Neural Eigenfluid Bases: Spectral Objectives, Gauge, and Topology

Baiming Zhang; Shiying Xiong; Zhecheng Wang

physics.flu-dynOct 24, 2026

Neural Eigenfluid Bases: Spectral Objectives, Gauge, and Topology

Authors: Baiming Zhang, Shiying Xiong, Zhecheng Wang

Organizations: Zhejiang University · University of Toronto

Abstract

Reduced-order models simplify fluid simulation by representing complex flows with a small number of spatial modes. Building on Laplacian Eigenfluids, we introduce Neural Eigenfluid Bases (NEB), a neural representation whose basis functions can be evaluated anywhere in space. We enforce incompressibility and impermeable boundaries directly through differential lifts and potential boundary conditions. Geometry-local orthogonalization and Rayleigh minimization then learn an ordered basis. We connect learning in potential and velocity spaces, and derive a spectral-variance relation explaining why these objectives weight approximate modes differently, even though their exact positive spectra agree. Gauge fixing reduces the number of spatial derivatives needed for potential learning, while a Dirichlet energy loss learns independent circulation fields around obstacles. With 1024 modes, both objectives keep velocity errors below 1% in periodic vortex merger. For random flows around obstacles, potential-based learning captures more initial energy and gives lower final errors than numerical Laplacian bases of the same size. The reduced state is about a thousand times smaller than the full reference model. Three-dimensional tests separate spatial reconstruction errors from errors in coefficient evolution. Precomputed Galerkin operators allow millisecond-scale updates when the same basis is reused across simulations. This provides a physically constrained basis for continuous flow reconstruction and efficient reduced-order simulation.

Figures & tables

Figure 1: Shape-dependent training and coefficient-only inference. (a) Geometry parameters vary obstacle position, scale and orientation within a fixed rounded-square outer boundary. A hard boundary envelope acts on geometry-conditioned potentials. Paired, oppositely oriented red and blue marks depict the antisymmetric tensor in the Neural Conservation Laws (NCL) construction ( Richter-Powell et al., 2022 ) ; its row-wise divergence generates divergence-free velocities. Gauge fixing and geometry-local causal orthogonalization produce normalized candidates within the divergence-free, impermeable trial space; the Rayleigh contours depict their energy minimization. Training yields the parameter-to-basis neural map. (b) For a selected geometry, the map supplies positive modes, completed by harmonic circulation fields. Initial projection and geometry-specific preassembled mass, linear and quadratic operators support coefficient-only ROM stepping; spatial fields are reconstructed at selected times. The schematic uses $M$ for the velocity mass matrix, $A$ for the reduced linear operator, $Q$ for the quadratic advection tensor, and $t_{0},t_{1},t_{2}$ for selected output times.

Figure 2: Harmonic circulation completes the flow representation. (a) A base flow combined with positive modes gives an incomplete, zero-circulation description of a lifting airfoil, with an incorrect sharp-trailing-edge flow. (b) Adding harmonic circulation permits smooth outflow satisfying the Kutta condition and generates lift in the presence of the base flow. Streamlines schematically illustrate the decomposition in Eq. ( 19 ) .

Case	Comparison	Validation
Periodic	Learned spectral and kinematic bases; same-capacity Laplacian	Neural eigenfluid and potential-space feasibility; autonomous accuracy and decay.
Multi-hole	Random small-scale fields and large-scale vortices; fixed-budget allocation and lift conditioning	Topology-aware circulation completion, energy capture and harmonic conditioning.
Metaball	Velocity-basis capacity; full-grid and assembled coefficient operators	3D feasibility, capacity-dependent representation and coefficient-space acceleration.

Table 1: Validation goals and controlled comparisons.

Basis	Velocity MSE
Laplacian	$9.737\times 10^{-10}$
Velocity	$8.882\times 10^{-9}$
Potential	$1.619\times 10^{-7}$
NKB	$2.823\times 10^{-3}$

Table 2: Periodic vortex merger,

d=1.35

T=18\pi

K=1024

. Velocity mean squared error (MSE) averages squared velocity-component errors over all 241 saved

128^{2}

observation-grid states. Learned entries then average three seeds. Relative errors are visualized in Figure 4 ; spectral training costs appear in Table 22 , and NKB costs in Table 24 . NKB uses the 2000-update adaptation in Appendix G .

Figure 3: Potential: periodic vortex merger. Panels show the $K=1024$ , seed-0 Potential trajectory with vortex separation $d=1.35$ , from left to right at $t=0,9\pi,18\pi$ . Surface relief encodes smoothed vorticity magnitude on a common scale; white accents trace vortex structure. Height is a display coordinate for the two-dimensional flow.

