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Hard-Trace First-Order Residual Learning for Coupled Stokes-Brinkman-Darcy Flow

Zefeng Liu, Hongxing Rui

arXiv:2609.36793v1math.NAcs.NA

TL;DR

Coupled Stokes–Brinkman–Darcy flow requires different shared quantities at its two interfaces, and strong Brinkman drag weakly penalizes a low-order pressure direction. The paper combines a mixed first-order formulation, exact interface and traction constraints, and a boundary-mass-based pressure correction; corrected manufactured-solution errors and a filtration test provide its reported validation.

  • Problem

    SBD coupling requires full velocity sharing at Stokes–Brinkman but only normal-flux sharing at Brinkman–Darcy, while strong Brinkman drag weakly penalizes a low-order pressure perturbation.

    Source: §1 · pp. 2–3

  • Method

    A mixed first-order state, shared interface traces, hard native Brinkman–Darcy traction, and an integral-momentum pressure correction form the proposed solver.

    Source: §1 · p. 3, §6 · p. 33

  • Results

    Corrected upper-pressure relative L2 means range from 0.87% to 2.58% across six manufactured-solution combinations, and all four filtration runs satisfy the prescribed field and physical-consistency criteria.

    Source: §6 · p. 33

  • Takeaways & Limitations

    Paired controls distinguish the roles of exact kinematic sharing, Brinkman–Darcy traction enforcement, and pressure correction.

    Source: §6 · p. 33

    At the prescribed training budget, nonlinear Darcy pressure-trace tests do not establish accurate nonlinear-trace learning or general superiority of pressure-scale separation.

    Source: §5.4.2 · p. 28

Abstract

from arXiv · show

Stokes-Brinkman-Darcy (SBD) models connect free flow and porous flow through a Brinkman layer of finite thickness. The proposed method for solving these coupled equations consists of a mixed first-order formulation, shared interface traces and hard traction constraints, and a pressure correction based on boundary mass balance. Independent stresses, Darcy flux and pressure-gradient auxiliaries avoid second derivatives of network outputs. Boundary liftings and shared traces impose exterior and interface kinematic conditions exactly, while a stress map enforces native Brinkman-Darcy force balance. An integral momentum relation determines the pressure correction after training, without reference-pressure labels. Across two manufactured solutions and three permeabilities, mean corrected upper-pressure relative $L^2$ errors range from 0.87% to 2.58%; correction reduces both upper-pressure $L^2$ errors and the full Brinkman-pressure $H^1$ error in every run. Paired controls assess the contributions of interface sharing and traction enforcement. A non-manufactured filtration problem tests mixed-boundary adaptation against an independently refined finite-element reference, with all four runs satisfying the prescribed field and physical-consistency criteria.

1. Introduction

The paper studies full-dimensional Stokes–Brinkman–Darcy flow, where a finite-thickness Brinkman layer connects free and porous flow. It targets distinct interface-sharing requirements and pressure accuracy under strong Brinkman drag using exact constraints and a post-training correction.

1. Introduction

  • At Stokes–Brinkman, full velocity must be shared; at Brinkman–Darcy, only normal flux is shared, and sharing alone does not enforce force balance.Darcy tangential flux remains independent.Source: §1 · pp. 2–3
  • The pressure correction addresses a low-order pressure perturbation that survives exact kinematic constraints and pressure augmentation but is weakly penalized under strong Brinkman drag.An integral momentum relation and prescribed boundary mass balance determine the correction without reference-pressure labels.Source: §1 · p. 3

2. Mathematical model

  • The study considers a full-dimensional, three-rectangle configuration with constant isotropic coefficients, nonsymmetric pseudo-stress, and zero Stokes–Brinkman traction jump.A finite-thickness Brinkman layer connects the free-flow and Darcy regions.Source: §1 · p. 2, §2 · p. 3

2.1. Geometry, boundaries, and orientation

  • The domain is partitioned into Stokes, Brinkman, and Darcy rectangles, with horizontal interfaces and downward-pointing normals.Prescribed exterior data and coupling-determined interface traces are summarized in Figure 1.Source: §2.1 · pp. 3–4, Figure 1 · p. 4

