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

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

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

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

arXiv:2608.25160v1Aug 25, 2026math.NAcs.LGphysics.geo-ph

TL;DR

Waveform inversion must estimate heterogeneous wave speed despite cycle skipping in nonlinear oscillatory FWI. ROMNet uses a neural network to simplify the ROM-to-wave-speed mapping, and simulations show consistently better in-distribution results, useful out-of-distribution initial guesses, and lower subsequent inversion cost. Its estimates remain inaccurate for wave speeds far from the training distribution, limiting standalone use in such cases.

Problem
Cycle skipping impedes nonlinear least-squares FWI, while the ROM-to-wave-speed mapping remains less understood and requires time-consuming optimization.
Method
ROMNet trains a neural network to map an operator ROM to a nearby matrix with explicit, quadratic dependence on the wave speed.
Results
ROMNet performs consistently better in-distribution than the compared learning-based approaches, with some challenging out-of-distribution results slightly worse than expensive pure ROM inversion.
Takeaways & Limitations
ROMNet estimates can provide good initial guesses that reduce the iterations and computational cost of subsequent pure ROM-based inversion.
Takeaways & Limitations
ROMNet cannot accurately estimate wave speeds far from those used during training, although such estimates can still initialize pure ROM-based inversion.

Abstract

from arXiv · show

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".

1 Introduction

Waveform inversion estimates heterogeneous wave speeds from sensor measurements, but nonlinear oscillatory data fitting causes cycle skipping. ROM-based methods offer a data-driven alternative, while ROMNet uses machine learning to simplify ROM inversion and reduce its computational cost.

Problem setting: Waveform inversion estimates the wave speed of an inaccessible heterogeneous medium from wave measurements gathered at user-controlled sensors.The paper studies acoustic waves measured by an active source/receiver array in a constant-density medium.
Full waveform inversion: FWI formulates inversion as nonlinear least-squares optimization, but high-frequency band-limited data can produce cycle skipping and spurious local minima.Reliable local optimization requires sufficiently accurate prior kinematic information about the medium.
Full waveform inversion: The cycle-skipping problem remains largely open because complex applications often lack accurate kinematic information and no general theoretical guarantee establishes that one mitigation approach works.Proposed alternatives include changing the data-misfit metric or expanding the search space.
ROM-based inversion: ROM methods first map array response data to a physics-informed reduced order model and then estimate wave speed from that intermediary representation.The data-to-ROM mapping is nonlinear but understood and computable non-iteratively.
ROM-based inversion: ROMNet trains a neural network to transform an operator ROM into a nearby matrix with explicit, quadratic dependence on the wave speed.This targets the less-understood, optimization-based ROM-to-wave-speed step.
Evaluation: The study evaluates ROMNet on random Gaussian media and the GeoFWI dataset, comparing it with direct ROM inversion, Fourier-DeepONet, and InversionNet.The comparisons include in-distribution and out-of-distribution results.

2 Data-driven operator ROM and its dependence on the wave speed

The operator ROM is a Galerkin projection of the acoustic wave operator onto a snapshot-generated space, yet its matrices can be computed directly from array data. ROMNet addresses the wave-speed dependence introduced both explicitly by the operator and implicitly through the projection basis.

Operator ROM: The operator ROM is a Galerkin projection of the positive definite operator −c^2(x)∆ on a bounded domain.The domain can be interpreted as a causal truncation when its boundary cannot affect measurements during the observation interval.
Data-driven construction: Inner products are computed on a uniform time grid using functional calculus and a similarity transformation that avoids the c^-2(x) weighting.The time step is chosen according to the Nyquist sampling criterion for the probing pulse.
Snapshot space: The projection space is spanned by time snapshots of the wave field collected for all source excitations.Snapshots are sampled at uniform time instants, and their components define the projection space.
Data-driven construction: The Galerkin coefficients satisfy a reduced ordinary differential system whose unknown basis and operator are nevertheless recoverable through data-driven algebraic ROM construction.The data-driven construction uses array measurements and avoids requiring the unknown wave speed to compute the snapshots directly.
Data-driven construction: The snapshot Gramian is a symmetric positive definite mass matrix with Toeplitz+Hankel block structure, while its block Cholesky factor supplies the matrix R.The stiffness matrix is also data driven, with blocks determined from the measurements.
Dependence on wave speed: In the operator ROM, c(x) enters explicitly through the operator coefficient and implicitly through the c(x)-dependent orthonormal basis.ROMNet learns a map from the input ROM to another matrix that simplifies this combined dependence.

