The Neuro-Physical Inverter: A Modular Framework for Magnetotelluric Inversion Coupling Ensemble Conditioning with Residual Learning

Jae Deok Kim; Sai Ravela; Rob. L. Evans

cs.LGOct 2, 2026

The Neuro-Physical Inverter: A Modular Framework for Magnetotelluric Inversion Coupling Ensemble Conditioning with Residual Learning

Authors: Jae Deok Kim, Sai Ravela, Rob. L. Evans

Organizations: MIT-WHOI Joint Program in Oceanography/Applied Ocean Science and Engineering, Cambridge, MA 02139 USA · Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139 USA · Department of Geology and Geophysics, Woods Hole Oceanographic Institution, Woods Hole, MA 02543 USA

Abstract

We present the Neuro-Physical Inverter (NPI), a modular, uncertainty-aware framework for geophysical inversion that couples ensemble-based conditioning with constrained residual learning, demonstrated in the 1D magnetotelluric (MT) setting as a controlled testbed. The framework operates in two stages. An Ensemble-Conditional Gaussian Process (EnsCGP) conditions a prior ensemble of resistivity models on the observed response, producing a physically admissible reference ensemble. A residual-learning neural network then predicts targeted corrections to this reference, trained on synthetic data and fine-tuned per station for field application through a physics-coupled objective. Because an ensemble is conditioned, refined, and propagated through both stages, every estimate carries an associated ensemble spread. Synthetic experiments show that NPI systematically reduces ensemble-mean error without destabilizing the ensemble. Applied to broadband MT data from the Gabbs Valley geothermal region (Nevada, USA), NPI reduces the across-station mean misfit over the mid-period band while retaining comparable ensemble spread. The propagated ensemble yields a factor of uncertainty that serves as an operational measure of constraint within the assumed model class. Both stages are dimension-agnostic in formulation, and the design principles established here are intended to scale to higher-dimensional parameterizations.

Figures & tables

Fig. 1: NPI training and synthetic testing workflow. A geologic prior is projected into a low-rank GMRF, from which 1D resistivity profiles are sampled and forward-modeled to generate synthetic MT responses, split into training (red) and testing (blue) subsets. In training, noisy responses are conditioned through EnsCGP, and the ensemble-mean responses and the residuals between true and EnsCGP ensemble-mean log-resistivity profiles form the ResNet-1D inputs and targets. In testing, the trained ResNet-1D refines the EnsCGP ensemble-mean estimates for evaluation against the known truth.

Fig. 2: Adaptation of the NPI to real MT data. From an observed MT response, EnsCGP produces an ensemble of physically admissible resistivity models and model-consistent responses. The ensemble-mean response is then passed through the pre-trained ResNet-1D to predict a depth-dependent residual correction. During fine-tuning, the network is updated under a physics-coupled objective (Equation 8 ) enforcing minimum-change refinement around the EnsCGP estimate. At each optimization step, the network’s resistivity estimate is forward modeled and its misfit against the observed response drives the parameter update. After convergence, the fine-tuned network is applied to each ensemble member, yielding a refined ensemble.

Fig. 3: Inversion results for three representative synthetic test examples (a)–(c), compared across EnsCGP, NPI, and RTO-TKO (left to right within each row). For each method, the left panel shows the true resistivity model (black), ensemble mean (blue), and 95% interval (shaded), together with the end-to-end point estimate (orange dashed). Right panels show the corresponding forward-modeled apparent resistivity and phase against the noise-contaminated observations (black, with error bars).

Method	RMS	nRMS	mean $U$	90%	95%	Percentile	Spread/error
	Accuracy		Ensemble
EnsCGP	0.131	1.355	1.85	88.6%	94.1%	42	0.97
NPI	0.111	0.962	2.13	96.8%	98.6%	52	1.40
RTO-TKO	0.135	1.011	2.62	94.7%	97.1%	48	1.42
End-to-end	0.104	0.961	–	–	–	–	–

TABLE I: Performance on the Synthetic Test Set

Fig. 4: Gabbs Valley study area, Nevada, USA. MT stations (inverted black triangles) collected by the USGS over a satellite basemap with Quaternary fault traces (magenta) [ 79 ] . Stars mark modern geothermal systems: Don A. Campbell (red), Gabbs Valley-Southeast (cyan), Rawhide (orange). Colored circles are wells binned by measured or geothermometry-derived temperature (legend). Inset: study-area location (red box) in the western United States.