2.2. Subdomain equations

  • The regional equations are reformulated as mixed systems with independent upper-region pseudo-stress and Darcy flux, requiring first derivatives only.The original Stokes and Brinkman momentum terms combine into the divergence of pseudo-stress.Source: §2.2 · pp. 4–5

2.3. Exterior and interface data

  • Exterior Dirichlet data are prescribed, while interface laws impose Stokes–Brinkman velocity and pseudotraction continuity and Brinkman–Darcy normal mass continuity with force balance.Darcy pressure on the Brinkman–Darcy interface is an unknown trace determined through coupling.Source: §2.3 · p. 6

2.4. Dimensionless model parameters and pressure anchoring

  • The dimensionless study uses prescribed nominal model parameters, and coupled boundary conditions anchor Darcy, Brinkman, and Stokes pressure levels.The pressure errors therefore require no fitted subdomain-mean subtraction.Source: §2.4 · p. 7, Table 1 · p. 7

3. Hard-trace first-order residual method

The method combines a mixed first-order state with shared interface traces and hard field maps, then corrects a weakly controlled pressure direction using an integral momentum relation. Its construction enforces selected kinematic and traction conditions algebraically, while its analytical guarantees remain conditional.

3.1. First-order state and pressure auxiliaries

  • Independent upper-region stresses and pressure-gradient auxiliaries, together with Darcy flux and pressure, form a mixed state that retains first-order differentiation.The pressure-gradient augmentation requires smooth fields, incompressibility, and constant isotropic coefficients.Source: §3.1 · pp. 7, 10

3.2. Shared traces and hard field maps

  • A shared Stokes–Brinkman velocity trace enforces full velocity continuity, while the Brinkman–Darcy map shares only normal flux and leaves Darcy tangential flux independent.Trace endpoints use compatible prescribed side data.Source: §3.2 · pp. 10–11

3.3. Darcy pressure-scale separation

  • Only the interior Darcy pressure bubble is scaled by μ/K_D, avoiding amplification of random interface-trace initialization.The parameterization does not assume that the physical trace is O(1).Source: §3.3 · p. 12

3.4. Residual objective and implementation

  • The residual objective averages each group over points and components, uses fixed physical-scaling weights, and separately records identically zero kinematic groups.Loss magnitudes are not directly comparable across formulations.Source: §3.4 · p. 13

3.5. Hard BD traction map

  • The hard Brinkman–Darcy stress lifting imposes native traction exactly while leaving the original stress output free away from the interface.Traction reconstructed from velocity and pressure still requires an independent check.Source: §3.5 · pp. 13–14

3.6. Consistent low-order pressure correction

  • The post-training correction uses boundary mass balance and an integral momentum relation to update pressure, stress, and auxiliaries along a low-order pressure direction without exact-pressure labels.The correction leaves velocities, flux, Darcy pressure, BD tractions, and SB traction jumps unchanged.Source: §3.6 · p. 14

3.7. Construction properties and analytical scope

  • The field maps are algebraically admissible and the augmented residual is continuously consistent for sufficiently smooth solutions under the stated assumptions.The resulting continuous-error control is conditional and does not establish permeability-uniform stability, finite-sample error control, or optimizer convergence.Source: §3.7 · p. 15

4. Experimental settings and evaluation protocol

The experiments define comparison configurations and budgets, construct two manufactured solutions, and evaluate checkpoints against independent field and interface criteria. The protocol distinguishes configuration controls, test-solution scope, and checkpoint-level acceptance.

4.1. Comparison configurations

  • Five seeds compare PINN, soft first order, and kinematic hard traces; the complete method spans six MMS–permeability combinations, with three reused seeds per combination and paired controls for BD traction or interface sharing.The 18 complete-method runs are not 18 independent replicates; raw and corrected refer to states before and after correction of one trained checkpoint.Source: §4.1 · p. 15

4.2. Networks, sampling and budgets

  • Baseline comparisons match nominal total budgets, not parameter count, optimization path, or elapsed time, and whole-configuration gains cannot be assigned to any one component.PINN reconstructs stress and flux and requires second derivatives, whereas soft first order predicts stress and flux independently; the hard-trace configuration also changes auxiliaries, pressure scaling, and selected residual scaling.Source: §4.2 · p. 16

