Data-Free Finite-Volume Convolutional Neural Networks for Steady Compressible Euler Flow around a Circular Cylinder

Ryosuke Yano

Study	Spatial formulation	Prescribed information or additional constraints	Reported problem class
Mao et al. [ 5 ]	Euler residual collocation	Initial/boundary states; additional observations for inverse problems	One- and two-dimensional Euler flows, including shock-tube and oblique-shock problems
Jagtap et al. [ 8 ]	Euler residuals, including domain-decomposed networks	Density-gradient and boundary observations; entropy and positivity constraints	Inverse supersonic flows: expansion, oblique shocks, and detached bow shocks
Liu et al. [ 9 ]	Equation-weighted differential residuals	Rankine–Hugoniot jump and global conservation constraints	Burgers equation and one- and two-dimensional Euler discontinuities
Wassing et al. [ 6 ]	Euler residuals with adaptive artificial viscosity	Boundary data; viscosity-reduction and training schedules	Parametric stationary two-dimensional subsonic and supersonic Euler flows
Ferrer-Sánchez et al. [ 10 ]	Gradient-dependent residual weighting	Initial/boundary data; reduced residual weighting near discontinuities	Riemann problems in relativistic hydrodynamics
Chaumet and Giesselmann [ 13 ]	Weak residual norms with learned test functions	Initial/boundary and entropy conditions	Scalar conservation laws and the Euler Sod shock tube

Table 1: Qualitative comparison of neural formulations for shock-containing compressible flows and conservation laws. The rows describe the cited methods and problem classes; numerical accuracy and timing are not ranked across different benchmarks. The present row describes the FV-CNN formulation used here.

Figure 1: FV-CNN architecture and residual evaluation. (a) Nodewise coordinates are Fourier-encoded into 64 channels, passed through six hybrid blocks, and mapped by one linear head to four raw outputs. Decoding gives density, two Cartesian momenta, and pressure; velocity, energy, and temperature are derived. (b) A depthwise radial convolution, the sigmoid linear unit (SiLU), and channel mixing precede a residual addition and layer normalization (LN, labeled LayerNorm); an azimuthal convolution and SiLU precede a second residual addition and LayerNorm. The kernel sizes are

K_{r}=15

and

K_{\theta}=(3,1)

; both convolutions use replicate padding. The indices

(j,i)

denote azimuthal and radial directions. (c) Shared-face fluxes define the FV residual and training objective:

A_{K}

is cell area,

\ell_{f}

is face length, and

\sigma_{Kf}

accounts for orientation. AD gives parameter gradients. Flux modifications, residual weights and physical priors are specified in the text.

$Ma_{\infty}$	$2$	$3$	$5$	$7$	$9$	$11$	$13$	$15$
$d$	$1.5$	$0.6$	$0.6$	$0.6$	$0.6$	$0.6$	$0.6$	$0.6$
$R_{\rm side}$	$4$	$2$	$2$	$2$	$2$	$2$	$2$	$2$
$x_{\rm fix}$	$-1.3$	$-1.0$	$-0.815$	$-0.792$	$-0.784$	$-0.781$	$-0.778$	$-0.777$
$b$ (implied)	$0.1797$	$0.1757$	$0.0824$	$0.0797$	$0.0795$	$0.0804$	$0.0796$	$0.0800$
$\lambda_{P}$	$1$	$60$	$120$	$200$	$50$	$20$	$20$	$50$
$\lambda_{0}$	$2\times 10^{6}$	$10^{4}$	$10^{4}$	$10^{4}$	$2000$	$10^{4}$	$7500$	$10^{4}$

Table 2: Conditions of the main FV-CNN computations at all eight Mach numbers. The columns identify the runs used in the flow-field and stagnation-line comparisons. Initial training and later refinement are distinguished within this single table; common settings are specified below.

Factor	Baseline	Tested alternatives
Dissipation $\alpha_{0}=\alpha_{\rm f}$	$1.5$	$1.0,\ 2.0$
TV coefficient $\lambda_{\rm TV}$	$0.1$	$0,\ 0.01$
PDE coefficient $\lambda_{P}$	$50$	$25,\ 100$
Stagnation coefficient $\lambda_{0}$	$10000$	$0,\ 5000,\ 20000$
Fixing boundary $x_{\rm fix}$	$-0.777$	$-0.9,\ -1.0$
Fixing coefficient $\lambda_{m}$	$1000$	$0$

Table 3:

\mathrm{Ma}=15

sensitivity experiments. Each row varies the named factor against the same baseline while retaining the other settings. All use 15000 AdamW updates from the shared seed-42 untrained state; no L-BFGS or entropy cleanup is used. Mach-scaling controls are listed separately in Table 4 .

