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arxiv: 2604.02335 · v1 · submitted 2026-01-19 · 💻 cs.LG · cs.NA· math.NA

Convolutional Surrogate for 3D Discrete Fracture-Matrix Tensor Upscaling

Martin \v{S}petl\'ik , Jan B\v{r}ezina This is my paper

Pith reviewed 2026-05-16 12:42 UTC · model grok-4.3

classification 💻 cs.LG cs.NAmath.NA
keywords surrogate modelingconvolutional neural networkdiscrete fracture matrixhydraulic conductivity tensornumerical homogenizationupscalinggroundwater flowvoxelized domains
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The pith

A 3D convolutional neural network surrogate predicts equivalent hydraulic conductivity tensors for fractured rock domains with normalized errors below 0.22 and over 100x speedup on GPU.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a surrogate model to replace expensive 3D discrete fracture-matrix simulations when upscaling conductivity tensors for groundwater flow modeling. It combines a convolutional network with feed-forward layers to process voxelized domains of matrix and fracture conductivities, where fractures are sampled from statistical distributions. Three separate models are trained for different fracture-to-matrix conductivity contrasts. The approach is tested on varied fracture parameters and matrix correlation lengths, then applied to macro-scale problems such as tensor computation and outflow prediction. If the surrogate maintains accuracy, it allows multilevel Monte Carlo workflows that would otherwise require prohibitive fine-scale computations.

Core claim

The central claim is that a 3D convolutional surrogate trained on discrete fracture-matrix simulation data can predict the equivalent conductivity tensor Keq from voxelized inputs with normalized root-mean-square errors below 0.22 in most cases, while delivering speedups exceeding 100x during GPU inference in two macro-scale demonstration problems that compare surrogate outputs against direct numerical homogenization.

What carries the argument

A 3D convolutional neural network followed by feed-forward layers that ingests voxelized random fields of matrix and fracture conductivities and outputs the six independent components of the equivalent conductivity tensor.

If this is right

  • Surrogate predictions can substitute for numerical homogenization inside multilevel Monte Carlo frameworks without substantial accuracy loss.
  • In macro-scale problems the surrogate reproduces both equivalent conductivity tensors and domain outflow values at far lower cost than direct simulation.
  • GPU-based inference reduces the wall-clock time of repeated upscaling operations by more than two orders of magnitude.
  • Separate models trained for each conductivity contrast allow the method to handle different geological regimes.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same voxel-to-tensor mapping could be retrained for other heterogeneous media such as porous rocks with inclusions or layered sediments.
  • If the statistical fracture distributions match field data only loosely, the surrogate error may rise sharply and require online fine-tuning or ensemble correction.
  • Embedding the surrogate inside an adaptive mesh-refinement loop might allow on-the-fly upscaling at every coarse-scale time step.

Load-bearing premise

Fracture networks generated from the chosen statistical distributions adequately represent real geological sites and the trained surrogate generalizes to fracture configurations and conductivity contrasts outside the three training regimes.

What would settle it

Apply the surrogate to a fracture network whose size, orientation, aperture, or conductivity contrast lies outside the training distributions and measure whether the normalized root-mean-square error on the predicted tensor exceeds 0.22.

Figures

Figures reproduced from arXiv: 2604.02335 by Jan B\v{r}ezina, Martin \v{S}petl\'ik.

