math.NASep 17, 2026

Domain-decomposed Evolutional Deep Neural Network with Random Features for Transient Pressure Diffusion with Discontinuous and High-Contrast Coefficients

Authors: Peiqi Li, Jie Chen, Hui Zhang, Simon Hands

Organizations: Department of Applied Mathematics, Xi’an Jiaotong-Liverpool University, Suzhou, 215123, China · Department of Mathematical Sciences, University of Liverpool, Liverpool, L69 3BX, United Kingdom

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Abstract

Transient pressure diffusion in heterogeneous porous media becomes difficult to resolve efficiently when permeability is discontinuous and spans several orders of magnitude. We develop a domain-decomposed random-feature evolutional deep neural network (DD RF-EDNN) that separates spatial approximation from temporal evolution. Permeability-informed random features are compressed into an orthonormal pressure space, and a conservative finite-volume operator is projected onto this space so that only the reduced coordinates are advanced in time. This formulation retains the dissipative structure of the discrete flow problem without repeated neural-network optimization. The analysis quantifies dictionary and singular-value truncation errors, proves contractivity under an energy-consistent dissipativity correction, and derives a conditional grid-level estimate for homogeneous dynamics and time-independent data admitting a steady lifting. Experiments on low-permeability inclusions, high-conductivity channels and blocks, and three-dimensional grids demonstrate consistent accuracy across distinct coefficient structures. The method reaches a final-time relative L2L^2 error of 9.54×10−49.54\times10^{-4} in the principal benchmark and maintains errors on the order of 10−210^{-2} at a permeability contrast of 10310^3. Numerical diagnostics further show that the observed error balance and temporal convergence are consistent with the analysis. These results support DD RF-EDNN as a structure-preserving and interpretable reduced formulation for transient simulations on prescribed heterogeneous media.