3 The ROMNet approach

ROMNet exploits the apparent weak dependence of the orthonormal basis on reflectivity to transform the ROM into a form with simpler wave-speed dependence. The resulting neural-network-assisted inversion aims to reduce optimization cost while retaining ROM-based advantages.

Motivation: The data-to-ROM mapping involves nonlinear operations including a matrix square root and inverse, making ROM-to-wave-speed estimation computationally demanding.The mass matrix depends linearly on the data, but its square root and inverse introduce nonlinear dependence.
Motivation: ROMNet is motivated by the observation that the orthonormal basis V changes more slowly with reflectivity than the snapshots U.This behavior is supported by one- and two-dimensional numerical illustrations, though higher-dimensional analysis remains unavailable.
ROMNet approach: ROMNet uses a neural network to map the ROM matrix to a nearby matrix with simpler, approximately quadratic dependence on the unknown wave speed.The transformed matrix is then fitted using a fixed reference basis, avoiding repeated wave-equation solves during optimization.
ROMNet algorithm: The algorithm starts from the data-driven ROM matrix A_ROM, applies the learned transformation, and estimates the wave speed through optimization using selected diagonals.The procedure is summarized as a two-stage inversion pipeline, with ROMNet and direct ROM branches sharing the initial data-driven matrix.
Resolution analysis: In a two-parameter landscape, FWI has 123 grid local minima, whereas ROMNet and the oracle each have a single global minimum with a better basin of attraction than ROM.The comparison uses objective functions based on selected diagonals of the ROM matrices and disregards errors introduced at the first algorithm stage.
Resolution analysis: The resolution analysis concludes that features at scales much smaller than the central wavelength cannot be determined, while variations at scales comparable to or larger than it remain feasible.The numerical parametrization uses smaller-support basis functions but regularizes the resulting over-parameterized optimization.

4 Numerical results

Numerical experiments evaluate ROMNet against direct ROM inversion, an oracle, Fourier-DeepONet, and InversionNet on random-Gaussian media and out-of-distribution probes. ROMNet performs strongly among learning-based methods while reducing optimization cost relative to direct ROM inversion.

Evaluation setup: The evaluation compares ROMNet with direct ROM inversion, an oracle, Fourier-DeepONet, and InversionNet.The oracle removes Stage 1 error and helps quantify the Stage 2 Gauss-Newton performance.
Computational cost: ROMNet optimization uses 10 Gauss-Newton iterations versus 40 for ROM inversion, with less than 1% error change between the tenth and fortieth iteration.ROMNet’s objective can be evaluated directly, whereas ROM inversion requires solving the wave equation at each current search speed.
Random-Gaussian dataset: Random-Gaussian training media use wave speeds formed by random superpositions of isotropic Gaussian functions around a constant reference speed.Amplitudes and standard deviations are randomized, with boundary tapering and clipping to a prescribed speed range.
In-distribution evaluation: ROMNet achieves the smallest error among the three learning-based methods on 11 held-out random-Gaussian test media.The reported metrics include mean relative L2 error, MAE, RMSE, and SSIM.
In-distribution evaluation: On representative held-out samples, ROMNet performs close to direct ROM inversion and the oracle, and better than the other learning-based methods.The recovered wave-speed panels report each estimate’s L2 relative error.
Out-of-distribution evaluation: On three piecewise-constant out-of-distribution probes, ROMNet has the lowest error among learning-based methods but remains less accurate than ROM inversion and the oracle.The disk, vertical-bar, and horizontal-bar probes are identified well despite the accuracy gap.
Windowing parameter: The ROMNet error saturates at γ = 3, while ROM inversion shows no significant advantage from using γ > 5.Most simulations use γ = 5; Table 3 averages ROMNet and oracle over 11 media but ROM over 3.
Parametrization basis: Using a less smooth piecewise-bilinear hat basis produces qualitatively similar reconstructions to the Gaussian basis.The comparison uses the same grid for centering the basis functions.

4.2 GeoFWI

GeoFWI evaluates ROMNet on geological velocity models containing layered structures, faults, folds, and salt bodies. ROMNet outperforms the other learning-based methods, especially for salt-body media, but can require ROM-based refinement outside its training distribution.