4.3. Manufactured solutions

  • Manufactured solutions provide closed-form sources, boundary and interface data, and fields for error evaluation; their data are generated by substituting the fields into the continuous equations.The exact pseudostresses are defined by Equation (3), rather than fitted independently.Source: §4.3 · p. 17

4.3.1. MMS1: exponential single-frequency field

  • MMS1 uses Ruan and Rybak’s exponential analytical fields, with the Stokes and Brinkman fields restricted from the same smooth upper solution.Its parameters are set to ω=1 and φ=0.Source: §4.3.1 · p. 17

4.3.2. MMS2: polynomial field with two-component interface flow

  • MMS2 is an in-work polynomial test with a nonzero linear Darcy interface-pressure trace and both tangential Brinkman velocity and normal through-flow at ΓBD.The governing and interface equations for both manufactured examples have independently tested maximum absolute residuals of approximately 2.22 × 10^-16.Source: §4.3.2 · pp. 17–18

4.4. Independent evaluation and acceptance

  • Checkpoints are evaluated without retraining on independent regional grids and interfaces; pressure means are not removed, and each checkpoint must satisfy every applicable strict, study-specific criterion.Errors aggregate regional maxima before seed averaging; interface RMS uses physical, unscaled units, and the extended MMS and BDF acceptance families are specified separately.Source: §4.4 · pp. 18–19, Table 3 · p. 19

5. Results and validation

Across manufactured tests, kinematic hard traces outperform soft baselines, and the corrected complete method retains low pressure errors across permeabilities. The complete method also meets the prescribed criteria in all four mixed-boundary filtration runs.

5.1. Experimental system and overall accuracy

  • For both MMS cases, kinematic hard traces had lower five-seed mean errors than both soft baselines across all six fields and passed joint criteria in 5/5 seeds, versus 0/5.Benchmark errors were reported over five seeds per method and case.Source: §5.1 · p. 19, Figure 4 · p. 21, Table 4 · p. 21

5.2. Spatial pressure recovery and interface consistency

  • At KB=10^-6 in MMS2, correction cut Stokes/Brinkman pressure L2 errors from 10.15%/6.39% to 0.77%/1.29%; the BD normal-flow jump was zero at stored precision in MMS2.The pressure correction coefficient came from the physical moment, with exact pressure used only to evaluate errors.Source: §5.2 · p. 22, Figure 5 · p. 23

5.3. Permeability adaptation and spatial error

  • Across six MMS–permeability combinations, mean corrected upper-pressure L2 errors were 0.87–2.58% and full Brinkman-pressure H1 errors 4.77–8.03%; correction reduced both upper-pressure L2 errors in every run.Correction also reduced full Brinkman-pressure H1 and normal-derivative errors in every run; all corrected states passed 14 criteria on both grids.Source: §5.3 · p. 25, Table 5 · p. 25

5.4.1. Hard BD traction

  • At KB=10^-4 and 10^-6, hard BD improved both pressure L2 errors and full Brinkman-pressure H1 in every pair, passing 3/3 runs versus 0/3 for soft BD.At KB=10^-2, both configurations passed 3/3 runs.Source: §5.4.1 · p. 27, Figure 10 · p. 28

5.4.2. Shared and released interfaces

  • About 4.90% corrected full Brinkman-pressure H1 means occurred in both shared and released configurations; nonlinear pressure-trace variants did not pass the inherited screen in any of three runs.Released-interface SB velocity and BD normal-jump RMS means were 0.00083 and 0.00220, unchanged by correction; hard sharing makes kinematic jumps zero.Source: §5.4.2 · pp. 27–28, Table 6 · p. 27, Figure 10 · p. 28

5.5. Boundary-driven filtration (BDF)

  • All four BDF runs met 18 criteria on both grids before and after correction; mean Stokes/Brinkman pressure L2 errors fell from 1.4808%/2.5330% to 1.4504%/2.5181%.The main BDF result is joint satisfaction of field, traction, and mass-balance criteria against an independently refined FEM numerical reference.Source: §5.5 · pp. 30, 32, Figure 12 · p. 32