Figure 4: Autonomous accuracy and decay in vortex merger. All panels use

K=1024

d=1.35

T=18\pi

. (a) Relative velocity (top) and vorticity (bottom) errors against Numerical GT, normalized by the corresponding time-dependent observation-grid

L^{2}

norm. Learned curves show three-seed means with one sample standard deviation (SD), on a symmetric-log scale with a linear interval below

0.01\%

. Open and filled markers indicate initial and final states. (b) Kinetic energy,

E=\frac{1}{2}\langle|u|^{2}\rangle

. (c) Enstrophy,

Z=\frac{1}{2}\langle\omega^{2}\rangle

, where

\omega=\partial_{x}u_{y}-\partial_{y}u_{x}

is scalar vorticity,

u_{x},u_{y}

are velocity components, and

\langle\cdot\rangle

denotes the spatial mean. All five methods appear in the upper decay panels, using seed-0 learned trajectories and common initial GT normalization. Distinct line styles and sampled markers distinguish nearly overlapping curves. The lower panels of (b,c) show

100(E-E_{\rm GT})/E_{\rm GT}

and

100(Z-Z_{\rm GT})/Z_{\rm GT}

for Laplacian, velocity and potential bases. Curves use all 241 saved

128^{2}

states from

256^{2}

operator quadrature and maximum step

0.005

Figure 5: Initial energy captured by the original

K=1024

representations. Circles and squares denote the two predetermined physical fields in each topology; vertical ticks denote their mean. The annotations give mean captured-energy percentages and state-coordinate compression

N_{\rm ref}/K

, where

N_{\rm ref}

is the number of independent coordinates of the reference state. The complete set of learned branches, raw/scaled harmonic coordinates, numerical controls, and NKBs use the same analytic initial fields.

Holes	Basis	$\Delta\eta$	$\Delta e(1)$
1	Laplacian	+0.004	+0.020
1	Potential	-0.045	+0.092
2	Laplacian	+0.018	+0.087
2	Potential	+0.102	-0.243
3	Laplacian	-0.014	+0.021
3	Potential	+0.305	+0.942

Table 3: Paired changes under harmonic allocation on the six qualified random fields at total

K=1024

. Each entry averages the two fixed fields in its topology. Initial capture

\eta

uses the analytic-target full-mass projection; final velocity error uses the qualified numerical trajectory. Gains are

\Delta\eta=\eta_{PH}-\eta_{P}

and

\Delta e(1)=e_{P}(1)-e_{PH}(1)

in percentage points, so positive values favor completion. Absolute capture and trajectories are shown in Figures 5 and 12 .

Holes	Numerical $P$	Learned $P$
1	32.542	32.124
2	34.197	33.963
3	36.685	36.707

Table 4: Positive-only controls for the random large-vortex ensemble: final relative velocity error (%), total

K=1024

T=1

, averaged over 20 initial states. Learned positive modes use potential training, seed 1. Completed bases and raw/scaled lift comparisons are visualized in Figure 17 .

Figure 6: Random large-vortex flow at $K=1024$ . Rows: one-, two-, and three-hole domains. Columns: numerical reference, Laplacian, potential- and velocity-trained bases at $t=1$ , seed 1, state 00. Reduced bases have $1024-h$ positive and $h$ harmonic coordinates; the learned objectives share raw $H_{1}$ and scaled $H_{2},H_{3}$ . Errors use the common level-2 reference. Overhead views share height and foam mappings within each geometry ( Appendix H ).

Figure 7: Positive spectral modes across obstacle topologies. Panels (a–c) show one-, two-, and three-hole domains. Within each panel, rows compare Laplacian, potential-trained, and velocity-trained fields; columns show modes 1–8. Learned fields follow the positive-mode selection order in the total- $32$ continuous piecewise-linear finite-element (P1) basis, whose $h$ harmonic coordinates are separate. The shared color scale shows area-centered streamfunction divided by its per-mode maximum magnitude. Signs and rotations within repeated eigenspaces preserve the span.

Figure 8: Three-dimensional autonomous flow at $K=1024$ . Rows show Numerical GT, Laplacian, Velocity, and NKB; columns show $t=0,0.60,1.16$ . Each time pairs speed $|u|$ (blue) with pointwise vector error $|u-u_{\rm GT}|$ (orange–red) on the offset $z=0.31640625$ slice, which exposes the fused Metaball silhouette. Filled contours interpolate the saved nodal values and are clipped to the original geometry’s smooth boundary. Speed, error, and in-plane arrow lengths use separate fixed scales across all methods and times. Percentages in the upper-left corner of the vector-error panels denote full-volume relative velocity error, evaluated on all $1.6\,\mathrm{M}$ fluid nodes. Here $\Delta t$ is the integration time step; ROM trajectories use $\Delta t=0.000625$ and the common NS reference uses $\Delta t=0.0025$ .