Figure 1
Figure 1. Figure 1: Comparison of the fine DFM model fracture network ℎ,𝐿, with ℎ < 5 (left), and the corresponding coarse DFM model fracture network 𝐻,𝐿, with 𝐻 < 10 (right). The illustration also shows the homogenization blocks of size 𝑙 = 15 and their overlap 𝑙∕2. 2. The fracture network and the hydraulic conductivity field 𝑲ℎ are generated on an enlarged domain of side 𝐿 +𝑙, enabling homogenization blocks to capture bou… view at source ↗
Figure 2
Figure 2. Figure 2: Left: The fine DFM model (ℎ < 5). Orange fractures (ℎ,𝐻 ) and the hydraulic conductivity tensor field (𝑲ℎ ) are homogenized using overlapping square blocks of size 𝑙 = 15. Right: Corresponding coarse DFM model (𝐻 < 10) with homogenized hydraulic conductivity tensor component 𝐾𝑥𝑥. Notice the reduced range in the upscaled field. Black fractures (𝐻,𝐿) are not affected by homogenization. Only the 𝐾𝑥𝑥 compone… view at source ↗
Figure 3
Figure 3. Figure 3: An illustration of an input hydraulic conductivity tensors (𝐾𝑥𝑥 component) on a mesh and its voxelized representation. 3.1. Datasets analysis The distributions of components of equivalent hydraulic conductivity tensors for datasets , , and  are depicted in [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Distributions of 𝑲𝑒𝑞 components for datasets of different 𝐾𝑓 ∕𝐾𝑚 , Dataset  for 𝐾𝑓 ∕𝐾𝑚 = 1 × 103 , Dataset  for 𝐾𝑓 ∕𝐾𝑚 = 1 × 105 , and Dataset  for 𝐾𝑓 ∕𝐾𝑚 = 1 × 107 . Diagonal components are shown on a log10 scale. M. Špetlík, J. Březina: Preprint submitted to Elsevier Page 12 of 30 [PITH_FULL_IMAGE:figures/full_fig_p012_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: The prediction accuracy of the trained surrogates for the components of 𝑲𝑒𝑞 evaluated on test datasets. Diagonal components are shown on a log10 scale. M. Špetlík, J. Březina: Preprint submitted to Elsevier Page 16 of 30 [PITH_FULL_IMAGE:figures/full_fig_p016_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Effect of the SRF correlation length 𝜆 on the prediction accuracy of the surrogate model. 5.4. Prediction accuracy on datasets of different DFN settings To evaluate the prediction accuracy across varying DFN parameter settings, we generated four additional test datasets, each comprising 3,500 samples with 𝜆 = 10 and 𝑃30 = 0.0015 [PITH_FULL_IMAGE:figures/full_fig_p018_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Impact of number of fractures (𝑃30) on surrogate prediction accuracy. 5.5. Computational cost reduction Surrogates are primarily employed to accelerate numerical homogenization. To quantify this benefit, we compare the computational cost of standard numerical homogenization (𝐶𝐻 - CPU time) with that of surrogate inference (𝐶𝑆 - GPU time), excluding training time. We examine cases with varying domain sizes … view at source ↗
Figure 8
Figure 8. Figure 8: The prediction accuracy of the trained surrogates for diagonal and off-diagonal components of 𝐊𝑒𝑞 evaluated on test datasets with different DFNs. Diagonal components are shown on a log10 scale. M. Špetlík, J. Březina: Preprint submitted to Elsevier Page 21 of 30 [PITH_FULL_IMAGE:figures/full_fig_p021_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: Comparison of computational time for numerical homogenization 𝐶𝐻 (CPU run) and its surrogate counterpart 𝐶 𝑆 (GPU used for inference). Data shown on a log10 scale. M. Špetlík, J. Březina: Preprint submitted to Elsevier Page 22 of 30 [PITH_FULL_IMAGE:figures/full_fig_p022_9.png] view at source ↗
read the original abstract

Modeling groundwater flow in three-dimensional fractured crystalline media requires accounting for strong spatial heterogeneity induced by fractures. Fine-scale discrete fracture-matrix (DFM) simulations can capture this complexity but are computationally expensive, especially when repeated evaluations are needed. To address this, we aim to employ a multilevel Monte Carlo (MLMC) framework in which numerical homogenization is used to upscale sub-resolution fracture effects when transitioning between accuracy levels. To reduce the cost of conventional 3D numerical homogenization, we develop a surrogate model that predicts the equivalent hydraulic conductivity tensor Keq from a voxelized 3D domain representing tensor-valued random fields of matrix and fracture conductivities. Fracture size, orientation, and aperture are sampled from distributions informed by natural observations. The surrogate architecture combines a 3D convolutional neural network with feed-forward layers, enabling it to capture both local spatial features and global interactions. Three surrogates are trained on data generated by DFM simulations, each corresponding to a different fracture-to-matrix conductivity contrast. Performance is evaluated across a wide range of fracture network parameters and matrix-field correlation lengths. The trained models achieve high accuracy, with normalized root-mean-square errors below 0.22 across most test cases. Practical applicability is demonstrated by comparing numerically homogenized conductivities with surrogate predictions in two macro-scale problems: computing equivalent conductivity tensors and predicting outflow from a constrained 3D domain. In both cases, surrogate-based upscaling preserves accuracy while substantially reducing computational cost, achieving speedups exceeding 100x when inference is performed on a GPU.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The manuscript develops a 3D CNN surrogate that predicts the six independent components of the equivalent hydraulic conductivity tensor Keq from voxelized 3D fracture-matrix conductivity fields. Three separate models are trained on DFM simulation data for distinct fracture-to-matrix conductivity contrasts; performance is reported via NRMSE on held-out test sets spanning fracture statistics and matrix correlation lengths. The surrogate is then substituted into two macro-scale test problems (tensor homogenization and outflow prediction) to demonstrate accuracy preservation and >100x GPU speedup relative to direct numerical homogenization.

