Learning Discrete Riemannian Metrics for Physical Fields with Cochain-Frame Equivariance

Dongzhe Zheng; Christine Allen-Blanchette

cs.LGMay 14, 2026

Learning Discrete Riemannian Metrics for Physical Fields with Cochain-Frame Equivariance

Authors: Dongzhe Zheng, Christine Allen-Blanchette

Organizations: 1Princeton University

Abstract

Physical fields on meshes require a separation between topology and geometry: differential identities should remain exact, while geometry, material response, and anisotropic coupling are learned from data. Existing neural surrogates often mix these roles inside unconstrained message passing. We introduce Riemannian Hodge Message Passing (RHMP), which turns this separation into an architectural principle. RHMP fixes the cellular coboundaries ( $d_k$ ) determined by oriented incidence and learns symmetric positive-definite cochain metrics ( $H_k$ ) for geometry-dependent propagation. The learned metric defines how cochain fields are measured and coupled; its invariance to orthogonal changes of the hidden feature basis yields cochain-frame-equivariant propagation. RHMP implements this principle with metric-weighted Hodge blocks ( $d_k^\top H_{k+1}d_k$ ), yielding exact cochain-complex identities ( $d_{k+1}d_k=0$ ), nonnegative Hodge energies, positive-semidefinite operators, and exact Abelian curvature invariance. Across seven physical benchmarks spanning fluids, electromagnetism, gauge fields, and variable-mesh CFD, RHMP performs strongly across these systems, with the largest gains when topology, learned geometry, and field structure interact.

Figures & tables

Figure 1: Metric invariance and cochain-frame equivariance. Applying an orthogonal change of hidden cochain frame $R\in O(C)$ before or after an RHMP layer $\Phi$ gives the same result. The frame action is on feature channels, while $d_{k}$ and $H_{k}$ act on cell indices.

Figure 2: Navier–Stokes vorticity. Left blocks summarize the inputs; right blocks compare ground truth, RHMP, and representative baselines. The bottom row shows the error maps.

Task	Metric	GCN	GAT	SchNet	EGNN	MPSN	SCCNN	GaugeEqCNN	GEM-CNN	CW Net	Cliff-SMPN	DeepONet	RHMP
		Graph				Topological		Geometric				Operator
NS Vort	SSIM	.617	.591	.625	.616	.617	.610	.944	.966	.921	.737	.625	.994
	Pearson	.159	.095	.082	.092	.134	.028	.900	.941	.880	.548	.226	.991
	NRMSE	.0350	.0356	.0349	.0350	.0336	.0336	.0126	.0096	.0140	.0271	.0349	.0054
Torus Adv	SSIM	.777	.863	.793	.780	.766	.763	.940	.961	.932	.893	.289	.990
	Pearson	.785	.898	.784	.789	.786	.785	.959	.972	.934	.894	.131	.991

Table 2: Quantitative results on the six fixed-mesh tasks, three metrics, and 11 baselines whose native definition admits irregular meshes. SSIM (

\uparrow

), Pearson (

\uparrow

), NRMSE (

\downarrow

). Bold = best, underline = second best among node-target methods. All entries use seed

42

and the first 100 test samples. On Maxwell, RHMP uses the physical solve layer (Appendix E.4 ). MPSN, SCCNN, and Clifford-SMPN use edge-averaged targets (Appendix E ). Comparison against FNO on the structured-grid setting (NS Vorticity) is discussed inline in § 5.1 ; variable-mesh generalization (AirfRANS) is reported separately in Table 3 . Baselines grouped by family: Graph , Topological , Geometric , Operator .

Figure 3: Torus advection–diffusion. Left blocks summarize the inputs; right blocks compare ground truth, RHMP, and representative baselines. The bottom row shows the error maps.

Figure 5: Maxwell magnetostatics. Left blocks summarize the inputs; right blocks compare ground truth, RHMP, and representative baselines.

Figure 6: U(1) Wilson-loop curvature. Left blocks summarize the inputs; right blocks compare ground truth, RHMP, and representative baselines. The bottom row shows the error maps.

Appendix figures & tables24 assets

Supplementary material from the paper’s appendix.

Appendix

Figure 7: The triangle complex $K$ and its coboundary matrices. Left: three vertices (blue), three oriented edges (orange, arrows indicate orientation), and one face (green, counterclockwise). Right: the coboundary matrices $d_{0}$ (discrete gradient, $3\times 3$ ) and $d_{1}$ (discrete curl, $1\times 3$ ), with row/column labels matching the mesh elements. The identity $d_{1}d_{0}=0$ holds by construction: every vertex on the face boundary appears in exactly two boundary edges with opposite signs.