Figures & tables

Figure 1: Material-adapted domain decomposition and random-feature construction. (a) Material subregions {Ωm}\{\Omega_{m}\} and their interface set Γ\Gamma . (b) Subregion-based feature allocation with additional enrichment near high-contrast interfaces. The schematic is independent of the benchmark geometry.
StageArithmetic costPrincipal storage or reuse
Random-feature evaluationO(NhM0)O(N_{h}M_{0})O(NhM0)O(N_{h}M_{0}) for the feature matrix
First thin SVDO(NhM02)O(N_{h}M_{0}^{2})Offline; produces the rank- rr compression
Second thin SVDO(Nhr2)O(N_{h}r^{2})Offline; produces the rank- kk pressure basis
Reduced-operator constructionO(nnz⁡(Lh)k+Nhk2)O(\operatorname{nnz}(L_{h})k+N_{h}k^{2})Offline; O(Nhk+k2)O(N_{h}k+k^{2}) working storage
Crank–Nicolson factorizationO(k3)O(k^{3})Performed once for fixed Δt\Delta t and B~\widetilde{B}
Reduced time integrationO(Ntk2)O(N_{t}k^{2})Online; two vectors and the factored k×kk\times k matrix
Table 1: Leading computational costs of the offline and online stages.
SymbolDefinitionPurpose
EpL2E_{p}^{L^{2}}∥ph(T)−pM(T)∥2∥ph(T)∥2\displaystyle\frac{\|\mathbf{p}_{h}(T)-\mathbf{p}_{M}(T)\|_{2}}{\|\mathbf{p}_{h}(T)\|_{2}}Final-time relative pressure error
EpL∞E_{p}^{L^{\infty}}∥ph(T)−pM(T)∥∞∥ph(T)∥∞\displaystyle\frac{\|\mathbf{p}_{h}(T)-\mathbf{p}_{M}(T)\|_{\infty}}{\|\mathbf{p}_{h}(T)\|_{\infty}}Final-time maximum relative pressure error
ρΔt\rho_{\Delta t}ρ ⁣[(Ik−Δt2B~)−1(Ik+Δt2B~)]\displaystyle\rho\!\left[\left(I_{k}-\frac{\Delta t}{2}\widetilde{B}\right)^{-1}\left(I_{k}+\frac{\Delta t}{2}\widetilde{B}\right)\right]One-step stability diagnostic
Table 2: Summary of reported error metrics and stability diagnostic used in the numerical experiments.
Figure 2: Central low-permeability benchmark ( A=0.2A=0.2 in the inclusion, A=1A=1 elsewhere, and S=1S=1 ). Finite-volume reference, DD RF-EDNN prediction, and absolute error at representative times on the 65×6565\times 65 grid.
Figure 3: Final-time baseline comparison for the central low-permeability benchmark. The panels show the finite-volume reference, predictions from DD RF-EDNN and the four baselines, and their absolute errors at T=0.05T=0.05 .
MethodDimensionEpL2E_{p}^{L^{2}}EpL∞E_{p}^{L^{\infty}}ρΔt\rho_{\Delta t}Runtime (s)
Ours4699.54×10−49.54\times 10^{-4}1.08×10−31.08\times 10^{-3}0.998114.60
Global RF-EDNN6003.16×10−23.16\times 10^{-2}1.02×10−11.02\times 10^{-1}1.00008.76
RFM6004.25×10−24.25\times 10^{-2}1.47×10−11.47\times 10^{-1}0.99801.57
PINN–9.52×10−29.52\times 10^{-2}7.10×10−27.10\times 10^{-2}–69.41
XPINN–3.11×10−23.11\times 10^{-2}1.30×10−11.30\times 10^{-1}–165.52
Table 3: Final-time baseline comparison for the central low-permeability benchmark on the 65×6565\times 65 grid.
Figure 4: Ablation and robustness analysis for the central low-permeability benchmark. The panels examine pressure-space whitening, recurrent correction, random-feature sampling, feature-space size, permeability contrast, and grid/time-step settings.
SettingDimensionEpL2E_{p}^{L^{2}}EpL∞E_{p}^{L^{\infty}}ρΔt\rho_{\Delta t}
Ours4699.54×10−49.54\times 10^{-4}1.08×10−31.08\times 10^{-3}0.9981
No whitening4699.63×10−49.63\times 10^{-4}1.15×10−31.15\times 10^{-3}1.0000
With recurrent correction4699.59×10−49.59\times 10^{-4}1.08×10−31.08\times 10^{-3}0.9981
Small RF space4099.67×10−49.67\times 10^{-4}1.34×10−31.34\times 10^{-3}0.9981
Rich RF space5289.52×10−49.52\times 10^{-4}1.11×10−31.11\times 10^{-3}0.9981
Strong contrast, Ain=0.01A_{\rm in}=0.014698.37×10−48.37\times 10^{-4}5.44×10−45.44\times 10^{-4}0.9995
