ROMNet: a hybrid reduced order modeling and machine learning approach to waveform inversion

Liliana Borcea; Alexander Mamonov; Kui Ren; Haizhao Yang; Chugang Yi

math.NAAug 25, 2026

ROMNet: a hybrid reduced order modeling and machine learning approach to waveform inversion

Authors: Liliana Borcea, Alexander Mamonov, Kui Ren, Haizhao Yang, Chugang Yi

Organizations: Applied Physics and Applied Mathematics, Columbia University, New York, NY, 10027. · Department of Mathematics, University of Houston, Houston, TX 77204-3008. · Department of Mathematics, University of Maryland, College Park, MD 20742.

Abstract

Waveform inversion seeks to estimate the wave speed of a heterogeneous, inaccessible medium, from time-resolved measurements of the waves at user controlled sensors. We consider this inverse problem for acoustic waves and an active array of source/receiver sensors that emit probing signals and measure the generated pressure waves. The forward map, from the wave speed to the measurements, is nonlinear and oscillatory. The oscillations cause cycle skipping, the main impediment to using the standard, nonlinear least-squares data fitting formulation, known as full waveform inversion (FWI). A recently introduced alternative waveform inversion approach computes from the measurements an algebraic surrogate of the wave operator, a reduced order model (ROM) matrix, which is then used to estimate the wave speed. The mapping from the measurements to the ROM is nonlinear, but well understood. It is computed efficiently, in a non-iterative manner. The nonlinear mapping from the ROM to the wave speed is less understood, and its approximation involves time-consuming optimization. Our goal in this paper is to use a neural network to map the ROM matrix to a nearby one, that has a simpler and explicit dependence on the wave speed. This simplifies and reduces the computational cost of the ROM-based waveform inversion. We introduce the methodology, called ROMNet, and test it with numerical simulations, using two training data sets: The first set consists of random media with variations of the wave speed modeled by a superposition of Gaussians with random amplitudes and standard deviations. The second is the publicly available GeoFWI dataset introduced for benchmarking FWI using deep learning. We compare the performance of ROMNet with the direct ROM-based inversion and with two representative deep learning approaches to FWI: Fourier-DeepONet" and InversionNet".

Figures & tables

Figure 1: One-dimensional simulations in a homogeneous and layered medium.The domain is $[0,2.5]$ km, with homogeneous Dirichlet boundary conditions, and the source is at $x=0.06$ km. The homogeneous medium has a wave speed $3$ km/s. The layered medium has the same wave speed, except in the interval $[0.35,2.1]$ km, which is divided into four equal layers, with wave speeds $3.5,2.5,3.5,2.5$ km/s, from top to bottom. The snapshots in the homogeneous medium (a) and the heterogeneous medium (b). Orthogonalized snapshots in the homogeneous medium (c) and the heterogeneous medium (d). The abscissa is the discrete time $t_{j}=j\tau$ , for $j=0,\ldots,n-1$ . The ordinate is the travel time $T$ . For each $t_{j}$ , we use colors to plot the amplitudes of $u_{j}(x(T))$ and $v_{j}(x(T))$ , respectively.

Figure 2: Two dimensional simulations. The reference medium is homogeneous, with wave speed $c_{o}({{\itbf x}})\equiv\bar{c}=3$ km/s. The true speed $c({{\itbf x}})$ differs from it only inside two thin horizontal bars outlined by the dashed rectangles, where the wave speed is $4$ km/s. The top two rows show the snapshots in the reference and heterogeneous media, respectively, at three time instants. The bottom two rows show the corresponding orthonormal basis vectors. The array is linear and lies at the top of the plotted domain, where we have a Dirichlet boundary condition. The subscript $o$ in the titles refers to $c_{o}$ , and the yellow triangles are the sensors in the array. The contour of the reflectors is outlined in all the pictures, including those corresponding to the reference medium, to aid in the interpretation.

Figure 3: ROMNet inversion (top branch) and the ROM inversion (bottom branch). Both start from the data-driven

\mathbfcal{A}^{{\scalebox{0.5}[0.4]{ROM}}}

. The network

\mathcal{G}_{\bm{\theta}}

is trained offline (dashed).