5.6. Conservation and computational cost

  • SB velocity jumps were algebraically zero; the MMS1 BD mass jump was (5.77 ± 0.21) × 10^-17, and pressure correction took 0.0506–0.0615 s.The complete BDF adaptation had mean training time 885.06 ± 2.07 s; CPU and GPU batches were not ranked for speed.Source: §5.6 · pp. 32–33

6. Conclusions

The method combines exact kinematic traces, native Brinkman–Darcy traction enforcement, and a first-order mixed state; pressure correction improves manufactured-solution pressure errors, while the filtration test meets its prescribed criteria. Evidence is limited to steady, two-dimensional rectangular domains with constant coefficients.

6. Conclusions

  • The mixed first-order method uses shared velocity and flux traces, exact kinematic constraints, native Brinkman–Darcy traction balance, and only first derivatives of network outputs.At SB it shares full velocity; at BD it enforces normal-flux continuity while allowing independent Darcy tangential flux.Source: §6 · p. 33
  • 0.87%–2.58% are the corrected upper-pressure L2 error means across six manufactured-solution combinations; correction lowers pressure L2 and full Brinkman-pressure H1 errors in every run.The corrected full Brinkman-pressure H1 means range from 4.77% to 8.03%.Source: §6 · p. 33
  • All four boundary-driven filtration runs satisfy the 18 prescribed acceptance criteria on both evaluation grids, alongside agreement with an independently refined finite-element reference.Pressure changes are small; the reported result is simultaneous recovery of flow fields and satisfaction of coupling and conservation criteria.Source: §6 · p. 33
  • The evidence covers steady two-dimensional rectangular domains with constant coefficients; it does not establish robust learning of more general interface-pressure traces or universal superiority of pressure-scale separation.The additional nonlinear Darcy pressure trace was not recovered accurately at the tested budget.Source: §6 · p. 33

CRediT authorship contribution statement

Zefeng Liu and Hongxing Rui’s contribution roles are reported separately.

CRediT authorship contribution statement

Declaration of generative AI and AI-assisted technologies in the manuscript preparation process

The authors disclose AI assistance limited to manuscript preparation tasks and retain responsibility for the scientific content and final manuscript.

Declaration of generative AI and AI-assisted technologies in the manuscript preparation process

Supplementary Material

The supplement documents implementation, conditional analysis, experiment protocols, and additional validation and controls. It specifies that its error estimate is conditional and identifies boundaries on the current stability and lifting results.

Supplementary Material

  • The supplement covers implementation and conditional analysis, experiment protocols and metrics, optimization diagnostics, BDF validation and controls, pressure-moment analysis, and complete-method MMS validation.It also includes pressure-correction controls for benchmarks and Brinkman–Darcy pairs.Source: Supplementary Material · p. 38

S1.1. Network output components and boundary assignments

S1.2. Conditional error estimate and its assumptions

  • The error estimate concerns the continuous residual under a stated stability assumption, with pseudo-stress rows in H(div) and interface residuals required in L2.Uniform stability in permeability or layer thickness and transfer from finite sampling to the bound remain open; the lifting assumes horizontal rectangular interfaces and compatible corners.Source: S1.2. Conditional error estimate and… · p. 39

S2.1. Experiment groups and seed identifiers

  • Run identifiers initialize weights and scrambled Sobol sampling; matched seeds pair initial weights and samples, but do not create additional independent replicates across coefficients.The regional networks have four hidden layers of width 64, and the MMS hard-trace and complete BDF configurations use three trace networks with three hidden layers of width 32.Source: S2.1. Experiment groups and seed… · p. 40, Table S2 · p. 40

S2.2. Compared formulations

  • The conventional strong-form PINN uses second derivatives and soft penalties; the soft first-order baseline removes second derivatives but keeps soft boundary penalties, unlike the kinematic hard-trace first-order model.The hard-trace model makes exterior velocity and Darcy-pressure data, SB velocity continuity, and BD normal-mass continuity identities of the trial map; its volume equations and two traction groups remain active in optimization.Source: S2.2. Compared formulations · p. 41

S2.3. Collocation sampling, seeds, and reproducibility

Fixed sampling, explicit optimizer accounting, and checkpoint-level evaluation rules define how the benchmark runs and their variability are assessed.