Figure 8: One upper Hodge step $d_{0}^{\top}H_{1}d_{0}x_{0}$ on the triangle of Figure 7 , with $x_{0}=(2.0,5.0,3.0)$ . Panel 1: the coboundary $d_{0}$ computes oriented differences along edges (discrete gradient, fixed). Panel 2: the diagonal metric $H_{1}=\mathrm{diag}(1.5,0.3,2.1)$ weights each edge’s gradient by a learned constitutive weight (red); edge thickness reflects the metric value. Panel 3: the adjoint $d_{0}^{\top}$ aggregates weighted gradients to vertices; arrows from each edge midpoint show the sign ( $-$ at source, $+$ at target). Inset: the same $d_{0}$ with a permuted metric $H_{1}^{\prime}=\mathrm{diag}(0.3,2.1,1.5)$ produces a different output, illustrating the topology–geometry separation.

Figure 9: Schematic of the cochain-complex data flow within one RHMP layer. The horizontal arrows are the fixed coboundaries $d_{0}$ and $d_{1}$ (oriented incidence; satisfy $d_{1}\circ d_{0}=0$ ). Each upper loop $d_{k}^{\top}H_{k+1}d_{k}$ lifts features to the adjacent cochain space via $d_{k}$ , re-weights them with the learned SPD metric $H_{k+1}$ , and projects back via $d_{k}^{\top}$ .

Variant	Params	Val $R^{2}$	Test $R^{2}$	NRMSE	SSIM	Pearson
scalar	0.011M	0.488	0.495	0.0235	0.7653	0.666
diagonal, $r{=}0$	0.125M	0.487	0.493	0.0236	0.7646	0.664
diagonal-plus-basis, $r{=}8$	0.066M	0.503	0.510	0.0232	0.7740	0.676
diagonal-plus-basis, $r{=}16$	0.122M	0.416	0.421	0.0252	0.7291	0.610
full Cholesky	36.573M	0.452	0.455	0.0244	0.7449	0.637

Appendix

Table 4: Metric parameterization sweep on

SU(2)

Yang–Mills (256-node mesh; 1000 train / 200 val / 500 test, 40 epochs). The diagonal-plus-low-rank-basis form with

r{=}8

is used in the ablation studies.

Metric quantity	Cells	Pearson	Slope	Intercept
Diagonal $\log(h_{e}/h_{e}^{\rm ref})$	$139{,}805$	$0.998$	$0.985$	$0.070$
Tensor $\tfrac{1}{2}\log\det\Sigma$	$91{,}550$	$0.996$	$1.039$	$0.005$

Appendix

Table 5: Constitutive calibration on 32 heterogeneous Poisson test meshes (seed

42

). Slopes and intercepts fit learned log coefficients against reference log conductivity, pooled over cells.

Method	SSIM	Pearson	NRMSE	$R^{2}_{0.55\mathrm{K}}$	$R^{2}_{1.9\mathrm{K}}$	$R^{2}_{6.9\mathrm{K}}$
GCN	.682	.837	.0377	.472	.703	.544
GAT	.901	.950	.0243	.698	.877	.600
SchNet	.863	.929	.0272	.455	.845	.558
EGNN	.936	.957	.0195	.679	.921	.678
MPSN	.874	.928	.0257	.781	.862	.681
SCCNN	.695	.837	.0376	.620	.705	.505

Appendix

Table 9: AirfRANS vorticity (100 test simulations, seed

42

). SSIM, Pearson, and NRMSE at the 1.9K-node training size;

R^{2}

on 0.55K-, 1.9K-, and 6.9K-node meshes, the first and the last without retraining. Bold = best, underline = second best.

		Same ellipsoid, nodes			Other semi-axes
Method	Orig.	4625	577	290	$2.1{:}1{:}0.7$	$1.6{:}1{:}0.5$
GCN	$-$ .009	$-$ .009	$-$ .009	$-$ .009	$-$ .008	$-$ .009
GAT	$-$ .010	$-$ .009	$-$ .010	$-$ .011	$-$ .009	$-$ .010
SchNet	$-$ .010	$-$ .010	$-$ .010	$-$ .010	$-$ .009	$-$ .009
EGNN	$-$ .010	$-$ .010	$-$ .010	$-$ .010	$-$ .009	$-$ .009
MPSN †	$-$ .014	$-$ .014	$-$ .014	$-$ .014	$-$ .013	$-$ .012

Appendix

Table 11: Zero-shot transfer on ellipsoid surface flow (

R^{2}

with one variance for the three vector components, 100 test samples, seed

42

, no retraining). The training ellipsoid has semi-axes

1.4{:}1{:}0.7

and 1157 nodes.