Fig. 5: Station-level comparison of the EnsCGP, NPI, and RTO-TKO ensembles for stations GV129 (a), GV137 (b), and GV161 (c). For each method, the left panel shows the ensemble mean (blue) and its 95% interval (shaded), and the right panels show the corresponding forward-modeled apparent resistivity and phase against the observed rotation-invariant data (black, with error bars).

Fig. 6: NPI ensemble-mean resistivity and spread across the Gabbs Valley survey. Panels (a), (c), (e): ensemble-mean estimates at $\sim$ 5, 15, and 25 km depth (colored circles) over depth slices of the regional 3D model [ 55 , 54 ] , shown for context only. Panels (b), (d), (f): the multiplicative factor of uncertainty $U$ (Equation 17 ), the width of the central 95% ensemble interval in $\log_{10}\rho$ , quantifying constraint within the assumed 1D model class. Larger $U$ indicates greater uncertainty. $U(z)=2$ , for example, means the 95% interval spans from half to twice the ensemble median resistivity at that depth; $U(z)=1$ corresponds to no spread.

Fig. 7: Across-station data misfit and forward-response ensemble spread for the EnsCGP (blue) and NPI (orange) stages over all 81 stations. Panels (a), (c): mean absolute percent error between observed and ensemble-mean apparent resistivity and phase. Panels (b), (d): mean relative 95% CI half-width (Equation 18 ). NPI reduces mid-band misfit in both apparent resistivity and phase relative to the EnsCGP baseline, while the relative ensemble spread remains comparable or modestly larger, showing the improved fit is not obtained by collapsing the ensemble.

Appendix figures & tables10 assets

Supplementary material from the paper’s appendix.

Appendix

Fig. 8: Architecture of the ResNet-1D. Two input channels (log apparent resistivity and phase) pass through a 1D convolutional stem, four residual blocks expanding and contracting the channel dimension (32 $\rightarrow$ 64 $\rightarrow$ 128 $\rightarrow$ 64 $\rightarrow$ 32), global average pooling, and a fully connected output module that produces a depth-dependent residual correction in log-resistivity space, added to the EnsCGP ensemble mean to give the refined NPI estimate. Insets show a residual block and the fully connected module.

Fig. 9: Synthetic dataset for training and evaluation. (a) Distribution of 1D resistivity models sampled from the reduced-rank GMRF prior; solid curve is the ensemble median, shaded region the 5–95% interval. (b–c) Corresponding clean forward-modeled apparent resistivity and phase responses, same median and percentile convention. The spread reflects geologic variability encoded in the prior.

Fig. 10: Prior sensitivity comparison at station GV09. (a) Regionally informed GMRF prior. (b) Homogeneous half-space prior with constant mean resistivity and depth-independent variance. In each row, the left pair of panels shows the prior resistivity ensemble (gray) with prior mean (red) and the corresponding forward-modeled apparent resistivity and phase against the observed rotation-invariant MT data (black dashed with error bars); the right pair shows the NPI-refined resistivity ensemble (gray) with NPI ensemble mean (red) and the corresponding forward responses against the observations. The NPI ensemble mean reproduces the observed responses under both priors, but ensemble spread behaves differently with depth in each case, discussed in the text.

Fig. 11: Regional 3D MT inversion cross-sections [ 54 ] compared with pseudo-2D sections from the NPI 1D inversions along profiles A-A’ (left) and B-B’, upper 5 km. Profiles A-A’ and B-B’ run approximately southwest–northeast and northwest–southeast across the Gabbs Valley survey area. (a, b) resistivity from the 3D inversion. (c, d) NPI pseudo-2D sections, stitched from ensemble-mean 1D estimates at stations projected onto each profile. (e, f) factor of uncertainty

U(z)

from the NPI ensembles. Station locations are marked at the top of each panel. A shallow conductive layer consistent with a geothermal clay cap is indicated in (c, d).

