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A Neural-network-based multiscale Hybridizable Discontinuous Galerkin method for solving PDEs in porous media

Tony Haines, Ke Shi

arXiv:2608.25850v1math.NA

TL;DR

Repeated fine-scale local solves make multiscale HDG costly for heterogeneous PDEs. The paper replaces coarse-block operator construction with neural surrogates while retaining global MsHDG coupling, achieving close lower-contrast solutions and total online speedups from 4.8 to 16.3.

  • Problem

    Constructing coarse-block operators requires repeated fine-scale local solves for heterogeneous coefficient configurations, creating a computational bottleneck.

  • Method

    NN-MsHDG trains networks to predict coarse-block Dirichlet-to-Neumann operators and source responses for standard global MsHDG assembly.

  • Results

    In lower-contrast two-dimensional tests, modeling errors were approximately 1.3% for n = 0, 1 and 3.0% for n = 2, while total online speedup increased from 4.8 to 16.3.

  • Takeaways & Limitations

    Neural surrogates can accelerate MsHDG while preserving its global coupling, but effective coarse spaces remain important for high-contrast heterogeneous problems.

  • Takeaways & Limitations

    For high contrast, learned-operator errors become more severe as the coarse trace space is enriched, and coefficient-adapted spectral trace spaces remain outside this study.

Abstract

from arXiv · show

We develop a neural-network-accelerated multiscale hybridizable discontinuous Galerkin method for elliptic problems with heterogeneous coefficients. The method preserves the standard MsHDG local-to-global structure: fine-scale HDG problems on coarse blocks define discrete Dirichlet-to-Neumann operators, which are assembled through the standard MsHDG global skeleton equations. To reduce the cost of constructing these local operators, we train a neural network on coefficient fields defined on a reference block and use the predicted operators in place of repeated fine-scale local solves. The numerical experiments assess both the accuracy and online efficiency of the resulting NN-MsHDG method. For two-dimensional binary permeability fields with moderate contrast, the neural method reproduces the standard MsHDG solution with relative modeling errors of a few percent while reducing the total online computational cost by factors of approximately 5 to 16, depending on the coarse trace dimension. In the high-contrast regime, the chosen polynomial coarse trace spaces already yield substantial discretization errors in the standard MsHDG method, indicating the need for more effective coarse spaces, such as coefficient-adapted spectral trace spaces. In addition, the learned local operators introduce modeling errors that become severe as the trace space is enriched. These results demonstrate the potential of neural surrogates for accelerating multiscale HDG computations while also highlighting the need for improved operator representations and a better understanding of error amplification in high-contrast problems.

1. Introduction

Heterogeneous PDEs make fine-scale local solves the main cost of multiscale HDG methods, despite their reduced global coupling. NN-MsHDG addresses this bottleneck by learning coarse-block Dirichlet-to-Neumann operators while preserving the standard MsHDG assembly and reconstruction procedures.

  • Motivation: Strong coefficient heterogeneity and contrast create the motivating computational challenge because constructing local coarse operators requires fine-scale solves for every coarse trace basis input and coefficient configuration.HDG reduces global unknowns to mesh-skeleton traces, but local operator construction remains expensive in heterogeneous media.
  • Contribution: NN-MsHDG learns each coarse block’s discrete DtN matrix and source vector from its local coefficient field, replacing repeated fine-scale local solves.The network predicts independent entries of S_i(α) and components of g_i(α, f), which are then used as local coarse-block operators.
  • Method: The predicted operators are assembled in the standard MsHDG coarse skeleton system, leaving global hybridized coupling and local reconstruction unchanged.This retains the local-to-global structure of multiscale HDG while accelerating only the expensive operator-construction stage.
  • Novelty: The approach differs from element-level learning and conforming finite-element substructuring by learning coarse-block multiscale HDG operators rather than fine-element solvers or nodal-trace DtN maps.This distinction defines the method’s central novelty and its placement within hybrid neural-numerical discretizations.
  • Scope: The framework is demonstrated on stationary Darcy flow but is presented as applicable beyond this prototype to time-dependent or nonlinear problems requiring repeated local-operator updates.Darcy flow provides a clean setting because its HDG local solver naturally defines a Dirichlet-to-Neumann map.

2. Review of the MsHDG methods for PDEs in heterogeneous media

MsHDG decomposes heterogeneous PDEs into independently solved fine-scale local problems whose Dirichlet-to-Neumann responses are coupled through a coarse-skeleton trace system. Its main computational bottleneck is repeatedly constructing discrete local operators for each coarse block and coefficient configuration.