Figure 4: (a) True model $c({{\itbf x}})$ and the linear array located near the top boundary. The abscissa and ordinate are in km and the colorbar is in km/s. (b)–(c) Decimal logarithms of the FWI and ROM objective functions $\mathcal{O}^{{\scalebox{0.5}[0.4]{FWI}}}$ , $\mathcal{O}^{{\scalebox{0.5}[0.4]{ROM}}}$ . (d)–(e) plot the decimal logarithms of the objective functions $\mathcal{O}^{{\scalebox{0.5}[0.4]{ML}}}$ , for $\gamma=5$ : In (d) we use the learned approximation of $\mathcal{F}^{{\scalebox{0.5}[0.4]{ML}}}$ while in (e) we use the exact $\mathcal{F}^{{\scalebox{0.5}[0.4]{ML}}}$ . The axes in (b)–(e) are: the contrast of the wave speed in the abscissa and the interface position in the ordinate. The global minimum is indicated with ${\color[rgb]{1,0,1}\bigcirc}$ and $\times$ marks the minimum of each objective on the search grid. The true parameters are not included in the search.

method	relative $L^{2}$ error
ROMNet	0.0297
Fourier-DeepONet	0.0332
InversionNet	0.0419
ROM	0.0293
Oracle	0.0232

Table 1: Mean relative

L^{2}

error for all methods, over11 held-out test samples.

method	rel. $L^{2}$	MAE	RMSE	SSIM
ROMNet	0.0297	0.0476	0.0890	0.7443
Fourier-DeepONet	0.0332	0.0517	0.0995	0.7438
InversionNet	0.0419	0.0676	0.1258	0.5902

Table 2: Complementary error metrics over the 11 held-out test samples.

Figure 5: Recovered wave speed on two held-out Random-Gaussian test samples. From left to right, the columns show ground truth $c_{\rm true}$ , ROMNet, Fourier-DeepONet, InversionNet, Oracle, and the ROM inversion results. The $L^{2}$ relative error of each estimate is printed in its panel.

Figure 6: Recovered velocity on three out-of-distribution probes. From left to right: true $c({{\itbf x}})$ , ROMNet, Fourier-DeepONet, InversionNet, Oracle, and the ROM inversion results. The $L^{2}$ relative error of each estimate is printed in its panel.

	$\gamma=1$	$\gamma=2$	$\gamma=3$	$\gamma=4$	$\gamma=5$
ROMNet	0.0304	0.0299	0.0298	0.0297	0.0297
Oracle	0.0241	0.0233	0.0231	0.0231	0.0232
ROM	0.0301	0.0293	0.0292	0.0291	0.0289

Table 3: Effect of the windowing parameter

\gamma

on the mean

L^{2}

relative error of the reconstruction. The ROMNet and Oracle rows are averaged over the same

11

held-out test media as Table 1 , and the ROM row over

3

of them, which is why its value at

\gamma=5

differs slightly from the one in Table 1 .

Figure 7: From left to right: true $c({{\itbf x}})$ , the hat-basis inversion with ROMNet and the Oracle; the Gaussian-dictionary inversion with ROMNet and the Oracle.

category	sample size	ROMNet	Fourier- DeepONet	InversionNet	ROM	Oracle
fold	15	0.0264	0.0284	0.0528	0.0235	0.0180
fault	15	0.0382	0.0459	0.0799	0.0252	0.0186
salt	15	0.0428	0.0523	0.0967	0.0530	0.0241
overall	45	0.0358	0.0422	0.0765	0.0339	0.0202

Table 4: Mean

L^{2}

relative error on

15

random samples per structural category of the GeoFWI test set (

45

in total). Best learning-based method in each row is in bold.

Figure 8: Three velocity models with folded layering. From left to right: True $c({{\itbf x}})$ , ROMNet, Fourier-DeepONet, InversionNet, Oracle and ROM inversion.

Figure 9: Three velocity models with faulted layering. Columns as in Figure 8 .

Figure 10: Three velocity models containing a salt body. Columns as in Figure 8 .

Figure 11: GeoFWI: the initial wave speed of the ROM inversion (smoothed true $c({{\itbf x}})$ , middle column) and the reference $c_{o}({{\itbf x}})$ of ROMNet (right column), for a medium with a salt body (top) and a folded medium (bottom). The true wave speeds are in the left column. Both initial wave speeds depend only on depth; the ROM inversion uses a different one for each medium, whereas $c_{o}({{\itbf x}})$ is the same for all media.

