In the reported configurations, NEMO evaluated a reduced parameter-to-observable operator in 0.012 seconds for the contaminant test and 0.006 seconds for the hypersonic-pressure test. The corresponding true reduced operators, computed with PDE solvers, took 541 seconds and 5.6 seconds, respectively, for reported speedups of more than 45,000-fold and more than 900-fold.
Those timings are hardware- and implementation-specific: the true operators were computed in parallel, while NEMO was evaluated on a single core.
The study asks whether NEMO can solve inverse problems in real time, using observations to infer model inputs. It learns the map from model parameters to the reduced parameter-to-observable operator itself, and its reduced-basis formulation provides a closed-form least-squares inverse.
The inverse can also be paired with a rapidly evaluated, closed-form posterior covariance, a reduced-subspace measure of uncertainty, for real-time uncertainty quantification.
Two numerical testbeds
For the contaminant application, 400 model-parameter samples formed the training and validation set. Separate held-out tests used 100 independent initial-condition samples and 100 independent velocity-field samples, and training was repeated in five independent experiments.
Representative contaminant reconstructions from NEMO were described as nearly identical to reconstructions from the true reduced operator, although the figure's colorbar range was truncated for visual clarity. The NEMO subspace solution approached the true solution as the learned operator approximation improved.
For the hypersonic application, the training data comprised 1,052 pressure- and temperature-coordinate samples, with 242 reserved for validation. Its test set combined 50 CFD-computed pressure fields with 100 model-parameter samples to produce 5,000 test samples.
Hypersonic pressure reconstructions were reported to have low relative errors, particularly at low observation noise. The benefits diminished as observation noise increased, and higher regularization was required.
The online calculation
An online inverse comparison put NEMO's runtime at 0.0021 seconds for the contaminant case and 0.0017 seconds for the hypersonic case. Across the tested regularization settings, reported speedups over forward MIONet ranged from more than 10-fold to more than 56-fold in the contaminant test and from more than 19-fold to more than 34-fold in the hypersonic test.
Those figures came from selected architectures, hardware and L-BFGS settings, so they describe the tested configurations rather than every neural-operator design.
With the same amount of training data, NEMO had the lowest reported relative errors among the three tested approaches—NEMO, forward MIONet and direct inverse MIONet—although most errors for all three were below 10%.
What the evidence leaves open
The paper's mathematical analysis reports consistency between NEMO learning and inverse-problem accuracy. It also claims network complexity independent of ambient discretization dimension for smooth infinite-dimensional maps, under the stated regularity and integrability assumptions. That is an analytical result rather than an empirical validation.
The evidence comes from numerical evaluation: PDE solves and neural-network training were used with test parameters not used in NEMO training, and no physical deployment is reported in the supplied evidence.
NEMO depends on a viable low-rank approximation, and network size and complexity may increase when the observation dimension or inversion subspace becomes large.
Paper data and sources
Original title: Real-time inverse solutions via neural matrix operators
Authors: Julie Pham, Thomas O'Leary-Roseberry, Omar Ghattas, Karen Willcox
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-25
DOI: Not available
Original paper · Full text