\dagger

: edge-averaged first component as in Table 2 . Bold = best.

Method	Orig.	Stretched	$64^{2}$	Rot.
GCN	$-$ .058	$-$ .103	$-$ .079	$-$ .057
GAT	$-$ .091	$-$ .159	$-$ .127	$-$ .055
SchNet	$-$ .049	$-$ .104	$-$ .192	$-$ .043
EGNN	$-$ .057	$-$ .104	$-$ .081	$-$ .054
MPSN †	$-$ .062	$-$ .111	$-$ .091	$-$ .059
SCCNN †	$-$ .062	$-$ .106	$-$ .079	$-$ .062

Appendix

Table 12: Zero-shot transfer on NS vorticity (

R^{2}

, 100 test samples, seed

42

, no retraining). Orig.: training grid; Stretched: a second stretched grid with two fine regions per axis;

64^{2}

: uniform grid with

64^{2}

nodes; Rot.: training grid rotated by

90^{\circ}

\dagger

: edge-averaged targets as in Table 2 . Bold = best.

Method	Orig.	$1.5I$	Held-out	Fine	Coarse	$30^{\circ}$	$90^{\circ}$
GCN	.002	.002	$-$ .008	$-$ .008	$-$ .001	$-$ .006	$-$ .014
GAT	.037	.037	$-$ .026	$-$ .003	$-$ .027	$-$ .001	$-$ .064
SchNet	$-$ .004	$-$ .005	$-$ .005	$-$ .007	$-$ .005	$-$ .006	$-$ .007
EGNN	.051	.049	$-$ .033	$-$ .002	$-$ .049	.025	$-$ .071
MPSN †	.065	.056	$-$ .051	.008	$-$ .053	.008	$-$ .086
SCCNN †	.128	.131	$-$ .120	.025	$-$ .163	.028	$-$ .104

Appendix

Table 14: Zero-shot transfer on Maxwell magnetostatics (

R^{2}

, 100 test samples, seed

42

, no retraining). Orig.: training distribution;

1.5I

: coil currents scaled by

1.5

; Held-out: E–E inductor cores, a family absent from training; Fine and Coarse: meshes with 7255 and 459 nodes;

30^{\circ}

90^{\circ}

: rigid rotations of the device.

\dagger

: edge-averaged targets as in Table 2 . Bold = best.

Mesh scale	Fwd (ms)	Bwd (ms)	Peak mem (GB)	Trainable params	Activations (M)
1K	16.7	23.7	0.04	$\sim$ 0.05M	0.31
5K	17.2	23.1	0.12	$\sim$ 0.05M	1.50
10K	21.2	25.8	0.22	$\sim$ 0.05M	2.98
100K	169.2	183.0	3.93	$\sim$ 0.05M	59.4

Appendix

Table 20: Wall-clock time and memory for RHMP at increasing mesh scale on a single NVIDIA RTX PRO 6000 Blackwell GPU (96 GB). Trainable params (weights of

\mathrm{MLP}_{H}

, the lifting encoder, and the message-passing transforms) are shared across all cells and fixed regardless of mesh size. Activations count per-pass scalar outputs (one

h_{i}

per

k

-cell plus intermediate features), which grow linearly with

n_{k}

as in any message-passing architecture.

Model	Params	Test $R^{2}$	Test NRMSE	Wall time
RHMP ( $C{=}16$ , 3 layers)	10.80M	0.9823	0.0033	113 s
CW Net ( $h{=}512$ , 6 layers)	11.30M	0.9231	0.0069	2181 s

Appendix

Table 21: Parameter-matched comparison on a 100K-cell

U(1)

Wilson loop. Both models trained end-to-end on a single GPU with the same

70/15/15

split. Wall time is total training time.

Method	NS Vort	Torus Adv	Ellips Flow	Maxwell	Wilson	Yang–Mills	Airfoil
GCN	$\pm\,0.029$	$\pm\,0.015$	$\pm\,0.010$	$\pm\,0.028$	$\pm\,0.016$	$\pm\,0.001$	$\pm\,0.018$
GAT	$\pm\,0.030$	$\pm\,0.013$	$\pm\,0.010$	$\pm\,0.025$	$\pm\,0.007$	$\pm\,0.002$	$\pm\,0.021$
SchNet	$\pm\,0.029$	$\pm\,0.013$	$\pm\,0.010$	$\pm\,0.028$	$\pm\,0.002$	$\pm\,0.002$	$\pm\,0.018$
EGNN	$\pm\,0.029$	$\pm\,0.016$	$\pm\,0.010$	$\pm\,0.025$	$\pm\,0.003$	$\pm\,0.002$	$\pm\,0.018$
MPSN	$\pm\,0.029$	$\pm\,0.017$	$\pm\,0.009$	$\pm\,0.026$	$\pm\,0.017$	$\pm\,0.005$	$\pm\,0.019$
SCCNN	$\pm\,0.029$	$\pm\,0.017$	$\pm\,0.009$	$\pm\,0.026$	$\pm\,0.010$	$\pm\,0.005$	$\pm\,0.018$