Figure 12: Two crops of the BP-2004 model. From left to right: True $c({{\itbf x}})$ , ROMNet, Fourier-DeepONet, InversionNet, Oracle, ROM.

Figure 13: The evolution of the estimated wave speed with the ROM-based inversion, when starting with the initial guess given by ROMNet. From left to right: True $c({{\itbf x}})$ , the ROMNet estimate (the same as in Fig. 12 ) and the estimates after $5$ , $10$ , $15$ and $20$ Gauss-Newton iterations.

	per iter.	total ( $K\!=\!40$ )	error
ROM	$19.4$ s	$14$ min	$0.0295$
ROMNet	$\mathbf{0.26}$ s	$\mathbf{10.5}$ s	$0.0296$

Table 5: Wall time and

L^{2}

relative error for ROM and ROMNet. Per-iteration times exclude the one-time operator build and compilation, and the totals are measured end-to-end sums, averaged over the three test samples on an idle GPU. The errors are averaged over the same three media, which is why they differ slightly from those of Table 1 , and the ROM inversion and ROMNet are both run for

K=40

iterations. The two methods use the same configuration described in appendix A.2 .

Figure 14: Random Gaussians dataset: $L^{2}$ relative error of the reconstruction per Gauss–Newton iteration, for the same configuration as in Table 5 .

Figure 15: GeoFWI dataset: ROMNet estimates for 3 test media, from noiseless data without and with the regularization ( 4.3 )–( 4.4 ), and from data with $1\%$ additive noise, with the regularization.

Appendix figures & tables3 assets

Supplementary material from the paper’s appendix.

Appendix

Random Gaussians
Domain, grid	$\Omega=[0,2.0]\times[0,2.5]$ km, uniform grid of $100\times 125$ points
Probing pulse	$\omega_{o}/(2\pi)=6$ Hz, $B/(2\pi)=4$ Hz
$\bar{c}$ and $\lambda_{\rm min}$	$\bar{c}=3$ km/s, $\lambda_{\rm min}=2\pi\bar{c}/(\omega_{o}+B)=0.3$ km
Array	$m=10$ sensors at depth $0.14$ km, at horizontal positions $0.30,0.456,\ldots,1.70$ km (spacing $0.156$ km, aperture $a=1.4$ km)
Time sampling	$\tau=0.040$ s, $n=16$ snapshots
Reference speed	$c_{o}({{\itbf x}})\equiv\bar{c}=3$ km/s

Appendix

Table 6: Setup for the Random Gaussians data set.

GeoFWI [ 40 ]
Domain, grid	$\Omega=[0,2.05]\times[0,3.0]$ km, uniform grid of $123\times 180$ points. GeoFWI models, given on $100\times 100$ grids, are resampled to this grid.
Probing pulse	$\omega_{o}/(2\pi)=6$ Hz, $B/(2\pi)=4$ Hz
$\bar{c}$ and $\lambda_{\rm min}$	$\bar{c}=1.5$ km/s (the smallest wave speed in the data set, in water), $\lambda_{\rm min}=2\pi\bar{c}/(\omega_{o}+B)=0.15$ km
Array	$m=10$ sensors at depth $0.15$ km, at horizontal positions $0.30,0.461,\ldots,1.75$ km (spacing $0.161$ km, aperture $a=1.45$ km)
Time sampling	$\tau=0.0435$ s, $n=40$ snapshots for ROMNet; the ROM inversion uses $n=56$ snapshots, with the domain extended below the imaging region, so that the bottom boundary has no effect.
Reference $c_{o}({{\itbf x}})$	the per-depth median of the training wave speeds; it equals $1.69$ km/s at the depth of the array and $3.83$ km/s at the bottom

Appendix

Table 7: Setup for the GeoFWI data set.

	Random Gaussians	GeoFWI
Number of functions	$28\times 28+1=N_{c}=785$	$20\times 32+1=N_{c}=641$
Spacing of the centers	$0.072$ km $\times$ $0.090$ km	$0.105$ km $\times$ $0.095$ km
$\rho_{x},\rho_{y}$ ( $0.7\times$ spacing)	$0.050$ km, $0.063$ km	$0.074$ km, $0.067$ km
Windowing $\gamma$	$5$	$5$

Appendix

Table 8: Setup of the Stage 2 (Gauss–Newton) inversion for the two data sets.

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