$Ma_{\infty}$	Scaling	$w_{m}$	$w_{E}$	$\lambda_{P}$	L-BFGS closure calls
$2$	on	$4$	$16$	$1$	$10002$
$2$	off	$1$	$1$	$1$	$10002$
$3$	on	$1/9$	$1/81$	$60$	$0$
$3$	off	$1$	$1$	$1$	$0$
$15$	on	$1/225$	$1/50625$	$50$	$0$
$15$	off	$1$	$1$	$50$	$0$

Table 4: Separate conditions for Mach-scaling comparisons. The coefficients

w_{m},w_{E}

multiply squared residuals. All rows have 15000 initial AdamW updates and no entropy cleanup. The

\mathrm{Ma}=15

rows use the sensitivity baseline, not the final production field in Table 2 .

Figure 2: Flow-field comparison at

Ma_{\infty}=2

. Top: contours of

\rho

,

p

,

T/T_{\infty}

, and local Mach number

M

, with AD-CNN in the first row and FV-CNN in the second. Solid neural and dashed CFD contours share colors and levels. Bottom: stagnation-line profiles for CFD (black dashed), FV-CNN (blue solid), and AD-CNN (orange dash-dot); circles show the imposed analytical density and pressure targets. Temperature uses the common freestream normalization, and every profile axis retains vertical margins.

Figure 3: Flow-field comparison at

Ma_{\infty}=3

. Top: contours of

\rho

,

p

,

T/T_{\infty}

, and local Mach number

M

, with AD-CNN in the first row and FV-CNN in the second. Solid neural and dashed CFD contours share colors and levels. Bottom: stagnation-line profiles for CFD (black dashed), FV-CNN (blue solid), and AD-CNN (orange dash-dot); circles show the imposed analytical density and pressure targets. Temperature uses the common freestream normalization, and every profile axis retains vertical margins.

Figure 4: Flow-field comparison at

Ma_{\infty}=5

. Top: contours of

\rho

,

p

,

T/T_{\infty}

, and local Mach number

M

, with AD-CNN in the first row and FV-CNN in the second. Solid neural and dashed CFD contours share colors and levels. Bottom: stagnation-line profiles for CFD (black dashed), FV-CNN (blue solid), and AD-CNN (orange dash-dot); circles show the imposed analytical density and pressure targets. Temperature uses the common freestream normalization, and every profile axis retains vertical margins.

Figure 5: Flow-field comparison at

Ma_{\infty}=7

. Top: contours of

\rho

,

p

,

T/T_{\infty}

, and local Mach number

M

, with AD-CNN in the first row and FV-CNN in the second. Solid neural and dashed CFD contours share colors and levels. Bottom: stagnation-line profiles for CFD (black dashed), FV-CNN (blue solid), and AD-CNN (orange dash-dot); circles show the imposed analytical density and pressure targets. Temperature uses the common freestream normalization, and every profile axis retains vertical margins.

Figure 6: Flow-field comparison at

Ma_{\infty}=9

. Top: contours of

\rho

,

p

,

T/T_{\infty}

, and local Mach number

M

, with AD-CNN in the first row and FV-CNN in the second. Solid neural and dashed CFD contours share colors and levels. Bottom: stagnation-line profiles for CFD (black dashed), FV-CNN (blue solid), and AD-CNN (orange dash-dot); circles show the imposed analytical density and pressure targets. Temperature uses the common freestream normalization, and every profile axis retains vertical margins.

Figure 7: Flow-field comparison at

Ma_{\infty}=11

. Top: contours of

\rho

,

p

,

T/T_{\infty}

, and local Mach number

M

, with AD-CNN in the first row and FV-CNN in the second. Solid neural and dashed CFD contours share colors and levels. Bottom: stagnation-line profiles for CFD (black dashed), FV-CNN (blue solid), and AD-CNN (orange dash-dot); circles show the imposed analytical density and pressure targets. Temperature uses the common freestream normalization, and every profile axis retains vertical margins.