Figure 9: Spatial representation and autonomous evolution in the three-dimensional Metaball domain. (a) Frozen-basis reconstruction on 20 independently randomized large-eddy initial fields, with 400 paired measurements across four methods and

K=32,64,128,256,512

. Shaded rectangles show mean

\pm

sample standard deviation (

n=20

\mathrm{ddof}=1

); whiskers show minima and maxima. The

K=1024

randomized-field results are in Appendix D.2 . (b) A separate, fixed initial field is used for the autonomous-flow comparison of Laplacian, Velocity, and NKB. Initial full-mass projection error (open circles,

t=0

) and final autonomous error (filled circles,

T=1.16

) at

K=128,256,512,1024

, with arrows connecting the two times. The three methods are offset horizontally in this left-to-right order within each capacity. All capacities use the same method colors. Both panels report relative velocity

L^{2}

errors in percent using all retained modes and all

1.6\,\mathrm{M}

fluid nodes. Autonomous trajectories store 30 states, with ROM step

0.000625

and common NS reference step

0.0025

; training costs appear in Tables 10 and 24 and the error decomposition in Appendix E.5 .

Appendix figures & tables37 assets

Supplementary material from the paper’s appendix.

Appendix

$K$	Matched Potential-FG	Laplacian
32	$99.928\pm 0.045$	$65.924\pm 4.039$
64	$98.690\pm 1.315$	$49.278\pm 4.782$
128	$95.348\pm 3.671$	$34.595\pm 3.852$
256	$91.001\pm 4.096$	$26.511\pm 3.946$

Appendix

Table 5: Matched Potential-FG reconstruction on the same 20 independent random large-eddy fields, after 2000 epochs and 80 scheduled validations. Relative velocity

L^{2}

errors are reported as mean

\pm

sample standard deviation (%); all

1.6\,\mathrm{M}

points and all

K

modes are retained. The Laplacian entries use the same fields and capacity.

$K$	Potential	Velocity	Laplacian	Potential-AL/FG
32	0.843112	0.839273	0.845281	0.917776 (AL)
64	0.515772	0.622647	0.521114	0.878704 (AL)
128	0.343194	0.400527	0.348509	0.869497 (AL)
256	0.235699	0.281845	0.242727	0.994937 (FG)
512	0.201610	0.209542	0.196530	—
1024	—	0.170499	0.178991	—

Appendix

Table 6: Initial relative velocity

L^{2}

errors on the original common initial field (

n=1

per method and capacity), using all retained modes and all

1.6\,\mathrm{M}

fluid nodes. Potential completes 2000 epochs and 80 scheduled validations through

K=512

. The controls use Potential-AL at

K=32,64,128

and Potential-FG at

K=256

. The

K=1024

Potential training run completes 25 optimizer updates; its update count is reported separately from the full-training representation results.

Method	Size	Harm.	Wall tan.	Forward	Reverse	Setup	Assess.
Fourier–MFS	$193^{3}$	0.0177	0.6904	49.07	138.60	$54.57^{\mathrm{R}}$	$23.82^{\mathrm{R}}$
Fourier–MFS, reuse	$193^{3}$	0.0177	0.6904	48.28	136.12	$62.86^{\mathrm{R}}$	$23.32^{\mathrm{R}}$
Spline 12	3,106	0.0181	0.0012	6.63	6.98	1.98	5.90
Spline 20	10,648	0.0015	0.0012	15.40	16.60	4.75	5.91
Spline 24 + two-level	17,086	0.0036	0.0012	71.00	78.96	11.69	5.84
Volume FMM 32	$32^{3}$	0.0242	4.8727	12.15	13.43	22.99	61.76

Appendix

Table 7: Gauge-solver accuracy and cost at the fixed checkpoint. Figure 10 plots the Poisson residual

\epsilon_{P}

. Harmonic overlap and tangential trace are percentages; all four timing columns are seconds. Size denotes grid size or retained spline degrees of freedom. Setup and assessment are excluded from forward/reverse. Superscript R denotes a synchronized measurement of the complete workload; forward/reverse columns exclude setup and assessment.

Arithmetic	$\epsilon_{P}$ (%)	Source forward (s)
Training arithmetic	7.989593	37.24
FP32 source, FP64 FFT	7.989595	37.24
FP64 source and FFT	7.989582	266.41

Appendix

Table 8: Floating-point precision on the same

193^{3}

grid. The first two rows use identical source samples. Source time measures one all-32 forward pass, including export.

Figure 10: Gauge-solver accuracy and cost. Left: all-32 forward-plus-reverse time versus independent Poisson residual. Right: per-column residuals for Fourier–MFS and the finest tested spline and volume spaces. All methods evaluate the same fixed neural checkpoint.

Figure 11: Accuracy, energy, and cost of coefficient-space dynamics. Red and blue denote

K=512

and

1024

. (a) Autonomous velocity errors against the shared refined NS reference on all

1.6\,\mathrm{M}

fluid nodes; the dotted gray curve assesses

K=512

against the original reference. (b) Fluid kinetic energy. (c) Synchronized NS/ROM online integration, offline tensor assembly, and three-frame output times. Both 2000-epoch models use full-mass initial projection, all retained coordinates, and

\Delta t=0.000625

. Time-step checks and online speed ratios are given in the adjacent text.