Significance. If the tensors remain physically valid, the work supplies a practical route to cheap sub-grid upscaling inside MLMC hierarchies for 3D fractured-media flow. The reported NRMSE values below 0.22 on wide test ranges and the end-to-end preservation of macro-scale quantities constitute concrete evidence of utility; the GPU inference speedup is a clear practical gain for repeated evaluations.

major comments (1)
  1. [§3] §3 (Surrogate Architecture): the network directly regresses the six independent entries of Keq with no Cholesky factorization, eigenvalue projection, or other post-processing to enforce symmetry and positive-definiteness. Because the Darcy problem requires a symmetric positive-definite tensor, any negative eigenvalue produced by the surrogate would render the macro-scale demonstrations in §5 ill-posed and would invalidate both the accuracy-preservation claim and the reported speedups.
minor comments (2)
  1. [Figure 4] Figure 4 and Table 2: the NRMSE values are given without accompanying standard deviations across the multiple random seeds or network initializations; adding these would strengthen the reproducibility statement.
  2. [§2.2] §2.2: the precise voxel resolution and padding strategy used to embed the fracture networks into the 3D grid should be stated explicitly so that readers can reproduce the input representation.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful review and for raising this important point about physical constraints on the predicted tensors. We address the comment directly below.

read point-by-point responses

  1. Referee: [§3] §3 (Surrogate Architecture): the network directly regresses the six independent entries of Keq with no Cholesky factorization, eigenvalue projection, or other post-processing to enforce symmetry and positive-definiteness. Because the Darcy problem requires a symmetric positive-definite tensor, any negative eigenvalue produced by the surrogate would render the macro-scale demonstrations in §5 ill-posed and would invalidate both the accuracy-preservation claim and the reported speedups.

    Authors: We agree that the conductivity tensor must remain symmetric and positive definite for the Darcy problem to be well-posed. Symmetry is already enforced by construction, as the network regresses only the six independent components. However, the original manuscript does not include any explicit mechanism (such as Cholesky factorization or eigenvalue projection) to guarantee positive-definiteness, nor does it report verification that all predicted tensors satisfy this property. We will therefore revise §3 to add a post-processing projection step that enforces positive-definiteness while preserving the six-component output format. In the revised version we will also quantify the frequency and magnitude of any adjustments required and confirm that the macro-scale results in §5 remain unchanged after projection. This addition directly addresses the concern and strengthens the applicability claims. revision: yes

Circularity Check

0 steps flagged

No circularity: standard supervised surrogate trained on independent DFM data

full rationale

The paper generates training data from separate DFM simulations, trains a 3D-CNN + feed-forward surrogate to regress the six Keq components, and reports NRMSE on held-out test cases drawn from the same distributions. Performance metrics are direct comparisons to simulation outputs rather than quantities recomputed from fitted parameters inside the model. No derivation step reduces by construction to its inputs, no load-bearing self-citations appear, and the central claims rest on empirical validation against external simulations. This is a conventional ML surrogate workflow with no self-definitional or fitted-input-called-prediction patterns.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claim rests on standard neural-network training assumptions plus the domain premise that statistical sampling of fractures produces representative training data. No new physical entities are introduced.

free parameters (1)
  • CNN architecture hyperparameters
    Network depth, filter sizes, and training schedule chosen to fit the surrogate to DFM data.
axioms (1)
  • domain assumption Voxelized 3D grids of matrix and fracture conductivities contain all information needed for accurate tensor upscaling.
    Invoked by the choice of input representation to the network.

pith-pipeline@v0.9.0 · 5587 in / 1277 out tokens · 77425 ms · 2026-05-16T12:42:41.617230+00:00 · methodology

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Reference graph

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