Table 4: Representative final-time ablation and robustness results for the central low-permeability benchmark.
Figure 5: High-contrast permeability fields. Left: four high-permeability square inclusions with Km=1K_{m}=1 and Kf=1000K_{f}=1000 . Right: stripe-like high-permeability fractures with Km=1K_{m}=1 , Kf=1000K_{f}=1000 , and six randomly generated fracture channels. The color scale is shown in logarithmic form.
Figure 6: Four high-permeability block benchmark ( Km=1K_{m}=1 , Kf=1000K_{f}=1000 , and S=50S=50 ). Finite-volume reference and DD RF-EDNN prediction at representative times on the 65×6565\times 65 grid.
Figure 7: Stripe-like high-permeability fracture benchmark ( Km=1K_{m}=1 , Kf=1000K_{f}=1000 , nf=6n_{f}=6 , and S=1S=1 ). Finite-volume reference and DD RF-EDNN prediction at representative times under left–right Dirichlet and top–bottom no-flow conditions.
CaseKf/KmK_{f}/K_{m}SSTTΔt\Delta tDim.EpL2E_{p}^{L^{2}}EpL∞E_{p}^{L^{\infty}}ρΔt\rho_{\Delta t}
Four high- KK blocks1000100050500.20.25.0×10−45.0{\times}10^{-4}4864861.76×10−21.76{\times}10^{-2}5.38×10−25.38{\times}10^{-2}0.9998
Stripe-like fractures10001000110.050.051.0×10−41.0{\times}10^{-4}4924922.33×10−22.33{\times}10^{-2}8.07×10−28.07{\times}10^{-2}0.9951
Table 5: Final-time results for the high-contrast permeability tests.
Figure 8: Three-dimensional central low-permeability benchmark on the largest tested grid. The panels show pressure snapshots on the central slice, comparing Ours with the finite-volume reference solution at representative time levels, together with the corresponding absolute errors. The right panels show final-time orthogonal slices of the predicted and reference pressure fields.
GridDoFDimensionEpL2E_{p}^{L^{2}}EpL∞E_{p}^{L^{\infty}}ρΔt\rho_{\Delta t}Runtime (s)
17317^{3}33755006.57×10−46.57\times 10^{-4}3.10×10−33.10\times 10^{-3}0.98557.70
25325^{3}121677501.21×10−31.21\times 10^{-3}6.97×10−36.97\times 10^{-3}0.985537.48
33333^{3}297919001.40×10−31.40\times 10^{-3}7.68×10−37.68\times 10^{-3}0.985586.04
Table 6: Three-dimensional scalability results at T=0.02T=0.02 .
Figure 9: Accuracy and scalability of the three-dimensional benchmark. The top panels compare the finite-volume reference solution, Ours, and the absolute error on the final-time central slice. The bottom panels report the relative L2L^{2} error, runtime, and retained reduced dimension as the number of spatial degrees of freedom increases.
Figure 10: Error decomposition and numerical verification. (a) Reduced-error components over time. (b) Projection and closure quantities. (c) Final-time errors and analytical bounds, with the finite-volume term estimated by Richardson extrapolation. (d) Crank–Nicolson temporal refinement. (e–h) Signed spatial estimates of the four error components, shown with independent symmetric color scales.
ComponentRealized relative magnitudeUpper estimateEffectivity
Finite-volume estimate1.568×10−21.568\times 10^{-2}––
Pressure space and closure1.670×10−41.670\times 10^{-4}2.461×10−22.461\times 10^{-2}1.47×1021.47\times 10^{2}
Regularization and dissipativity shift3.711×10−83.711\times 10^{-8}1.928×10−31.928\times 10^{-3}5.19×1045.19\times 10^{4}
Crank–Nicolson time integration3.085×10−73.085\times 10^{-7}4.743×10−24.743\times 10^{-2}1.54×1051.54\times 10^{5}
Estimated total1.565×10−21.565\times 10^{-2}8.965×10−28.965\times 10^{-2}–
Table 7: Final-time normalized error decomposition for the central low-permeability benchmark. Effectivity is the analytical bound divided by the realized magnitude; the total estimate includes the empirical finite-volume contribution.
Appendix figures & tables11 assets