Method	Relative error (%)
Laplacian	$23.00\pm 4.11$
Velocity	$23.03\pm 4.46$
NKB	$27.77\pm 1.82$

Appendix

Table 9: Randomized-field representation at

K=1024

. Each frozen basis projects the same 20 independent fields used in Figure 9 (a), with all

1.6\,\mathrm{M}

fluid points and all retained modes. Entries are relative velocity

L^{2}

error in percent, reported as mean

\pm

sample standard deviation.

$K$	Method	Normal training	State recovery	Boundary to all basis
32	Laplacian	—	—	3467.56
32	Velocity	3042.38	0.00	267.91
32	Potential	8170.08	200.31	255.50
32	Potential-AL	6316.24	0.00	259.52
32	Potential-FG (matched)	7332.09	0.00	291.39
64	Laplacian	—	—	2545.51

Appendix

Table 10: Three-dimensional preparation costs in seconds. Training totals cover 2000 updates, scheduled validation and checkpoint I/O. Basis generation runs from the original parameterized boundary to all

1.6\,\mathrm{M}

points and all

K

fields, including streamed-output I/O; Laplacian also includes coefficient construction, interstage waits, setup, paging and background load. A dash means that the stage is not applicable. Superscript

*

marks an estimate, not a completed run; projected runs assume uninterrupted execution. “incl.” means recovery is already included in the training total and was not timed separately;

\dagger

marks these inclusive counters. Estimates and the measured short

K=1024

endpoint are explained below. Matched Potential-FG rows use the same training protocol.

$K$	Geometry	Learned	Numerical
128	Training geometry, independent points	.398468	.346917
256	Training geometry, independent points	.281388	.241861
256	Unseen geometry 2	.751690	.222570
256	Unseen geometry 3	.744673	.254720
256	Unseen geometry 4	.770144	.230048

Appendix

Table 11: Held-out velocity reconstruction on disjoint 16,384-point assessment sets. Fixed-domain learned checkpoints complete 2000 epochs. Numerical bases use the same capacity, fit and assessment point sets, and target fields. Geometries 2 and 3 use radial Piola transport; geometry 4 uses a globally qualified quadratic approximation of its original smooth boundary and consistent physical finite-element matrices.

Case	Capacity	Reference / assessment	Autonomous horizon
Periodic	1024	$256^{2}$ DNS / $128^{2}$ observations	241 frames, $T=18\pi$
Multi-hole P1	32	Common circulation-aware P1 mesh	51 frames, $T=1$
Multi-hole analytic	1024	Finest level-2 reference, common points	51 frames, $T=1$
Metaball	128–1024	$128^{3}$ , $1.6\,\mathrm{M}$ fluid nodes	30 frames, $T=1.16$

Appendix

Table 12: Primary comparison protocols. DNS denotes direct numerical simulation. The multi-hole P1 and analytic high-capacity stages are separate discretizations.

Holes	Potential $P$	Potential $PH$	Numerical $PH$
1	53.23	40.04	39.76
2	52.99	40.49	40.13
3	54.59	40.49	39.72

Appendix

Table 13: Harmonic completion for random large-vortex flows at total

K=32

. Final relative velocity error (%) over the same 20 initial states per geometry; learned entries average three training seeds. Completed bases replace

h

positive modes with

h

harmonic modes. These are the low-capacity results summarized in Section 3.2 .

Holes	P3 nodes	$N_{\rm ref}$	$K$	$\eta_{P}$ (%)	$\eta_{PH}$ (%)
1	881,793	872,578	32	2.27	2.13
1	881,793	872,578	1024	75.32	75.34
2	669,533	656,077	32	2.37	2.16
2	669,533	656,077	1024	79.66	79.73
3	1,390,768	1,368,343	32	1.79	1.60
3	1,390,768	1,368,343	1024	54.48	54.52

Appendix

Table 14: Initial kinetic-energy capture and reference state dimensions for 20 direct random-streamfunction fields per topology. P3 nodes count the reference mesh;

N_{\rm ref}

counts independent interior scalar coordinates plus

h

circulations after boundary constraints and outer-gauge fixing.

K

is the reduced state dimension, and

\eta_{P},\eta_{PH}

are mean captured-energy percentages for numerical

P_{K}

and

P_{K-h}+H_{h}

. Energy fractions are evaluated per field before averaging. The common assessment measure, basis discretizations, and initial reference qualification are specified in Appendix E.4 .

Holes	$K$	$\overline{Q}_{B,P}$	$\overline{Q}_{B,PH}$
1	32	618.62	581.29
1	1024	641.84	641.99
2	32	485.62	442.04
2	1024	510.41	510.82
3	32	767.13	685.73
3	1024	728.05	728.60

Appendix

Table 15: Derived energy–compression products for the numerical representations in Table 14 . Energy fractions are evaluated per field before averaging; all 20 draws per topology are retained.