Figure 8: Flow-field comparison at

Ma_{\infty}=13

. Top: contours of

\rho

,

p

,

T/T_{\infty}

, and local Mach number

M

, with AD-CNN in the first row and FV-CNN in the second. Solid neural and dashed CFD contours share colors and levels. Bottom: stagnation-line profiles for CFD (black dashed), FV-CNN (blue solid), and AD-CNN (orange dash-dot); circles show the imposed analytical density and pressure targets. Temperature uses the common freestream normalization, and every profile axis retains vertical margins.

Figure 9: Flow-field comparison at

Ma_{\infty}=15

. Top: contours of

\rho

,

p

,

T/T_{\infty}

, and local Mach number

M

, with AD-CNN in the first row and FV-CNN in the second. Solid neural and dashed CFD contours share colors and levels. Bottom: stagnation-line profiles for CFD (black dashed), FV-CNN (blue solid), and AD-CNN (orange dash-dot); circles show the imposed analytical density and pressure targets. Temperature uses the common freestream normalization, and every profile axis retains vertical margins.

$Ma_{\infty}$	$\rho$	$p$	$T/T_{\infty}$	$u$	$v$
2	9.2 / 4.9 / 55.4	12.8 / 7.5 / 46.4	4.0 / 2.6 / 21.1	5.5	13.3
3	10.1 / 2.1 / 42.1	11.9 / 1.6 / 41.7	9.4 / 2.4 / 41.9	8.0	21.7
5	5.4 / 1.0 / 28.6	6.2 / 1.4 / 33.5	6.1 / 1.3 / 61.0	2.5	8.2
7	18.4 / 4.0 / 57.6	19.7 / 1.7 / 54.5	20.8 / 3.1 / 71.9	8.3	34.9
9	8.5 / 1.3 / 48.4	9.5 / 1.3 / 54.1	9.8 / 1.5 / 42.0	3.3	15.1
11	6.3 / 1.4 / 38.1	7.7 / 1.3 / 42.0	11.2 / 0.5 / 73.5	3.2	10.7

Table 5: Main FV-CNN discrepancies on all

96\times 160

coordinate-matched CFD nodes. Entries for

\rho,p,T

give

100e_{2}

/ five-percent trimmed

100e_{2}

/

100e_{\infty}

; the two velocity columns give full-domain

100e_{2}

. All entries are percentages.

\mathrm{Ma}=2

uses its distinct larger physical domain.

Figure 10: Area-weighted field discrepancies on common physical support. Each panel compares the same nodes and CFD reference for the two neural fields. At

\mathrm{Ma}=2

only, the common support contains 10,579 nodes inside the smaller historical AD-CNN domain; Table 5 instead uses all 15,360 native FV-CNN nodes. At the other Mach numbers the supports coincide. Temperature uses

T/T_{\infty}

; its relative norm equals that for

T

because the normalization is common. These historical comparisons include the velocity conversion in Eq. ( 3.1 ).

Figure 11: Common physical support and CFD-based regions for the operator-consistent Euler diagnostic. The solid curve is

r_{s}^{\rm CFD}

and the dashed curves are radial offsets

\pm 0.10D

. Gray denotes excluded nodes, including the unsupported outer part of the

\mathrm{Ma}=2

historical AD-CNN domain and three boundary layers. Upper panels: reference-defined upstream, band, and downstream classes. Lower panels: the additional common mask excludes all three located-front bands; blue nodes are retained for the solid curves in Fig. 12 , yellow nodes are excluded by at least one front band, and gray nodes are outside the common interior support. The three colored curves mark the located centers. Within each diagnostic, exactly the same nodes are used for all methods.

Figure 12: Operator-consistent normalized physical-flux residuals on common nodes. Dashed/circles: outside the CFD-centered band. Solid/squares: outside the union of bands centered on all three fields. Every band has radial half-width

0.10D

; the same mask is applied to all methods within each comparison. The vertical axes are logarithmic. The diagnostic differs from the CFD evolution operator, so its finite value is not a zero-error benchmark.

Figure 13: Complete-boundary physical-flux imbalance over 15 curved wedges. The plotted

\mathcal{B}_{c}

is

\mathcal{B}_{c}^{a}

in Eq. ( 3.6 ) for the indicated band-center choice. Dashed/circles use the common CFD-centered volumes at

h=0.10D

; solid/squares use each field’s own center at that width. Bars show the minimum and maximum self-centered values across the seven tested widths

0.06\leq h/D\leq 0.12

, not statistical uncertainty. The self-centered volumes differ between fields. All four faces are included and the axes are logarithmic.