Dataset: GeoFWI generates velocity models from geological rules and includes folds, faults, stacked layers, and salt bodies absent from the random-Gaussian dataset.The evaluation uses a shallow-water subset whose near-surface speeds remain close to water.
Evaluation setup: The GeoFWI test evaluates mean relative L2 error on 45 samples, with 15 samples from each of nearly layered, faulted, and salt-body categories.Figures 8–10 show representative recovered velocities for the three categories.
Results: ROMNet has smaller error than all other learning-based methods across the GeoFWI evaluation.ROMNet and Fourier-DeepONet are comparable for nearly layered and faulted media, while InversionNet is clearly worse.
Results: For media with a salt body, ROMNet performs significantly better than the other learning-based methods.The salt-body examples are shown in Figure 10 alongside the competing reconstructions.
Out-of-distribution evaluation: On BP-2004 crops with heavy salt structures outside the GeoFWI training distribution, none of the learning-based methods reconstructs the wave speed well.The salt fraction is approximately 0.25–0.38, compared with approximately 0.06 for a representative GeoFWI salt model.
Out-of-distribution evaluation: ROMNet alone may be insufficient for accurate wave-speed estimates far from the training set, but it can provide a better starting value for ROM-based inversion.Starting from ROMNet outputs yields better estimates during Gauss-Newton refinement than the 40-iteration result initialized otherwise.

4.3 Computational cost and acceleration

ROMNet reduces the computational cost of ROM-based inversion by replacing repeated wave-equation solves with a neural-network-assisted matrix construction and inexpensive iterations. In the reported configuration, this yields a large end-to-end speedup while preserving a comparable error level after 40 iterations.

Cost controls: Time windowing and array sub-apertures can control ROM cost by reducing the temporal dimension n and sensor dimension m.The dominant ROM operation is the block-Cholesky factorization of the data matrix.
Source of the cost: ROM inversion repeatedly re-solves the wave equation for all m sources to build its matrix during each optimization iteration.Its Jacobian and objective-function evaluations account for the cost difference relative to ROMNet.
ROMNet computation: ROMNet evaluates the Stage 1 network once and assembles its matrix per iteration using numerical quadrature, with automatic differentiation providing the Jacobian.The regularized normal equations are solved by conjugate gradients using matrix-free Jacobian application.
End-to-end acceleration: 14 min versus 10.5 s for K = 40 iterations: ROMNet substantially reduces end-to-end inversion time.The one-time Stage 1 network evaluation adds only 6 ms.

4.4 Dealing with noise

Noise can disrupt the positive-definite algebraic structure required by ROMNet, so the method regularizes the data-derived matrices before forming the ROM. In the reported GeoFWI tests, regularized noisy-data estimates were practically as accurate as regularized noiseless-data estimates.

Noise issue: Noisy data can make the mass matrix nonsymmetric and non-positive-definite, threatening the algebraic structure required by ROMNet.Symmetry can be restored directly, but positive definiteness requires careful regularization.
Regularization: Adding a positive-definite block-diagonal term shifts the mass-matrix spectrum to positive eigenvalues, enabling block-Cholesky factorization and inversion.The procedure preserves the desired Toeplitz+Hankel block structure.
Evaluation setup: Table 5 reports wall time and L2 relative error averaged over three test samples, with both methods run for K = 40 iterations.Per-iteration times exclude one-time operator build and compilation.
Regularized pipeline: The regularized procedure computes the ROM from the regularized mass matrix and modifies the stiffness matrix accordingly.This changes Algorithm 1 at the ROM-construction step.
Numerical result: 1% additive noise: regularized ROMNet estimates for three test media had practically the same quality as estimates from noiseless data.The comparison is reported for the GeoFWI dataset and uses the regularization procedure.

5 Summary

ROMNet combines data-driven reduced order modeling with machine learning to estimate wave speed from active-array backscattering data. Numerical simulations show that it is the most accurate learning-based method in-distribution, while also providing useful initial guesses for expensive ROM inversion outside the training distribution.

Method: ROMNet maps a ROM matrix to another matrix with simpler, explicit quadratic dependence on the unknown wave speed.The resulting wave-speed estimation remains an optimization problem but is easier and orders of magnitude cheaper than direct ROM inversion.
Method: ROMNet estimates wave speed from backscattering data gathered by an active sensor array using a hybrid ROM and machine-learning approach.The ROM is an algebraic matrix surrogate of the wave equation operator.
Evaluation: ROMNet was evaluated against Fourier-DeepONet and InversionNet on random-Gaussian and GeoFWI training sets.The methods were trained on the same datasets for comparison.
Results: ROMNet was the most accurate learning-based method, with consistently better in-training-distribution results and some good challenging out-of-distribution results.Some out-of-distribution results were slightly worse than those of the more expensive pure ROM-based inversion.
Scope and consequence: For wave speeds far from training examples, ROMNet estimates are not accurate enough as final reconstructions but provide good initial guesses for pure ROM inversion.These guesses lead to convergence in fewer iterations and reduce computational cost.

A Details of the numerical implementation

The numerical implementation section introduces the details of the simulations used to evaluate ROMNet.

Numerical implementation: The paper provides detailed numerical-simulation settings for its ROMNet assessment.