Appendix

Table 24: SSIM bootstrap std (1000 resamples of the evaluation samples for the configuration specified above). Each row a method, each column a task.

Method	NS Vort	Torus Adv	Ellips Flow	Maxwell	Wilson	Yang–Mills	Airfoil
GCN	$\pm\,0.014$	$\pm\,0.014$	$\pm\,0.003$	$\pm\,0.013$	$\pm\,0.015$	$\pm\,0.005$	$\pm\,0.005$
GAT	$\pm\,0.011$	$\pm\,0.008$	$\pm\,0.006$	$\pm\,0.017$	$\pm\,0.028$	$\pm\,0.006$	$\pm\,0.006$
SchNet	$\pm\,0.012$	$\pm\,0.014$	$\pm\,0.008$	$\pm\,0.005$	$\pm\,0.016$	$\pm\,0.006$	$\pm\,0.014$
EGNN	$\pm\,0.014$	$\pm\,0.013$	$\pm\,0.007$	$\pm\,0.016$	$\pm\,0.022$	$\pm\,0.006$	$\pm\,0.014$
MPSN	$\pm\,0.014$	$\pm\,0.014$	$\pm\,0.006$	$\pm\,0.023$	$\pm\,0.027$	$\pm\,0.011$	$\pm\,0.004$
SCCNN	$\pm\,0.004$	$\pm\,0.014$	$\pm\,0.007$	$\pm\,0.027$	$\pm\,0.023$	$\pm\,0.011$	$\pm\,0.004$

Appendix

Table 25: Pearson bootstrap std (1000 resamples of the evaluation samples for the configuration specified above). Each row a method, each column a task.

Method	NS Vort	Torus Adv	Ellips Flow	Maxwell	Wilson	Yang–Mills	Airfoil
GCN	$\pm\,0.0054$	$\pm\,0.0017$	$\pm\,0.0015$	$\pm\,0.0032$	$\pm\,0.0010$	$\pm\,0.0001$	$\pm\,0.0010$
GAT	$\pm\,0.0055$	$\pm\,0.0014$	$\pm\,0.0015$	$\pm\,0.0032$	$\pm\,0.0006$	$\pm\,0.0001$	$\pm\,0.0010$
SchNet	$\pm\,0.0054$	$\pm\,0.0017$	$\pm\,0.0015$	$\pm\,0.0032$	$\pm\,0.0003$	$\pm\,0.0001$	$\pm\,0.0010$
EGNN	$\pm\,0.0054$	$\pm\,0.0017$	$\pm\,0.0015$	$\pm\,0.0031$	$\pm\,0.0004$	$\pm\,0.0001$	$\pm\,0.0010$
MPSN	$\pm\,0.0051$	$\pm\,0.0017$	$\pm\,0.0019$	$\pm\,0.0028$	$\pm\,0.0013$	$\pm\,0.0002$	$\pm\,0.0010$
SCCNN	$\pm\,0.0050$	$\pm\,0.0017$	$\pm\,0.0019$	$\pm\,0.0028$	$\pm\,0.0007$	$\pm\,0.0001$	$\pm\,0.0010$

Appendix

Table 26: NRMSE bootstrap std (1000 resamples of the evaluation samples for the configuration specified above). Each row a method, each column a task.

