A neural-network surrogate showed large online speedups in two-dimensional numerical tests, while its accuracy depended on the contrast in the simulated permeability fields. In lower-contrast tests, NN-MsHDG stayed close to standard MsHDG, the method used for comparison. In high-contrast tests, it did not reliably match the standard global solution, even though local diagnostics were much smaller than the resulting global discrepancies.
How the shortcut works
NN-MsHDG maps a local coefficient field to coarse-block Dirichlet-to-Neumann, or DtN, matrices that summarize each block's boundary response, along with source vectors. Those predicted quantities enter the standard MsHDG global skeleton assembly, taking the place of local fine-scale HDG solves. Both methods used the same fine and coarse discretizations, stabilization set to 1 on every fine-grid face, and global skeleton coupling; only the local DtN matrices and source vectors were replaced.
For each contrast regime, the study generated 5,000 independent two-dimensional local coefficient realizations and divided them into an 80% training set and a 20% validation set. The physical tests used a 5 by 5 coarse-block array, with each coarse block divided into 32 by 32 fine squares. The two-dimensional surrogate had three convolutional layers and two fully connected hidden layers, used 3 by 3 kernels, preserved an 8 by 8 spatial resolution, and ended with a linear output layer after GELU activations.
Speed held in milder fields
At lower contrast, the mean relative error of standard MsHDG against the fine-scale reference fell from 9.782% at trace level 0 to 6.683% at level 1 and 3.525% at level 2. NN-MsHDG recorded 9.637%, 6.534% and 4.516% at the same three increasingly enriched trace spaces. The mean modeling error between the neural and standard solutions was 1.351%, 1.338% and 3.001%, respectively. In other words, the learned solution stayed close at the lower two levels, with a wider gap at the richest tested space. Sample standard deviations were reported alongside the means.
The timing advantage grew as the trace space was enriched. Assembly speedup rose from 8.397-fold at level 0 to 15.811-fold at level 1 and 30.530-fold at level 2. Total online speedup increased from 4.824-fold to 8.669-fold and then 16.341-fold. These timing values were means from repeated runs.
High contrast exposed the weak point
High contrast produced a different pattern. Standard MsHDG's mean relative errors were 94.864%, 84.478% and 43.394% across trace levels 0, 1 and 2. The corresponding NN-MsHDG errors were 96.005%, 95.584% and 94.442%. The mean modeling error between the two solutions grew from 27.945% to 69.909% and then 90.271%. Enriching the trace space improved the standard method's error against the fine-scale reference, but the learned solution remained far from the standard one in this regime.
The comparison also has a built-in limitation: the tested piecewise-linear coarse trace spaces remained inaccurate in the high-contrast setting. Even at the finest tested trace level, standard MsHDG had an error of approximately 43% against the fine-scale reference, limiting how cleanly the neural model's global error can be interpreted.
Local checks did not resolve the discrepancy. On held-out high-contrast reference-block data, action errors for smooth and random probes were 2.00% and 2.33% at level 0, 0.974% and 1.06% at level 1, and 0.796% and 0.899% at level 2. Those local errors were much smaller than the global modeling discrepancies, showing that small local action errors did not guarantee an accurate assembled solution. Computational savings nevertheless persisted: assembly speedup increased from approximately 8.4-fold to 30.7-fold, while total online speedup rose from approximately 4.8-fold to 16.8-fold.
What the timing means
A separate one-dimensional verification gave a cleaner accuracy check. Mean modeling error stayed below 10^-2 across all tested coefficient resolutions, coarse partitions and both contrast regimes. But the example was not a speedup demonstration: measured total online speedup was approximately 0.98 to 1.00, indicating no meaningful acceleration.
The strongest timing results therefore describe the online phase under the tested two-dimensional setup, not the full cost of preparing the surrogate. Timing used 15 repeated runs on the same workstation, with MATLAB parallel processing disabled, and excluded offline dataset generation and neural-network training from the reported wall-clock times.
The document is identified as arXiv:2608.25850v1, dated 26 Aug 2026. It describes implementations using MATLAB R2026a, Python 3.11.7 and PyTorch 2.2.2, but public data, code and supplement availability were not reported. No funding statement appears in the supplied text.
Paper data and sources
Original title: A Neural-network-based multiscale Hybridizable Discontinuous Galerkin method for solving PDEs in porous media
Authors: Tony Haines, Ke Shi
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
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