ROMNet, a neural-network-assisted reduced-order method, performed better than Fourier-DeepONet and InversionNet in the reported held-out wave-speed reconstruction tests. It did not have the lowest errors overall: direct ROM inversion scored lower, and the oracle comparison scored lower still. The result makes ROMNet the strongest of the tested learning-based methods, while leaving the direct ROM approach ahead on the reported accuracy measures.
The main measure was mean relative L2 error, a way of summarizing the mismatch between a reconstructed wave-speed field and its reference; lower values indicate a closer reconstruction. The comparisons were made on held-out cases, with validation media used for parameter selection and the test set kept out of training.
The idea behind ROMNet
At the heart of ROMNet is a transformation of an input operator reduced-order model into a nearby matrix with an explicit quadratic dependence on wave speed. The described two-stage procedure evaluates the Stage 1 network once and requires no wave-equation solve.
Fourier-DeepONet and InversionNet were used as learning-based comparators, trained on the same velocity sets as ROMNet. The evaluation used validation media for parameter selection, while the held-out test set was not used in training and supplied the reported results.
The Random-Gaussians data were organized into training, validation and test portions, while the GeoFWI water-shallow subset was reported with its own three-way split. The performance figures therefore came from reserved evaluation cases rather than from the data used to fit the models.
The first benchmark
On 11 held-out Random-Gaussians test cases, ROMNet's mean relative L2 error was 0.0297, compared with 0.0332 for Fourier-DeepONet and 0.0419 for InversionNet. Direct ROM was at 0.0293, and the oracle at 0.0232. On this measure, ROMNet's advantage was over the two learning baselines; it did not extend to the other two comparisons.
The complementary error measures kept the same broad ordering among the learning-based models. Across the same 11 cases, ROMNet's mean absolute error was 0.0476 and its root-mean-square error was 0.0890. Fourier-DeepONet recorded 0.0517 and 0.0995, while InversionNet recorded 0.0676 and 0.1258. Both measures were lower for ROMNet than for either learning comparator.
Structural similarity added a small distinction between the top two learned systems. ROMNet's SSIM was 0.7443, just above Fourier-DeepONet's 0.7438 and above InversionNet's 0.5902. SSIM is a pattern-similarity score, and this measure also placed ROMNet first among the three learning-based methods.
A test outside the training pattern
The analysis then moved beyond the training pattern, using three piecewise-constant inclusion probes. ROMNet had the lowest error among the learning-based methods and identified the inclusions. Direct ROM inversion and the oracle were more accurate.
These probes reinforce a narrower reading of the headline result: ROMNet led within the learned group, but the direct ROM and oracle comparisons remained more accurate. The inclusion tests show recovery by ROMNet without putting it ahead of every method included in the comparison.
The same ranking on GeoFWI
The GeoFWI water-shallow evaluation tested 45 structural samples, with 15 samples in each category. ROMNet's overall mean L2 relative error was 0.0358. Fourier-DeepONet scored 0.0422 and InversionNet 0.0765, while direct ROM scored 0.0339 and the oracle 0.0202.
Again, ROMNet was the best learning-based method but not the best-scoring method overall. Its error was lower than both learning baselines and higher than direct ROM and the oracle. Read alongside the Random-Gaussians results, the GeoFWI figures point to a consistent ranking rather than a universal performance claim.
What the comparison shows
Taken together, the reported evidence supports a precise conclusion. ROMNet was the leading learning-based method on the Random-Gaussians metrics, the three inclusion probes and the GeoFWI structural test. Direct ROM remained more accurate than ROMNet in the reported relative-error comparisons, and the oracle remained lower still.
The study therefore reports a better result for ROMNet than the two learning-based comparators under the stated tests, not a result that displaces every other method. The held-out design makes the comparison a test on reserved cases, so the strongest supported claim is that ROMNet led the learned group in these reported benchmarks.
Paper data and sources
Original title: ROMNet: a hybrid reduced order modeling and machine learning approach to waveform inversion
Authors: Liliana Borcea, Alexander Mamonov, Kui Ren et al.
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
Original paper · Full text