A.1 Setup

The setup uses a finite-difference acoustic wave solver on a rectangular domain with homogeneous Dirichlet boundaries, an active linear sensor array, and a Gaussian search dictionary for wave-speed parametrization.

A.1 Setup: The Laplacian uses a five-point finite-difference stencil on a rectangular grid with homogeneous Dirichlet boundary conditions.The second time derivative is approximated on a fine grid with step τ/20.
A.1 Setup: The array response matrix is formed from numerically approximated pressure evaluated at sensor locations.Data matrices use convolution evaluated with the Fast Fourier transform.
A.1 Setup: The linear sensor array is placed horizontally near the top boundary, with uniformly spaced sensors across aperture a.The aperture and array depth are specified in the experimental tables.
A.1 Setup: The search dictionary parametrizes the medium with Gaussian basis functions whose centers lie on a uniform grid covering the domain.The number of functions and grid points are specified in Table 8.

A.2 Gauss-Newton iteration

The ROM-based inversion uses regularized Gauss–Newton optimization, with matrix-free Jacobian operations and causal time-windowing that estimates the medium progressively deeper from the array.

A.2 Gauss-Newton iteration: The inversion solves a Tikhonov-regularized minimization problem with Gauss–Newton iterations.The regularization term is fixed during the iterations.
A.2 Gauss-Newton iteration: The regularization parameter is fixed at µ = 2 × 10^-3σ2_max, where σmax is estimated from the first-iteration Jacobian.The estimate uses eight power iterations.
A.2 Gauss-Newton iteration: The Jacobian and its transpose are applied through automatic-differentiation jvp and vjp operations inside a conjugate-gradient normal-equation solve.Each CG step requires one jvp and one vjp, equivalent to two wave-equation solves.
A.2 Gauss-Newton iteration: ROMNet assembles the learned matrix by numerical quadrature from a reference-medium quantity, so each iteration avoids solving the wave equation.Its Jacobian is obtained by automatic differentiation of the quadrature map and applied matrix-free with the same damping and line search.
A.2 Gauss-Newton iteration: Time-windowing enables layer-peeling estimation from the vicinity of the array toward greater depths.For random Gaussian simulations, windows grow from 2τ to 16τ in increments of 2τ, with three Gauss–Newton iterations per intermediate window.
A.2 Gauss-Newton iteration: The learned matrix and operator ROM are constructed causally, matching the progressive depth dependence of their block rows.This causal structure is used in the ROMNet Gauss–Newton procedure.

GeoFWI [40]

The GeoFWI experiments resample the public models onto a 123 × 180 grid and use a ten-sensor array, 6 Hz probing pulse, and 40 ROMNet snapshots.

GeoFWI [40]: GeoFWI models provided on 100 × 100 grids are resampled to a rectangular 123 × 180 grid spanning [0, 2.05] × [0, 3.0] km.
GeoFWI [40]: The experiment uses ten sensors at depth 0.15 km, with 0.161 km spacing and a 1.45 km aperture.The horizontal sensor positions span 0.30 to 1.75 km.
GeoFWI [40]: The probing pulse has center frequency 6 Hz and bandwidth 4 Hz, with minimum wave speed 1.5 km/s and minimum wavelength 0.15 km.
GeoFWI [40]: Time sampling is τ = 0.0435 s, and ROMNet uses n = 40 snapshots.The remaining inversion setup is specified in the accompanying experimental tables.

A.3 Neural network and training

The neural network maps normalized ROM-derived inputs to normalized targets using causal token processing and gated outputs, trained with scale-normalized error and regularized optimization.

A.3 Neural network and training: The network wθ maps normalized input x to normalized target y, while prediction composes it with input normalization and output de-normalization.
A.3 Neural network and training: ROM residual entries are organized into time-ordered tokens, with zero padding giving every block row the same length.Each token contains the corresponding block-row entries of the residual.
A.3 Neural network and training: Eight residual convolutional blocks enforce causal dependence, so output token j depends only on tokens i ≤ j.This matches the ROM’s causal dependence on the medium depth reached by the corresponding time.
A.3 Neural network and training: A shared small network maps each token representation to output blocks, which are multiplied by a gate computed from the de-normalized input.The gate lies in [0, 1).
A.3 Neural network and training: At the reference medium, the gate and network output vanish for matching input block rows, so prediction reduces to the training-target mean µY.All network linear maps have no bias.
A.3 Neural network and training: Training minimizes mean squared error over output entries and the training set, with target scaling that makes media contribute comparably.A quadratic output penalty of 10^-5 is added, and AdamW trains for 400 epochs with validation-based parameter selection.