Supplementary material from the paper’s appendix.

Appendix

ComponentReproducible implementation rule
Activation and boundary factorσ(s)=tanh⁡(s)\sigma(s)=\tanh(s) and bΩ(x,y)=x(1−x)y(1−y)b_{\Omega}(x,y)=x(1-x)y(1-y) .
High-contrast base featuresϕj(x)=bΩ(x)tanh⁡(wj⊤x+βj)\phi_{j}(x)=b_{\Omega}(x)\tanh(w_{j}^{\top}x+\beta_{j}) , with independent wj∼N(0,32I2)w_{j}\sim\mathcal{N}(0,3^{2}I_{2}) and βj∼U[−2,2]\beta_{j}\sim\mathcal{U}[-2,2] .
Mask enrichmentΦ0=[Φ, mΦs, (1−m)Φs]\Phi_{0}=[\Phi,\,m\Phi_{s},\,(1-m)\Phi_{s}] . The first Ms=round⁡(ϑM)M_{s}=\operatorname{round}(\vartheta M) base features are duplicated, where MM is the base count and ϑ\vartheta is the split fraction.
High-contrast maskCells satisfying K≥KmKfK\geq\sqrt{K_{m}K_{f}} are selected and the resulting binary mask is expanded by one Manhattan grid layer.
SVD truncationA singular vector is retained when σj>τsvdσ1\sigma_{j}>\tau_{\rm svd}\sigma_{1} ; the retained dimension is additionally limited by the case-specific rank cap.
Dissipativity correctionμ2(B)=λmax⁡((B+B⊤)/2)\mu_{2}(B)=\lambda_{\max}((B+B^{\top})/2) and αdiss=max⁡{0,μ2(B)+δ}\alpha_{\mathrm{diss}}=\max\{0,\mu_{2}(B)+\delta\} with δ=10−4\delta=10^{-4} , followed by B~=B−αdissI\widetilde{B}=B-\alpha_{\mathrm{diss}}I .
Appendix
Table 8: Random-feature construction and truncation rules for the two-dimensional experiments. Here mm is the expanded material mask, and Φs\Phi_{s} contains the first MsM_{s} base-feature columns.
QuantityValueSampling or implementation detail
Random seed2026One generator is advanced in the row order shown below.
Inner-region features200w∼N(0,1.82I2)w\sim\mathcal{N}(0,1.8^{2}I_{2}) , β∼U[−2,2]\beta\sim\mathcal{U}[-2,2] .
Outer-region features240w∼N(0,32I2)w\sim\mathcal{N}(0,3^{2}I_{2}) , β∼U[−2,2]\beta\sim\mathcal{U}[-2,2] .
Side-localized features1515 per groupwτ∼N(0,2.52)w_{\tau}\sim\mathcal{N}(0,2.5^{2}) , wn∼N(0,1.52)w_{n}\sim\mathcal{N}(0,1.5^{2}) , β∼U[−2,2]\beta\sim\mathcal{U}[-2,2] .
Localization lengthℓΓ=0.055\ell_{\Gamma}=0.055Gaussian normal window exp⁡[−(ρΓ/ℓΓ)2]\exp[-(\rho_{\Gamma}/\ell_{\Gamma})^{2}] .
Original feature countM0=560M_{0}=560200+240+8×15200+240+8\times 15 .
Appendix
Table 9: Random-feature sampling and compression parameters for the central low-permeability benchmark. The side-localized count is specified per side and per material region, giving eight groups in total.
ParameterFour high- KK blocksStripe-like fractures
Grid65×6565\times 6565×6565\times 65
Permeability dataKm=1K_{m}=1 , Kf=1000K_{f}=1000Km=1K_{m}=1 , Kf=1000K_{f}=1000 , nf=6n_{f}=6
GeometryCenters {1/4,3/4}2\{1/4,3/4\}^{2} , half-width 0.0750.075Axis-aligned random channels
Storage coefficientS=50S=50S=1S=1
TT ; Δt\Delta t0.20.2 ; 5×10−45\times 10^{-4}0.050.05 ; 10−410^{-4}
Permeability seed–2026
Appendix
Table 10: Case-specific random-feature and evolution parameters for the two high-contrast benchmarks. A dash denotes a deterministic permeability geometry.
SettingDimensionCond.EpL2E_{p}^{L^{2}}EpL∞E_{p}^{L^{\infty}}ρΔt\rho_{\Delta t}Runtime (s)
Ours4691.00×1001.00\times 10^{0}9.54×10−49.54\times 10^{-4}1.08×10−31.08\times 10^{-3}0.998122.87
No whitening4696.82×1086.82\times 10^{8}9.63×10−49.63\times 10^{-4}1.15×10−31.15\times 10^{-3}1.000022.99
With recurrent correction4691.00×1001.00\times 10^{0}9.59×10−49.59\times 10^{-4}1.08×10−31.08\times 10^{-3}0.998123.24
Appendix
Table 11: Structural ablation results on the central low-permeability benchmark. The condition number is reported for the reduced coordinate system used in each variant.