Holes	Basis	$\eta_{P}$ (%)	$\eta_{PH}$ (%)	$\Delta\eta$	Improved
1	Laplacian	2.269	2.132	-0.137	2/20
1	Potential	2.252	2.178	-0.075	5/20
1	Velocity	2.269	2.145	-0.124	2/20
2	Laplacian	2.369	2.156	-0.213	3/20
2	Potential	2.370	2.174	-0.197	3/20
2	Velocity	2.345	2.216	-0.129	3/20

Appendix

Table 16: Initial kinetic-energy capture at total

K=32

, in percent.

\Delta\eta

is the mean captured-energy change in percentage points. Positive values favor completion. Learned energy fractions average three seeds within each of the 20 independent fields; the last column counts fields with higher

PH

capture.

Holes	Basis	$\bar{e}_{P}$	$\bar{e}_{PH}$	$\bar{\Delta}$	$\Delta(T)$
1	Laplacian	99.28325	99.29712	-0.01387	-0.01336
1	Potential	99.26077	99.26889	-0.00812	-0.00733
1	Velocity	99.27255	99.29087	-0.01832	-0.01817
2	Laplacian	98.75970	98.79192	-0.03221	-0.02997
2	Potential	98.79390	98.84504	-0.05114	-0.05093
2	Velocity	98.81309	98.91302	-0.09993	-0.10682

Appendix

Table 17: Random-field dynamics for index 0 in each of two domains.

\bar{e}=\int_{0}^{1}e_{u}(t)\,\mathrm{d}t

is evaluated by the trapezoidal rule over 51 frames. Errors are percentages and differences are percentage points. Learned entries average the three training seeds of the same physical field.

Figure 12: All six autonomous random-field comparisons at

K=1024

against qualified references. Each panel is one predetermined physical field, and every curve uses all 51 saved times and the original common mask. The numerical control has the same total coordinate count and the corresponding P or P+H allocation.

Figure 13: Energy projected onto each representation’s explicit harmonic span as a fraction of its instantaneous total energy. The projection measures occupation of each representation’s explicit circulation coordinates.

$K$	Best representation	In-span evolution
128	67.0459	44.5447
256	55.5999	45.6442
512	43.9777	49.7599
1024	35.0357	49.6553

Appendix

Table 18: Final orthogonal error decomposition for the velocity-trained Eigenfluid spans,

T=1.16

. Entries are relative velocity errors (%), normalized by the common full-fluid NS state. Representation and in-span terms add in square, as in Eq. ( 51 ) . The

K=512,1024

entries extend the decomposition using saved full-mass matrices, all 30 reference loads, and autonomous coefficients; their final total errors also agree with direct saved-field differences.

Figure 14: First eight periodic modes. Rows compare analytic Fourier modes and generalized Ritz modes from the potential- and velocity-trained $K=1024$ spans; columns show modes 1–8. Potential Ritz fields use centered potential mass and gradient energy, while velocity Ritz fields use velocity mass and curl energy. Color shows mean-zero streamfunction with per-mode normalization; all 1024 channels contribute to each learned Ritz field.

Figure 15: First eight velocity fields at common $K=32$ . Rows compare Laplacian, Velocity, Potential, and Potential-AL; columns show modes 1–8. All methods use the same $z=0.50390625$ slice and per-mode maximum-magnitude normalization of $u_{z}$ . Laplacian fields follow numerical eigenvalue order; learned fields follow their saved channel order, without a display-only Ritz rotation.

Case	Initial velocity error (%)
Periodic	$2.493\pm 0.138$
One hole	20.760
Two holes	22.566
Three holes	22.201

Appendix

Table 19: Initial full-mass projection errors for the 2D NKB adaptation at

K=1024

(%). Periodic entries are mean

\pm

sample SD over three seeds; multi-hole entries average 20 initial states. Trajectory comparisons are shown in Figure 16 , 3D initial errors in Figure 9 (b), and NKB training costs in Table 24 .

$K$	Mean error (%)	$\kappa(M)$	RHS discrepancy	Time-step difference
128	90.170	$5.51\times 10^{5}$	$2.37\times 10^{-10}$	$9.39\times 10^{-12}$
256	85.572	$5.58\times 10^{6}$	$2.04\times 10^{-9}$	$2.14\times 10^{-11}$
512	78.442	$1.17\times 10^{8}$	$5.33\times 10^{-8}$	$3.86\times 10^{-11}$
1024	66.826	$4.33\times 10^{9}$	$9.92\times 10^{-8}$	$5.00\times 10^{-11}$

Appendix

Table 20: Completed 3D NKB trajectories and numerical checks. Mean error is relative velocity

L^{2}

in percent. Right-hand-side (RHS) discrepancy compares the assembled coefficient RHS with the projected full-grid RHS at

0.2c_{0}

and

c_{0}

, taking the larger relative Euclidean error. The final column is the maximum relative mass-norm trajectory difference between

\Delta t=0.00125

and

0.000625

. These checks measure operator agreement and time-step sensitivity on the fixed

128^{3}

grid.