$h/D$	CFD	FV-CNN	AD-CNN
Common CFD-centered bands
0.06	0.0052 / 0.0048	0.1296 / 0.1305	0.2007 / 0.2007
0.07	0.0051 / 0.0048	0.1078 / 0.1091	0.2175 / 0.2174
0.08	0.0050 / 0.0047	0.0773 / 0.0788	0.2305 / 0.2305
0.09	0.0048 / 0.0046	0.0435 / 0.0447	0.2400 / 0.2399
0.10	0.0047 / 0.0045	0.0193 / 0.0208	0.2464 / 0.2461

Table 6:

\mathrm{Ma}=15

finite-band dependence on center and radial half-width. Entries are mass / energy RMS from 15 complete wedges; these dimensionless ratios are not percentages. Both center choices use the same locator and quadrature.

Figure 14: Complementary diagnostics of the adopted fields: angular RMS shock displacement from CFD; mean local density 10–90% transition width; entropy increase across two CFD-centered stagnation-line sample points; and downstream total-enthalpy RMS. Lengths are normalized by

D

. The width and sampling definitions are given in the text.

Figure 15: Mach-scaling comparisons on the stagnation streamline. Rows:

\mathrm{Ma}=2,3,15

; columns:

\rho,u,T/T_{\infty},p

. Gray dashed: CFD; blue solid: scaling on; orange dash-dot: scaling off.

\mathrm{Ma}=2

uses its own physical domain; its common seed-42 initialization is verified by reconstruction from the recorded code. GPU trajectory repeatability remains unmeasured.

\mathrm{Ma}=3

also changes

\lambda_{P}

from 60 to 1.

\mathrm{Ma}=15

changes only Mach weighting within the common-initialization sensitivity protocol. No entropy continuation is used in these scaling tests.

$Ma_{\infty}$	Scaling	$\rho$	$u$	$T/T_{\infty}$	$p$
2	on	4.14	5.26	3.14	4.78
2	off	17.65	17.89	10.67	22.44
3	on	3.54	3.90	3.12	4.01
3	off	41.22	51.36	32.52	50.28
15	on	13.87	8.73	11.61	16.41
15	off	58.05	49.88	89.61	94.97

Table 7: Stagnation-line discrepancies

100e_{2,q}^{\rm SSL}

with CFD for scaling on/off, in percent. The physical-

x

trapezoidal norms use the domain of each Mach case. These are profile norms, distinct from the full-field main-result norms. Conditions and comparability limits are specified in the text and the separate scaling-settings table.

Figure 16: Complete

\mathrm{Ma}=15

initialization experiment: five area-weighted full-domain discrepancies and the prespecified density-dispersion comparison. Open circles: 15000 AdamW updates; filled squares: the subsequent 10000 entropy-penalized updates. All ten new seeds and the unchanged archived seed 42 (shaded) are shown. Diamonds and thick whiskers give the eleven-run mean and sample standard deviation; thin capped whiskers span the minimum and maximum. Dashed lines are historical single-run AD-CNN values, not AD ensemble statistics. Panel (f) compares the final density dispersion with the historical adopted-field gap, in percentage points.

Figure 17: Main FV-CNN loss histories. Blue curves show the initial composite total; orange solid curves show the total after adding the upstream entropy penalty in

\mathrm{Ma}=7

–15. The

\mathrm{Ma}=5

black curve shows the further base-objective stage and its orange dashed curve shows only the base part during entropy continuation, not a combined entropy-stage objective. Vertical dotted lines mark changes of stage. Curves on opposite sides of a stage change are not connected. The abscissa preserves the recorded labels: initial losses have zero-based labels 0–14,900, whereas the inherited entropy logs have labels 15,001–25,000. The last initial point is not an evaluation of the final saved 15,000-update field. The

\mathrm{Ma}=2

L-BFGS history has a separate counter in Fig. 18 (a).