SeedDimensionEpL2E_{p}^{L^{2}}EpL∞E_{p}^{L^{\infty}}EbandE_{\rm band}ρΔt\rho_{\Delta t}Runtime (s)
20224699.64×10−49.64\times 10^{-4}1.35×10−31.35\times 10^{-3}1.01×10−31.01\times 10^{-3}0.998122.04
20234699.53×10−49.53\times 10^{-4}1.10×10−31.10\times 10^{-3}9.80×10−49.80\times 10^{-4}0.998120.10
20244699.62×10−49.62\times 10^{-4}1.24×10−31.24\times 10^{-3}1.01×10−31.01\times 10^{-3}0.998114.70
20254699.69×10−49.69\times 10^{-4}1.47×10−31.47\times 10^{-3}1.04×10−31.04\times 10^{-3}0.998115.53
20264699.54×10−49.54\times 10^{-4}1.08×10−31.08\times 10^{-3}9.78×10−49.78\times 10^{-4}0.998115.23
Appendix
Table 12: Random-seed robustness results for the proposed method on the central low-permeability benchmark.
SettingRF sizeDimensionEpL2E_{p}^{L^{2}}EpL∞E_{p}^{L^{\infty}}ρΔt\rho_{\Delta t}Runtime (s)
Small RF space4004099.67×10−49.67\times 10^{-4}1.34×10−31.34\times 10^{-3}0.998113.59
Default RF space5004699.54×10−49.54\times 10^{-4}1.08×10−31.08\times 10^{-3}0.998115.80
Rich RF space6005289.52×10−49.52\times 10^{-4}1.11×10−31.11\times 10^{-3}0.998116.93
Appendix
Table 13: Feature-space sensitivity results on the central low-permeability benchmark. The column “RF size” denotes the original random-feature dictionary size before pressure-space compression.
AinA_{\rm in}DimensionEpL2E_{p}^{L^{2}}EpL∞E_{p}^{L^{\infty}}EbandE_{\rm band}ρΔt\rho_{\Delta t}Runtime (s)
0.304699.65×10−49.65\times 10^{-4}1.13×10−31.13\times 10^{-3}9.90×10−49.90\times 10^{-4}0.998114.64
0.204699.54×10−49.54\times 10^{-4}1.08×10−31.08\times 10^{-3}9.78×10−49.78\times 10^{-4}0.998114.01
0.104699.20×10−49.20\times 10^{-4}8.91×10−48.91\times 10^{-4}9.22×10−49.22\times 10^{-4}0.998113.81
0.054698.93×10−48.93\times 10^{-4}7.10×10−47.10\times 10^{-4}8.70×10−48.70\times 10^{-4}0.998415.74
0.014698.37×10−48.37\times 10^{-4}5.44×10−45.44\times 10^{-4}6.73×10−46.73\times 10^{-4}0.999514.10
Appendix
Table 14: Permeability-contrast robustness results obtained by varying the inclusion permeability AinA_{\rm in} .
NNΔt\Delta tDimensionEpL2E_{p}^{L^{2}}EpL∞E_{p}^{L^{\infty}}ρΔt\rho_{\Delta t}Runtime (s)
332.0×10−42.0\times 10^{-4}3491.83×10−31.83\times 10^{-3}1.59×10−31.59\times 10^{-3}0.99621.71
491.5×10−41.5\times 10^{-4}3811.42×10−31.42\times 10^{-3}1.44×10−31.44\times 10^{-3}0.99714.77
651.0×10−41.0\times 10^{-4}4699.54×10−49.54\times 10^{-4}1.08×10−31.08\times 10^{-3}0.998113.84
Appendix
Table 15: Grid and time-step robustness results on the central low-permeability benchmark.
Figure 11: Schematic implementation of the PINN and XPINN baselines. The PINN baseline uses a single hard-boundary space-time network. The XPINN baseline uses separate subnetworks for the matrix and inclusion regions, with additional pressure-continuity and normal-flux-continuity penalties on the material boundary.
SettingPINNXPINN
Grid65×6565\times 6565×6565\times 65
Final time TT0.050.050.050.05
Reference time step10−410^{-4}10−410^{-4}
Network input(x,y,t)(x,y,t)(x,y,t)(x,y,t)
Hidden layers33 per subnet
Layer width4848 per subnet
Appendix
Table 16: Common training settings for the PINN and XPINN baselines on the central low-permeability benchmark.
SettingPINNXPINN
PDE residual points20481536 outer, 512 inner
Initial-condition points20481536 outer, 512 inner
Interface points–512
λpde\lambda_{\mathrm{pde}}1.0–
λpde,out\lambda_{\mathrm{pde,out}}–1.0
λpde,in\lambda_{\mathrm{pde,in}}–1.0
Appendix
Table 17: Sampling and loss weights for the PINN and XPINN baselines. Collocation points were randomly resampled during Adam training.
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