	Divergence $D$		Boundary normal $B$		Grid correction $P$
$K$	Initial	Final	Initial	Final	Initial	Final
128	0.447	0.328	0.395	0.212	0.401	0.312
256	0.472	0.385	0.431	0.272	0.373	0.519
512	0.430	0.391	0.368	0.222	0.304	0.470
1024	0.364	0.385	0.271	0.175	0.224	0.435

Appendix

Table 21: Initial and final 3D NKB kinematic residuals on the original

128^{3}

grid. All entries are dimensionless ratios, using the definitions above.

Figure 16: NKB and Eigenfluid under matched case assessment. (a) Vortex-merger mean trajectory error over 241 frames,

K=1024

, three learned seeds with one sample SD. (b) Final errors for the three topologies, averaged over 20 states. (c) Metaball capacity comparison against the same original reference. NKB uses the stated 2000-update, common-Galerkin adaptation. Numerical denotes the same-capacity Laplacian control.

Figure 17: Harmonic completion, lift conditioning, and objective choice. One-, two-, and three-hole trajectories use total-

1024

bases and average the same 20 random large-vortex states. Red/blue denote potential/velocity learning, dashed gray numerical completion, dotted red raw lifts, and dash-dot blue the raw-

H_{2}

velocity control. Matched learned bases share raw

H_{1}

, scaled

H_{2}

, and scaled

H_{3}

. Curve endpoints give the final-error comparison; positive-basis training costs appear in Table 22 .

Figure 18: Vortex merger at $K=1024$ . Columns compare Numerical GT, Laplacian, potential-trained, velocity-trained, and NKB bases; rows show $t=0,9\pi,18\pi$ . Learned fields use seed 0 and maximum step $0.005$ . Signed vorticity uses the common GT 99.5th-percentile scale; colorbar extensions indicate saturation. Annotations report velocity error over the full $128^{2}$ observation grid. All methods use the same $d=1.35$ initial velocity and evolve autonomously.

Figure 19: Three-dimensional flow structure. Numerical GT, Laplacian, and velocity-trained Eigenfluid $K=1024$ streamtubes at $t=0,0.60,1.16$ , colored by speed. The transparent shell follows the rounded domain boundary. Camera, seeding, material, and the $[0,0.8]$ speed range are shared. Centerline smoothing is a display operation.

Figure 20: Three-hole evolution at $K=1024$ . Rows compare Numerical GT, Laplacian, potential-trained, and velocity-trained bases; columns show $t=0,0.5,1$ , seed 1, state 00. Each basis contains 1021 positive and three harmonic coordinates; the learned bases share scaled $H_{3}$ . All panels share the height mapping, contour levels, camera, and material. Annotations give full-domain velocity error against the level-2 reference.

Case	$K$	Potential	Velocity
Periodic	1024	$106.428\pm 3.196$	$165.760\pm 1.636$
1 hole	1024	232.61	423.97
2 holes	1024	235.48	376.76
3 holes	1024	233.88	388.71

Appendix

Table 22: Two-dimensional spectral training wall time in minutes for complete 2000-update runs. Periodic entries are mean

\pm

sample SD over three seeds. Topology totals exclude the separately trained harmonic lift shared within each Potential/Velocity pair. Three-dimensional spectral totals are in Table 10 ; NKB totals and components are in Table 24 .

$K$	Fwd.	Rev.	Geometry	Ortho.	Validate	Save	Opt./sample
128	10.52	35.97	25.14	0.21	8.35	4.67	0.22
256	20.85	72.05	25.22	0.62	10.76	9.26	0.22
512	42.17	145.88	25.61	2.20	15.95	1.82	0.35
1024	89.94	298.41	29.88	8.86	28.61	3.77	0.67

Appendix

Table 23: Instrumented 3D Velocity-training components, in minutes, summed over 2000 committed updates and 80 scheduled validations. Fwd./Rev. include analytic derivatives and the reverse pass includes recomputation. The full wall total also includes initialization, progress logging, cleanup, and other uninstrumented overhead; the listed components identify the principal computational stages. Checkpoints are saved every update through

K=256

and every 25 updates at higher capacities.

Case	$K$	Seed	Updates	Validate	Total
Periodic	1024	0	151.75	3.17	159.13
Periodic	1024	1	151.59	3.19	158.98
Periodic	1024	2	151.45	3.17	158.78
1 hole	1024	1	153.04	3.28	160.96
2 holes	1024	1	154.53	3.33	162.52
3 holes	1024	1	159.76	3.52	167.91

Appendix

Table 24: NKB training timers, in seconds. Updates sum the 2000 measured optimization iterations, including sampling, analytic network forward/reverse, losses, Adam, and checkpoint writes within those iterations. Validation sums the 80 scheduled events. Total additionally includes preflight, initialization, and other run overhead. Topology seed is 1 and the 3D seed is 20260823; update totals measure the combined neural forward/reverse and optimization workload.