Fig. 12: Fine-tuning target sensitivity test. (a) Station GV06 and (b) station GV120, each at target nRMS = 1.50 (left, blue) and target nRMS = 1.00 (right, orange). In each, the left panel shows the resistivity ensemble mean and 95% interval, and the right panels compare forward-modeled apparent resistivity and phase against the observed data. Relaxing the target to 1.50 suppresses internal resistivity structure within the conductive zone, acting similarly to stronger regularization, though through early stopping rather than a loss weight, while the large-scale conductive feature persists under either target.

Fig. 13: Phase tensor maps for the Gabbs Valley MT dataset at six representative periods. Ellipse shape and orientation depict the phase tensor at each station [ 87 ] , and color indicates the skew angle $\beta$ , a measure of departure from 1D or 2D structure, with larger $|\beta|$ indicating stronger three-dimensional effects. Background is satellite imagery of the study area.

Component	Parameter	Value
EnsCGP	Ensemble size $N$	500
	Localization	500 nearest neighbors, cosine distance in data space
	Observation noise model	$\sigma(\ln\rho_{a})=0.10$ , $\sigma(\phi)=2.86^{\circ}$ (5% relative impedance error, Eq. 14)
ResNet-1D pretraining	Architecture	2-channel input, conv stem (32), four residual blocks (32 $\to$ 64 $\to$ 128 $\to$ 64 $\to$ 32), global average pool, FC(32 $\to$ 64 $\to$ 89)
	Optimizer	AdamW, learning rate $10^{-4}$ , weight decay $10^{-4}$
	Loss	Huber, $\delta=1.0$

Appendix

TABLE II: Method Parameters

Method	Time (s)	Ensemble	Per-member / per-draw cost	Execution
EnsCGP	1.03	500	0.00205 s per member	parallel, 4 worker processes
NPI	11.96	500	fine-tuning 11.01 s (independent of ensemble size); propagation and forward modeling 0.00190 s per member	fine-tuning single-threaded; propagation batched; per-member forward modeling serial
RTO-TKO	640.07	500 (+50 burn-in)	1.28 s per kept sample (1.16 s per draw including burn-in)	serial Markov chain; per-draw Jacobian assembly parallel, 16 threads
End-to-end †	5.19	n/a	n/a, single point estimate	single-threaded
† No ensemble; reported for completeness and not part of the EnsCGP, NPI, and RTO-TKO comparison in Section III-B3 .

Appendix

TABLE III: Computational Environment and Inference Time at Station GV129

Fig. 14: Representative examples from the capacity-matched ablation, compared across EnsCGP, NPI, and the end-to-end network with Monte Carlo dropout (left to right within each row). For each method, the left panel shows the true resistivity model (black), ensemble mean (colored), and 95% interval (shaded). Right panels show the corresponding forward-modeled apparent resistivity (top) and phase (bottom) against the observations (black, with error bars). (a) A representative in-distribution example. (b) A representative out-of-distribution example, drawn from the homogeneous half-space prior of Appendix C.

Setting	Method	RMS	nRMS	mean $U$	90%	95%	Percentile	Spread/error
		Accuracy		Ensemble
In distribution	EnsCGP	0.124	1.200	1.79	88.3%	93.8%	42	0.98
	NPI	0.114	1.058	1.83	93.1%	96.8%	53	1.10
	End-to-end (MC dropout)	0.191	1.798	1.58	59.2%	66.9%	47	0.52
Out of distribution	EnsCGP	0.196	1.608	1.68	72.7%	78.0%	44	0.55
	NPI	0.192	1.583	1.71	74.6%	79.9%	54	0.57

Appendix

TABLE IV: Capacity-Matched Ablation: EnsCGP, NPI, and End-to-End with Monte Carlo Dropout

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