  • Each local Dirichlet trace induces a Neumann or flux response, defining the Dirichlet-to-Neumann map used for Darcy flow, Stokes, and other second-order PDEs.The response represents normal Darcy flux, normal stress, or the corresponding natural boundary quantity, depending on the equation.
  • Global coupling selects the coarse trace whose locally computed fluxes satisfy conservation across every interior coarse interface.The transmission condition imposes continuity of the normal flux jump across the coarse skeleton.
  • MsHDG places the global unknown on the coarse skeleton, while fine-scale variables inside each coarse subdomain are eliminated through independent local solvers.Once the trace is determined, each subdomain solution can be recovered independently.
  • Fine-mesh HDG discretization converts each local map into a trace-induced matrix and source-induced vector, which are assembled into the global coarse-scale system.The fine mesh resolves local heterogeneity before the local contributions are assembled.
  • Repeated fine-scale solves for local trace basis functions and source problems make discrete DtN construction costly, especially for systems, high-order or three-dimensional meshes, time dependence, and nonlinear iterations.These costs motivate replacing repeated local computations with efficient surrogates.

3. Darcy flow and the multiscale partitions

The paper formulates Darcy flow with heterogeneous positive coefficients through an MsHDG discretization on coarse blocks with fine-scale HDG meshes and coarse skeleton traces. The formulation is well posed under stated stabilization conditions, while the study replaces repeated local operator solves with neural-network evaluations without changing the discretizations or global coupling.

  • Problem setting: The method targets second-order Darcy flow in bounded polyhedral domains with possibly highly heterogeneous coefficients, while also applying to diffusion, transport, and heat transfer.The coefficient satisfies κ(x) ≥ κ0 > 0, with homogeneous Dirichlet boundary conditions.
  • Multiscale partition: MsHDG partitions the domain into coarse blocks, resolves each block with a conforming fine triangulation, and couples blocks through coarse skeleton trace spaces.Interfaces are organized into a coarse skeleton with scale H, while fine elements and faces use scale h.
  • MsHDG formulation: The MsHDG scheme uses mixed HDG equations with numerical traces and fluxes, stabilized by bq_h,H · n = q_h · n + τ(u_h − bu_h,H), reducing to standard HDG for one block.The coarse trace space and fine-scale local spaces complete the formulation.
  • Well-posedness and coarse traces: Under compatible local spaces and stabilization positive on required fine and coarse faces, the MsHDG problem has a unique solution and permits flexible coarse trace spaces.The analysis identifies coarse-trace approximation quality, rather than polynomial form alone, as essential; the present work retains discontinuous piecewise-linear traces.
  • Neural operator replacement: The study tests neural surrogates for repeated fine-scale local HDG solves, generating coarse-block operators by network evaluations while preserving the same discretizations, stabilization τ = 1, and global skeleton coupling.The implementation does not claim the optimal global convergence estimate associated with the single-face stabilization assumption used in [9].

4. The Upscaling of the MsHDG method

MsHDG eliminates fine-scale variables within each coarse block and couples blocks through coarse-skeleton trace unknowns and normal-flux continuity. Its local Dirichlet-to-Neumann operators require repeated fine-scale matrix inversions, motivating a neural surrogate that predicts these operators from local permeability fields while retaining the standard global assembly.

  • Upscaling structure: MsHDG uses coarse-skeleton scalar traces as global unknowns, eliminates fine-scale block variables locally, and enforces normal numerical-flux continuity across coarse interfaces.The local problems separate imposed coarse traces from interior fine-skeleton unknowns, leaving only coarse traces in the global system.
  • Local operators: The exact upscaling requires constructing each coarse block’s discrete Dirichlet-to-Neumann matrix, which maps imposed coarse traces to boundary normal numerical fluxes.The global coarse system contains only trace unknowns, but its assembly depends on local DtN matrices and source-induced flux vectors.
  • Computational bottleneck: Repeated local operator construction is the computational bottleneck because each DtN matrix requires applying the inverse of a heterogeneous fine-scale matrix to all coarse-boundary trace basis inputs.The operator depends nonlinearly on the local permeability field through the inverse of the fine-scale matrix.
  • Neural-network surrogate: The proposed surrogate takes the local permeability field on a fixed reference block as input and predicts the local DtN matrix and source-induced flux vector for standard MsHDG assembly.Because coarse-block geometry and the source f ≡1 are fixed, only the local permeability field varies; network evaluation replaces repeated fine-scale matrix inversions.

5. Neural-network-based local solvers

The method learns a coefficient-to-local-operator map on a reference coarse block, predicting reduced Dirichlet-to-Neumann quantities for standard MsHDG global assembly. Fixed binary 8 × 8 coefficient images, trace-level-specific outputs, geometric transfer, and structure-aware losses define the training procedure.