Case	$K$	Basis	Basis eval.	Projection	Derived	$Q$
3D	512	Velocity	$454.46^{\mathrm{R}}$	18.50	3036.39	3645.51
3D	1024	Velocity	$679.90^{\mathrm{R}}$	41.25	10381.08	32434.41
Periodic s0	1024	Potential	143.21	1.73	incl.	m.f.
Periodic s1	1024	Potential	142.83	1.73	incl.	m.f.
Periodic s2	1024	Potential	143.98	1.63	incl.	m.f.
Periodic s0	1024	Velocity	144.22	1.63	incl.	m.f.

Appendix

Table 25: Offline costs in seconds. Basis gives all-node field-evaluation times; periodic entries include analytic derivatives. Projection includes the recorded reference preparation where bundled. Derived means field/linear-operator preparation, and

Q

denotes accepted assembly of the quadratic tensor

\mathcal{Q}

. The incl. periodic derivative fields are included in basis evaluation, and their nonlinear loads are matrix-free. n.r. marks components without a separate measurement. The

K=512

derived cost covers all 512 modes, basis-cache transposition, and linear-operator preparation. These columns exclude neural training and online integration.

$K$	Method	Execution	Coarse	Fine	Output
128	NKB	Coefficient	3.57	6.84	$22.30^{\mathrm{R}}$
256	NKB	Coefficient	4.90	9.78	$22.26^{\mathrm{R}}$
512	Velocity	Coefficient	3.85	7.03	20.41
512	Laplacian	Grid ROM	119.45	238.71	$0.26^{\mathrm{R}}$
512	NKB	Coefficient	8.29	16.58	7.98
1024	Velocity	Coefficient	24.30	48.59	32.72

Appendix

Table 26: 3D online costs in seconds,

T=1.16

. Coarse/fine steps are

0.00125/0.000625

. ROM output timers cover three selected fields; NS output timers cover its 30 saved fields. Laplacian online timers include the field operations required at every integration step, excluding assessment outside the stepping timer. Repeated NS rows are the measured reference runs for the two capacity experiments; n.a. indicates that NS has no retained-mode count.

Case	Method	$\Delta t$	Time (s)	Timer scope
Periodic s0	Potential	$0.005$	182.62	Integration + coefficient capture
Periodic s1	Potential	$0.005$	182.63	Integration + coefficient capture
Periodic s2	Potential	$0.005$	182.64	Integration + coefficient capture
Periodic s0	Velocity	$0.005$	185.97	Integration + coefficient capture
Periodic s1	Velocity	$0.005$	183.62	Integration + coefficient capture
Periodic s2	Velocity	$0.005$	181.99	Integration + coefficient capture

Appendix

Table 27: 2D measured execution intervals in seconds,

K=1024

. Each periodic method–seed pair is listed once. Periodic runs reach

T=18\pi

with 241 saved fields; topology runs reach

T=1

with 20 initial states. Periodic integration timers include synchronized coefficient capture and exclude field reconstruction and file writing. The matched potential, velocity and NKB runs use

256^{2}

operator points and

128^{2}

observations. Topology entries time their original 20-state batches.

Stage	1 hole	2 holes	3 holes
Boundary fields and assessment	60.05	70.07	70.04
Initial projection	20.01	40.03	40.02
Linear/operator preparation	874.27	977.54	978.13
Nonlinear tensor	3484.71	3504.59	3501.97
ROM execution	520.06	610.08	610.08
Reference fields and error assessment	570.13	2250.30	2280.28

Appendix

Table 28: High-capacity Velocity pipeline wall times, in seconds. Process-stage intervals are grouped by purpose for each 20-state topology experiment. They include process startup, I/O, and process monitoring, and are kept separate from the synchronized NKB coefficient timers in Table 27 . Shared harmonic training is excluded.

$K$	Velocity load	Velocity check	Refined proj.	Velocity eval.	NKB eval.	Lap. refresh
512	0.71	183.30	19.37	386.33	69.50	376.58
1024	3.36	438.64	35.32	761.22	2337.10	407.83

Appendix

Table 29: Additional 3D measured costs in seconds. Velocity check includes full-grid RHS and one-step comparisons; refined projection uses the matched-step NS reference. Velocity/NKB evaluation bundles include both trajectories, checks, and output and overlap the preceding component timers. NKB capacities time evaluation after operator preparation. Laplacian refresh includes both time steps and fresh FP64 projection. Bundle totals and individual components describe their respective timer boundaries.