S2.3. Collocation sampling, seeds, and reproducibility

  • Fixed scrambled Sobol point sets are shared across methods within each case–seed run and remain unchanged during Adam and L-BFGS.The seed also controls initialization and Sobol scrambling; the protocol does not claim bitwise reproducibility across software or hardware versions.Source: S2.3. Collocation sampling, seeds, and… · p. 42
  • Hard-trace exterior residuals vanish algebraically, while its BD residuals are diagnostic-only; the same sampled conditions train both soft formulations.The fixed collocation counts are given per region in Table S3.Source: S2.3. Collocation sampling, seeds, and… · p. 42, Table S3 · p. 42

S2.4. Optimization budget and hardware

  • All runs use 400 Adam updates before L-BFGS, whose iteration totals are nominal upper bounds rather than guaranteed accepted steps.Line-search closure evaluations may exceed accepted steps and are recorded separately; the hard-trace MMS2 schedule resets optimizer history between blocks.Source: S2.4. Optimization budget and hardware · pp. 42–43
  • Equal nominal optimizer budgets do not imply equal runtime or FLOPs because the hard-trace model is larger and differentiates more quantities.MMS training used float64 on the same RTX 3050 Ti GPU; independent frozen-checkpoint evaluation ran on CPU.Source: S2.4. Optimization budget and hardware · p. 43

S2.5. Additional norms and quadrature conventions

  • Reported field errors use tensor-product trapezoidal quadrature, while sampled maxima are not continuum suprema and pressure means are retained.Relative errors are undefined when the reference norm is at most 10^-12.Source: S2.5. Additional norms and quadrature… · pp. 43–44

S2.6. Cross-seed aggregation; S2.7. Evaluation families and diagnostic conventions

  • Each grid yields one checkpoint metric before aggregation; standard deviations describe run variation, not confidence intervals, and pass counts apply to individual checkpoints.All applicable strict acceptance criteria must hold for a checkpoint; a fine grid checks quadrature sensitivity of a fixed network, not training-resolution convergence.Source: S2.6. Cross-seed aggregation · p. 44, S2.7. Evaluation families and… · pp. 44, 46

S2.7. Evaluation families and diagnostic conventions

  • Native traction uses independently learned stress, whereas reconstructed traction uses velocity and pressure; momentum diagnostics using reconstructed stress are distinguished accordingly.Vector RMS and component-averaged RMS differ by a factor of √m for an m-component residual.Source: S2.7. Evaluation families and… · p. 46

S2.8. Baseline field and conservation displays

The supplement documents evaluation displays and protocols, then reports fixed-checkpoint robustness checks and complete BDF validation against a refined numerical reference.

S2.8. Baseline field and conservation displays

S2.9. Computational environment and timing scope

S2.10. Spatial-figure protocol

  • Main flow and contour figures use a fixed representative complete-method seed, while the three-cut and permeability-map figures use their separately specified seed groups.Figure fields come from saved grids, with no model reevaluation; paired BD and interface-release profiles concern different control cohorts.Source: S2.10. Spatial-figure protocol · pp. 47–48
  • The pressure-direction figure uses a fixed, prespecified MMS2 checkpoint; its coefficient comes from physical-moment cancellation, while exact pressure is used only to evaluate errors.In this run, unscaled Brinkman momentum component RMS slightly increases from 691.8125 to 692.1413, so moment cancellation does not imply pointwise momentum improvement.Source: S2.10. Spatial-figure protocol · p. 48

S3.1. Optimization endpoints

  • All six representative hard-trace field errors improve between audited L-BFGS endpoints, by factors 1.72–10.54 for MMS1 and 2.43–12.50 for MMS2.Only two L-BFGS endpoints were independently audited, so monotonic improvement is not claimed.Source: S3.1. Optimization endpoints · p. 49