  • Reduced operator representation: The network predicts the symmetric local DtN matrix and source vector rather than the full inverse matrix, reducing output dimension while retaining all information needed for global assembly.Only upper-triangular matrix entries are stored, while the source vector is retained in full.
  • Reference-block operator approximation: A reference-block neural network approximates the local Dirichlet-to-Neumann map and transfers it to congruent physical blocks through deterministic affine geometric scaling.Because all experimental coarse blocks are affine images of the reference square, one trained model can serve every coarse block without geometric inputs.
  • Training-data generation: Training uses 8 × 8 independently sampled binary permeability images, with local labels generated by fine-mesh HDG solves on a 32 × 32-square triangular discretization.Each coefficient realization is a one-channel normalized image and a 64-dimensional neural-network input; the source term is fixed as f ≡1.
  • Training configuration: For each contrast regime and trace level n = 0, 1, 2, separate networks are trained from 5000 realizations split into 80% training and 20% validation data.The input dimension remains fixed at 64, while trace-space and output dimensions increase with trace enrichment.
  • Network and objective: A convolutional network extracts spatial coefficient features, while training combines normalized data, operator-action, energy-consistency, and constant-trace nullspace losses.The architecture uses three convolutional layers followed by two fully connected hidden layers with GELU activations.

6. Numerical experiments

The experiments show that NN-MsHDG preserves standard MsHDG accuracy while substantially reducing online cost, although high-contrast cases expose limitations of the coarse trace space and learned local operators. A one-dimensional test verifies local operator learning and global assembly but provides no meaningful speedup.

  • One-dimensional verification: The one-dimensional experiment keeps global modeling error below 1% across tested resolutions, partitions, and contrasts, including {1, 10^4}, but its total online speedup is only approximately 0.98–1.00.It therefore verifies the learned DtN operator and global assembly rather than demonstrating computational acceleration; contrast alone does not explain the later two-dimensional difficulties.
  • Two-dimensional accuracy: In the lower-contrast experiment, standard MsHDG error decreases from approximately 9.8% to 3.5%, while NN-MsHDG remains nearly identical for n = 0 and n = 1 and has approximately 3.0% modeling error at n = 2.The direct modeling error is approximately 1.3% for n = 0 and n = 1; enriching the trace space doubles the local trace dimension from 16 to 32 and increases predicted entries from 152 to 560.
  • Two-dimensional efficiency: Total online speedup reaches approximately 16.3 at n = 2 in the lower-contrast experiment and 16.8 at high contrast, while assembly speedup reaches approximately 30.5 and 30.7, respectively.Network evaluations replace repeated local fine-scale solves while retaining standard MsHDG global coupling.
  • Computational bottleneck: For standard MsHDG, local-operator construction and global assembly consume approximately 90%, 95%, and 97% of total online time for n = 0, 1, and 2, respectively.This identifies repeated fine-scale local solves, rather than the global skeleton solve or interior reconstruction, as the dominant bottleneck.
  • High-contrast limitations: At high contrast, local DtN action errors remain on the order of 10^-2, but richer coefficient-dependent coarse spaces are needed because standard polynomial traces produce substantial discretization errors.Coefficient-adapted spectral trace spaces are identified as a possible alternative.

7. Concluding Remarks

The work develops a neural-network-accelerated MsHDG method that preserves the standard local-to-global structure while replacing repeated local operator construction with learned surrogates. Experiments demonstrate accurate behavior in moderate settings but expose high-contrast limitations and motivate improved trace spaces and stability analysis.

  • Method contribution: The method preserves MsHDG’s local-to-global structure by learning coarse-block Dirichlet-to-Neumann operators generated by fine-scale HDG solves.These local operators map coarse trace data to numerical flux responses and are assembled through the standard MsHDG framework.
  • Numerical findings: At N_c = 40, the mean E_model remains below 3×10^−3 for every tested one-dimensional coefficient resolution and contrast regime.Increasing N_κ makes local conductance learning more difficult, but assembled errors do not accumulate destructively.
  • Limitations: In the high-contrast regime κ ∈ {1, 10^4}, NN-MsHDG departs from standard MsHDG, with deterioration increasing as the coarse trace space is enriched.Strongly separated operator scales and perturbation-sensitive interface directions may explain this error amplification.
  • Future directions: Coefficient-adapted spectral trace spaces are proposed as a promising complement because polynomial traces may poorly represent heterogeneous interface behavior; see.Such spaces use local snapshots and coefficient-dependent spectral decompositions, but combining them with neural DtN approximation is nontrivial.
  • Future directions: Future analysis should relate local DtN approximation errors, assembled-skeleton conditioning, and global solution errors to stability guarantees for learned systems.Relevant tools may include operator norms and solution-relevant trace directions, alongside structure-preserving parameterizations.
  • Scope: The framework extends beyond Darcy to other elliptic and diffusion problems whenever HDG provides independent local elimination and globally coupled skeleton traces.Its essential ingredient is a local input-output map from trace and source data to numerical fluxes.
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