)	$\alpha$	$\nu$	Source
Capricorn	Conductivity	$\sigma_{\mathrm{regolith}}$	$[-2.5,\,0]$	[300, 900]	[120, 300]	[1.0, 2.5]	[2, 5]	Thorne and Tyler (1996) Anand and Paine (2002)
		$\sigma_{\mathrm{sediment}}$	$[-3.0,\,-0.1]$
		$\sigma_{\mathrm{basement}}$	$[-4.0,\,-1.5]$
	Geometry	$h_{\mathrm{regolith}}$	$[0,\,120]$
		$d_{\mathrm{basement}}$	$[40,\,400]$
Seward	Conductivity	$\sigma_{\mathrm{host}}$	$[-3.3,\,-2.6]$	[300, 900]	[120, 300]	[1.0, 3.0]	[1.5, 4]	Emond et al. (2024)

Table 1: Gaussian process hyperparameters of the five geological priors. Each entry is a uniform distribution

U[a,b]

sampled per realization.

\sigma

is the unit conductivity in

\log_{10}

S/m,

h

is a layer thickness,

d

is a top or interface depth, and

w

is a body width, all in meters.

\ell_{h}

and

\ell_{v}

are the horizontal and vertical correlation lengths in meters,

\alpha

is the horizontal-to-vertical anisotropy factor, and

\nu

is the Matérn smoothness, where a larger

\nu

gives a smoother field. (The Zeeland source is a frequency-domain survey, and only its conductivity structure is used.)

Figure 4: The neural operator surrogate and the sampler it drives. (a) The model encoder, with the eight lowest modes boxed. (b) The prior embedding, the query network and the output. The layered-earth baseline is dashed because the deployed checkpoints set it to zero. (c) The sampler. The gray fill marks the same operator, trained in (a) and (b) and called in (c).

Figure 5: Forward accuracy on the Capricorn test data. (a) Surrogate against forward solver at the fifteen gates, with the 1:1 line. (b) The median sounding, surrogate dashed and forward solver solid. The shaded strip marks gates whose median decay falls below the noise model.

Figure 6: Test error against training set size on Capricorn. Markers and bars are the mean and standard deviation of ten runs at each size, scored on the same 125 test soundings. The line is a power law fitted to all fifty runs.

Figure 7: The validity check on the four regions the operator has not learned, trained on the Capricorn prior alone. (a) The disagreement score of Equation 17 against dissimilarity, the median distance from a test model to the nearest Capricorn training model in a standardized 64-number lateral profile. Markers are the median over soundings and bars run from the tenth to the ninetieth percentile. The dashed line is the threshold

u_{95}

, the 95th percentile of the score on the Capricorn validation data. (b) The fraction of each region’s test soundings above the threshold, which are inverted with the forward solver.

Figure 8: Continual learning with and without replay of the earlier priors, across the five geological priors and four arrival orders. Each row is one order, the order our surrogate was trained in first, and each column is a position in the sequence, with the region named in its panel. Arrows run from the stage that taught the region to the end of the sequence.

Figure 9: Sampler-level equivalence on the Capricorn prior. The identical PPM-MCMC sampler runs with the forward solver and with the surrogate, at the same budget and seeds, on one measured sounding of line 1005801. Bands are the 10–90% posterior range of the volume-mean resistivity, lines the medians, pooled over thirty-two chains per operator. The quoted 96% is the median over depth of the surrogate range divided by the solver range.

Figure 10: The falsification test over the whole survey, before and after one turn of the prior revision. Each flown line is colored by its coverage, the fraction of its soundings falling inside the 95% envelope of the prior’s forward responses, and a line is falsified below the 90% level marked on the bar. (a) The deployed Capricorn prior, with the line the updated prior was built from marked in amber. (b) The updated prior.

Figure 11: Whole-survey inversion of Capricorn TEMPEST. The top row is the posterior-median resistivity and the bottom row the width of the posterior, at 38 m (a, c) and 238 m (b, d). The width is the factor in resistivity between the 10th and 90th percentiles, so a value of 2 fixes the resistivity to within a factor of two. All 2,131,667 inverted soundings are shown, plotted one in four for legibility. Panel (a) marks line 1005801, which grounds the prior, and lines 1015101 and 1013902, which it does not explain.

prior	model space (decades)	lines explained (of 190)	lines lost
deployed Capricorn	7.90	127	–
width control	10.53	139	19
regional, independent layers	11.75	176	4
two-component mixture	8.65	184	0

Table 2: The four priors compared. Model space is the width of the prior in decades of

\log_{10}\sigma

summed over layers, for the mixture the weighted mean of its two components. Lines explained and lines lost are counted over the 190 unseen flight lines.

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Accelerating Bayesian inverse design in computational fluid dynamics using neural operators

Bipin Tiwari, Omer San

Department of Mechanical and Aerospace Engineering, University of Tennessee, Knoxville,2026 TN 37996, USA

Neural Surrogate Modeling Bayesian Inverse Problems Posterior Sampling Inverse Design Neural Operators Computational Fluid Dynamics

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Bayesian inverse design provides a principled framework for inferring aerodynamic geometries from sparse flow observations while quantifying uncertainty. However, its practical use in computational fluid dynamics (CFD) is severely limited by the cost of repeated high-fidelity simulations required for gradient-based Markov chain Monte Carlo (MCMC) sampling. While surrogate models are commonly proposed to reduce this cost, their effect on posterior geometry and uncertainty, especially for shock-dominated flows, remains poorly understood. In this work, we demonstrate that neural operator surrogates can be embedded directly within the MCMC inference loop while preserving posterior structure. Using a fully Bayesian inverse formulation of quasi-one-dimensional nozzle flow, we demonstrate that geometry parameterization plays a decisive role in identifiability and posterior conditioning, with cubic B-splines yielding stable and physically meaningful uncertainty estimates. Building on this formulation, a Deep Operator Network trained on CFD-generated data is substituted for the CFD solver within a No-U-Turn Sampler, while keeping the likelihood model, priors, and sampling configuration unchanged. Across sparse to fully observed regimes, surrogate-based inference reproduces the posterior geometry and uncertainty trends of the CFD reference. As a result of surrogate integration, total inference time is reduced to under one second, corresponding to a speedup exceeding three orders of magnitude. In addition, a direct inverse neural operator is examined as a deterministic alternative for inverse design, enabling single-shot geometry reconstruction without posterior sampling. These results demonstrate that neural operator-accelerated Bayesian inference enables practical, uncertainty-aware inverse design workflows for aerodynamic applications.

Abstract

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