S3.2. Sampling sensitivity

  • In a one-seed, three-level sampling check, all settings meet stored diagnostic thresholds, but pressure and Darcy-flux errors vary nonmonotonically with sample count.The check is not a convergence-order or uncertainty study.Source: S3.2. Sampling sensitivity · p. 49

S3.3. Fixed-checkpoint evaluation grids

  • Across three evaluation grids, representative hard-model maximum velocity and pressure errors change by less than 0.6%, while Darcy-flux errors change by about 2.5%–3.6%.Field-gate decisions do not change; this check does not certify quadrature for all thirty checkpoints.Source: S3.3. Fixed-checkpoint evaluation grids · p. 49

S4.1. Complete BDF: verification and adverse changes

  • Both BDF states pass all 18 joint acceptance criteria on both grids in 4/4 seeds, with velocity, Darcy-field, and stated invariant checks also passing.The 14 MMS-style criteria also pass; independent moment and boundary-stress integrations differ by at most 6.22 × 10^-15.Source: S4.1. Complete BDF: verification and… · p. 50

S4.2. Complete-method training protocol

The supplement distinguishes complete-method BDF training from a staged soft-BD control and documents the numerical reference and validation limits.

S4.2. Complete-method training protocol

  • The complete BDF configuration keeps its physical objective fixed through all three optimizer blocks while adapting liftings, loss weights, and budgets to mixed boundaries.The method retains its first-order framework, hard-interface principles, and correction rule.Source: S4.2. Complete-method training protocol · pp. 50–51

S4.3. Soft-BD staged objective and validation protocol

  • The soft-BD control uses a prespecified three-stage schedule that changes constitutive and traction weights, with each reserved seed continuing only from its own preceding checkpoint.Its interface penalty checks the corresponding constitutive components algebraically; it is not a trace bound from an interior constitutive L2 norm or a parameter-uniform stability result.Source: S4.3. Soft-BD staged objective and… · pp. 51–52

S4.4. Independent finite-element reference and acceptance rule

  • The independent FEniCS reference uses Taylor–Hood upper-region fields, continuous linear Darcy pressure, GMRES interface iteration, and MUMPS subproblem solves.The largest six-field relative L2 change from mesh level 80 to 160 is 0.201% for Darcy flux, but adjacent-mesh changes are not rigorous errors against the continuous solution.Source: S4.4. Independent finite-element… · pp. 52–53, Table S9 · p. 53
  • The raw Darcy flux reconstructed from continuous pressure is not strictly H(div)-conservative, and the reference flux is also projected into a continuous vector space.Reported field errors are therefore relative to a converged numerical reference, and FEM and neural conservation values are not identical discretization measures.Source: S4.4. Independent finite-element… · p. 53

S4.5. Soft-BD BDF configuration: retrospective correction only

  • Retrospective soft-BD correction modestly lowers both pressure L2 means in 4/4 seeds, while Brinkman momentum component RMS rises on its separate evaluator.Velocities, Darcy fields, and tractions remain unchanged; this cohort is not the complete hard-BD method or a controlled hard-BD ablation.Source: S4.5. Soft-BD BDF configuration:… · pp. 53–54, Table S10 · p. 54

S4.6. Staged complementary traction mismatches

S4.7. Additional adverse momentum diagnostics

This block reports paired hard/soft BD diagnostics, a nonlinear Darcy pressure-trace test, and a geometry-based mean-flow relation.

S4.7. Additional adverse momentum diagnostics

  • Table S12 tabulates the number of seeds with momentum increases after correction.The supplied passage identifies the table's scope but does not state its cell values.Source: Table S12 · p. 55

S5.1. Frozen settings and integrity

S5.2. Cross-permeability paired diagnostics

  • Hard BD has lower full Brinkman-pressure H1 error in every paired run at each of three permeability coefficients, with trade-offs in pressure L2, reconstructed traction, and training time.At 10^-2, hard/soft means are 7.36%/7.93% for Stokes-pressure L2 and 4.80%/4.61% for Brinkman-pressure L2; training time rises by about 50.3%, 47.6%, and 48.5% across coefficients.Source: Table S15 · p. 56, S5.2. Cross-permeability paired… · p. 57, Table S16 · p. 57
  • The separate five-seed hard-BD diagnostics report mean Brinkman-pressure full H1 error of 24.982% and mean global exterior mass defect of 0.01221%.The mean raw Brinkman momentum component RMS is 749.902, whereas its scaled residual is not a raw-equation accuracy claim.Source: Table S14 · p. 56, S5.2. Cross-permeability paired… · p. 57