	Initial	Further base	Entropy	L-BFGS
$Ma_{\infty}$	AdamW	AdamW	continuation	iterations / calls
2	15,000	0	0	8,412 / 10,002
3	15,000	0	0	0 / 0
5	15,000	5,000	2,500	0 / 0
7	15,000	0	10,000	0 / 0
9	15,000	0	10,000	0 / 0

Table 8: Executed optimizer stages for the main FV-CNN fields. Counts in the AdamW columns describe progress through the saved field used in the main comparison. The

\mathrm{Ma}=5

entropy continuation uses its own optimizer and rollback mechanism; rejected attempts can add work beyond its progress counter.

Figure 18: Stage-specific optimization diagnostics. (a)

\mathrm{Ma}=2

accepted-state total and PDE component against closure evaluations; trial line-search losses are not shown. The PDE component already contains the case’s Mach-component weights and is not an independent physical Euler error. (b) Unweighted upstream entropy loss during the five fixed-coefficient continuations. The horizontal coordinate is the recorded global label minus 15,000. Neither panel is an error history against CFD.

$Ma_{\infty}$	Updates	$100e_{\rho}$	$100e_{p}$	$100R_{Z}$	Shock RMS/ $D$	$\mathcal{B}_{\rho}$
5	15,000	5.86	6.56	6.32	0.00345	0.00411
5	20,000	5.43	6.26	3.91	0.00359	0.00436
5	22,500	5.43	6.25	3.16	0.00352	0.00436
7	15,000	17.30	19.57	33.83	0.02957	0.00406
7	25,000	18.36	19.70	2.13	0.02985	0.00315
9	15,000	8.93	9.87	24.90	0.00856	0.00233

Table 9: Evaluation of the recorded continuation stages on identical CFD nodes and fixed physical masks.

e_{\rho}=e_{2,\rho}

and

e_{p}=e_{2,p}

are the area-weighted relative field discrepancies of Eq. ( 2.48 );

R_{Z}

is area-weighted RMS of

Z-1

in the fixed reference-defined upstream region. The last column is self-centered mass-balance RMS at

h=0.10D

.

Record	Time (minutes)	Scope
Historical AD-CNN, author estimate	$\approx 210$	Initial 15,000 updates
FV-CNN, author estimate	$\approx 16$	Initial 15,000 updates
FV-CNN $\mathrm{Ma}=2$ L-BFGS, author estimate	$\approx 30$	Nominal budget 10,000; counts in Table 8
MLP, contextual author estimate	$\approx 16$	Incomplete run metadata
FV sensitivity baseline, archived timer	$19.01$	Complete launched process
FV sensitivity suite, archived timers	$18.79$ – $26.10$	20 distinct factor runs

Table 10: Timing evidence and its recorded scope. Author estimates have no archived synchronized trace. Process timers refer to separate sensitivity runs and include setup and output. None of these entries is a matched-accuracy comparison.

$Ma_{\infty}$	Fine	Coarse	Coarse FV/AD ratio
2	4/4	16/16	0.290–0.960
3	4/4	16/16	0.199–0.398
5	4/4	16/16	0.073–0.118
7	4/4	16/16	0.061–0.083
9	4/4	16/16	0.060–0.140
11	4/4	16/16	0.054–0.106

Case/seed	$x_{\rm axis}/D$	$100e_{2,\rho}$	$\mathrm{RMS}_{\rm shock}/D$	$w_{10-90}/D$	Mass	$x$ -mom.	$y$ -mom.	Energy
CFD	$-0.70063$	$0.00$	$0.00000$	$0.01084$	$0.00465$	$0.00319$	$0.00298$	$0.00449$
AD-CNN	$-0.69780$	$20.69$	$0.00552$	$0.10620$	$0.26043$	$0.14389$	$0.07687$	$0.26035$
42	$-0.69154$	$17.29$	$0.02525$	$0.02543$	$0.00238$	$0.00475$	$0.00217$	$0.00755$
1	$-0.71096$	$12.46$	$0.00942$	$0.02439$	$0.00283$	$0.00619$	$0.00775$	$0.00858$
2	$-0.75044$	$42.57$	$0.07584$	$0.02381$	$0.00443$	$0.00886$	$0.01234$	$0.01254$
3	$-0.68802$	$17.45$	$0.02275$	$0.02540$	$0.00256$	$0.00509$	$0.00271$	$0.00757$

Data-Free Finite-Volume Convolutional Neural Networks for Steady Compressible Euler Flow around a Circular Cylinder

Abstract

Figures & tables

Appendix

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