$K$	Training only	+ $Q$ assembly	+ derived fields
512	123	154	179
1024	359	774	907

Appendix

Table 30: Reuse thresholds computed from original wall-time receipts, without rounding inputs. A reuse is one

T=1.16

, 1856-step trajectory on the same geometry, with the same time step and existing operators. The baseline is the numerical NS integration timer. Training-only counts set

T_{\rm off}=0

; adding recorded assembly or derived-field preparation charges those stages once. Derived-field costs include the recovered

K=512

preparation timers. Per-query projection/output and separately unrecorded stages lie outside these integration-only thresholds.

Reduced-Order Modeling

Reduced-order modeling builds compact approximations of complex systems to make simulation, prediction, and control more efficient.

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Oct 23, 2026

Solving and Learning Navier-Stokes Dynamics with the Neural Basis Method

Physics-informed machine learning has become a major direction in predictive modeling of fluid dynamics, combining governing physics with learned representations. Yet prevailing formulations neither guarantee that small PDE-residual losses imply small solution errors nor scale effectively with increasing problem size. Here we present the neural basis method (NBM) as a new approach for Navier--Stokes dynamics. NBM is built on a ``representation-then-projection'' principle: fixed neural bases define the representation space, while projection of the governing equations determines the expansion coefficients. The same bases define common coordinates for the parametric solution manifold, recasting operator learning as a parameter-to-coefficient mapping and enabling scalability by removing explicit spatial dependence. For incompressible flow, NBM yields a velocity-only formulation whose residual provides an a posteriori velocity-error estimate, establishing a certified, data-free training objective for coefficient learning. Theoretical analysis and numerical experiments demonstrate the accuracy, efficiency, generalization, and scalability of NBM, with substantial improvements over baselines.

Yuhe Wang, Zhu Wang, Min Wang

Institute for Scientific Computation, Texas A&M University, College Station, TX 77843, USA · Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA · Department of Mathematics, University of Houston, TX 77204, USA

Neural PDE Solvers Computational Fluid Dynamics PDE Operator Learning Physics-Informed Neural Operators

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Oct 20, 2026

Neural Operator Splitting for Fluid Dynamics in Complex Geometry

Solving dynamical systems in complex and diverse geometries remains challenging for both numerical solvers and learning-based surrogate models. We propose Neural Operator Splitting, a framework for learning fluid dynamics while preserving the physical structure of fractional-step numerical schemes. Instead of learning the full-step evolution monolithically, we decompose each time step into physically interpretable steps, and learn each operator separately using intermediate states from the underlying numerical solver. Guided by the linearity properties of the fractional steps, we introduce a conditional-linear Fourier neural operator (CL-FNO) whose linear state mappings are modulated by nonlinear conditioning on inputs such as boundary geometry. We further introduce two complementary strategies, symmetry sampling and corrective cycling, to mitigate error accumulation during autoregressive rollout. Experiments on 2D and 3D fluid flows with varying geometries and initial conditions show that our full model reduces mean relative

L^2

rollout error by 68% in 2D and 40% in 3D compared with a single full-step neural operator. At comparable error levels, neural-operator inference is approximately

10^3

times faster per step than the Monte Carlo solver.

Jianxiang Li, Xi Deng, Hong Chul Nam +1

New York University

PDE Operator Learning Neural Operators Computational Fluid Dynamics Fourier Neural Operator

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May 23, 2026

Learning Laplacian Eigenspace with Mass-Aware Neural Operators on Point Clouds

The eigendecomposition of the Laplace--Beltrami Operator (LBO) is fundamental to geometric analysis, yet computing its low-frequency eigenmodes remains a significant bottleneck due to the high cost of iterative solvers on large-scale data. To amortize this cost, we introduce the Neural Eigenspace Operator (NEO), a feed-forward framework designed to predict the spectrum directly from point clouds. Crucially, NEO circumvents the ill-posed nature of standard eigenvector regression, which suffers from intrinsic sign flips and rotation ambiguities, by learning the stable, invariant low-frequency subspace instead. Specifically, the network predicts a redundant set of basis functions whose span robustly covers the target eigenspace, allowing for the recovery of accurate eigenpairs via a lightweight Rayleigh--Ritz refinement. To handle irregular sampling, we propose a mass-aware neural operator that incorporates per-point area weights into attention-based aggregation, improving robustness to non-uniform densities and enabling zero-shot generalization across resolutions. Our approach achieves near-linear runtime scaling and substantial wall-clock speedups over iterative solvers at comparable accuracy, and exhibits strong zero-shot transfer to high-resolution point clouds. The resulting eigenpairs support standard spectral geometry tasks, while the raw basis functions provide effective point-wise features for downstream learning. Code: https://github.com/Adversarr/NEO.

Zherui Yang, Tao Du, Ligang Liu

University of Science and Technology of China, China · Tsinghua University, China · Shanghai Qi Zhi Institute, China +1

Point Cloud Learning Geometric Representation Learning Manifold Learning Spectral Methods Neural Operators

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Abstract

Reduced-Order Modeling

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