S5.3. Nonlinear Darcy pressure trace

  • Amplifying the trainable Darcy pressure-trace correction lowers mean error, but neither tested variant passes the joint screen.The analytical endpoint-linear interpolant has 13.6269% trace L2 error and is a mathematical reference, not a trained control.Source: S5.3. Nonlinear Darcy pressure trace · p. 59, Table S17 · p. 59

S6.1. Mean flow from exterior mass balance

  • The mean-flow expression uses geometry and prescribed exterior velocities only; its independent divergence-free-field check gives analytical and numerical means of −0.5016666667.No interior velocity labels enter the formula.Source: S6.1. Mean flow from exterior mass… · pp. 59–60

S6.2. Relation between the physical moment and learned momentum

The supplement explains why the boundary-mass-consistent physical moment can differ from approximate networks’ raw mean momentum, then documents validation and controls for the correction and interface constraints. Across prospective checkpoints, correction improves the reported pressure-derivative and low-permeability Brinkman-pressure errors while retaining the stated acceptance criteria.

S6.2. Relation between the physical moment and learned momentum

  • For exact incompressible solutions, the physical and raw mean momentum quantities agree and vanish; approximate velocities can diverge, with the discrepancy amplified by drag.Thus reducing the boundary-mass-consistent moment need not reduce pointwise momentum RMS.Source: S6.2. Relation between the physical… · p. 60

S6.3. Development, freezing and independent validation

  • Raw-momentum-mean fitting gave c≈−816.27 and severely worsened pressure, while the boundary-mass-consistent rule gave c≈0.606322 consistently at quadrature orders 16, 32, and 48.Seed 4301 was development-only; failed development records were retained but not counted as successful validation.Source: S6.3. Development, freezing and… · p. 60

S6.4. Equal-parameter interface-release control

  • The interface-release control retained 46,074 parameters and hard Brinkman-Darcy traction, but changed interface kinematics from identities to soft penalties; paired raw weights and samples matched despite different initial fields.The correction is limited to one low-order pressure direction and the stated steady, constant-coefficient setting with top/side velocity conditions; other geometries and mixed boundaries need a new derivation.Source: S6.4. Equal-parameter… · pp. 61–62

S7. Complete-method validation across MMS cases and permeabilities

  • The complete-method validation covers two manufactured solutions at three permeabilities, with order-32 integration fixing correction coefficients before reference-error evaluation and order 48 providing an independent check.For MMS2, the known boundary/forcing term reduces analytically to 1.6 and is checked by quadrature at all three permeabilities, rather than fitted to pressure errors.Source: S7. Complete-method validation across… · p. 64
  • All 12 prospective checkpoints had smaller Brinkman p_y relative L2 error after correction; all six low-permeability MMS1 runs had smaller Brinkman-pressure H1 seminorm error on both grids, and all raw/corrected states passed the 14 criteria on each grid.Because correction leaves ∂x p_B unchanged, the lower gradient-error seminorm also implies lower L2 error in ∂y p_B.Source: S7. Complete-method validation across… · p. 65

S7.1. Low-permeability MMS1: fine-grid and momentum diagnostics

  • All six additional low-permeability MMS1 runs passed the 14-gate rule for raw and corrected states, as well as the order-48 and invariance checks.The reported additions are fine-grid pressure checks and main/fine momentum diagnostics.Source: S7.1. Low-permeability MMS1: fine-grid… · p. 66

S8. Pressure-correction controls for benchmarks and BD pairs

  • The correction controls are separate from complete-method validation: ten kinematic-hard-trace benchmarks and eighteen hard/soft-BD pairs are evaluated before and after correction, with pressure means retained.The controls use extended MMS grids and component 14-gate diagnostics, while original benchmark acceptance remains on its original grid.Source: S8. Pressure-correction controls for… · pp. 66–67
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