Method	NS Vort	Torus Adv	Ellips Flow	Maxwell	Wilson	Yang–Mills	Airfoil
GCN	$\pm\,0.001$	$\pm\,0.000$	$\pm\,0.000$	$\pm\,0.000$	$\pm\,0.001$	$\pm\,0.000$	$\pm\,0.000$
GAT	$\pm\,0.029$	$\pm\,0.001$	$\pm\,0.000$	$\pm\,0.003$	$\pm\,0.007$	$\pm\,0.004$	$\pm\,0.014$
SchNet	$\pm\,0.002$	$\pm\,0.001$	$\pm\,0.000$	$\pm\,0.001$	$\pm\,0.001$	$\pm\,0.008$	$\pm\,0.010$
EGNN	$\pm\,0.022$	$\pm\,0.002$	$\pm\,0.000$	$\pm\,0.005$	$\pm\,0.003$	$\pm\,0.012$	$\pm\,0.001$
MPSN	$\pm\,0.004$	$\pm\,0.001$	$\pm\,0.000$	$\pm\,0.001$	$\pm\,0.001$	$\pm\,0.001$	$\pm\,0.004$
SCCNN	$\pm\,0.002$	$\pm\,0.003$	$\pm\,0.002$	$\pm\,0.010$	$\pm\,0.017$	$\pm\,0.002$	$\pm\,0.001$

Appendix

Table 27: SSIM cross-seed standard deviation (seeds 42, 1, 2).

Method	NS Vort	Torus Adv	Ellips Flow	Maxwell	Wilson	Yang–Mills	Airfoil
GCN	$\pm\,0.014$	$\pm\,0.001$	$\pm\,0.003$	$\pm\,0.002$	$\pm\,0.004$	$\pm\,0.000$	$\pm\,0.000$
GAT	$\pm\,0.029$	$\pm\,0.002$	$\pm\,0.003$	$\pm\,0.005$	$\pm\,0.026$	$\pm\,0.008$	$\pm\,0.002$
SchNet	$\pm\,0.016$	$\pm\,0.001$	$\pm\,0.012$	$\pm\,0.009$	$\pm\,0.004$	$\pm\,0.016$	$\pm\,0.010$
EGNN	$\pm\,0.044$	$\pm\,0.001$	$\pm\,0.008$	$\pm\,0.016$	$\pm\,0.016$	$\pm\,0.041$	$\pm\,0.006$
MPSN	$\pm\,0.014$	$\pm\,0.000$	$\pm\,0.008$	$\pm\,0.008$	$\pm\,0.021$	$\pm\,0.001$	$\pm\,0.009$
SCCNN	$\pm\,0.010$	$\pm\,0.002$	$\pm\,0.026$	$\pm\,0.006$	$\pm\,0.033$	$\pm\,0.004$	$\pm\,0.000$

Appendix

Table 28: Pearson cross-seed standard deviation (seeds 42, 1, 2).

Method	NS Vort	Torus Adv	Ellips Flow	Maxwell	Wilson	Yang–Mills	Airfoil
GCN	$\pm\,0.0000$	$\pm\,0.0000$	$\pm\,0.0000$	$\pm\,0.0000$	$\pm\,0.0001$	$\pm\,0.0000$	$\pm\,0.0000$
GAT	$\pm\,0.0002$	$\pm\,0.0001$	$\pm\,0.0000$	$\pm\,0.0000$	$\pm\,0.0004$	$\pm\,0.0001$	$\pm\,0.0001$
SchNet	$\pm\,0.0000$	$\pm\,0.0000$	$\pm\,0.0000$	$\pm\,0.0000$	$\pm\,0.0001$	$\pm\,0.0002$	$\pm\,0.0001$
EGNN	$\pm\,0.0002$	$\pm\,0.0000$	$\pm\,0.0000$	$\pm\,0.0002$	$\pm\,0.0002$	$\pm\,0.0002$	$\pm\,0.0000$
MPSN	$\pm\,0.0000$	$\pm\,0.0000$	$\pm\,0.0000$	$\pm\,0.0000$	$\pm\,0.0003$	$\pm\,0.0000$	$\pm\,0.0000$
SCCNN	$\pm\,0.0000$	$\pm\,0.0002$	$\pm\,0.0001$	$\pm\,0.0005$	$\pm\,0.0006$	$\pm\,0.0001$	$\pm\,0.0000$

Appendix

Table 29: NRMSE cross-seed standard deviation (seeds 42, 1, 2).

Panel of Figure 10 — Figure 10: Additional qualitative samples on Navier–Stokes vorticity (PDEBench). Each row contrasts the ground-truth vorticity with predictions from RHMP and the strongest cell-complex / manifold-gauge baselines.

Panel of Figure 11 — Figure 11: Additional qualitative samples on torus advection–diffusion. Each panel shows the scalar transport field on a genus-1 surface; RHMP preserves the localized plumes and sharp gradients that graph and cell-complex baselines smear out.

Panel of Figure 12 — Figure 12: Additional qualitative samples on ellipsoid coexact surface flow. Rows show $\|\mathbf{v}\|$ and the three Cartesian components of $\mathbf{v}_{\rm tan}=\mathbf{n}\times\nabla\psi$ ; CW-Net loses the sign structure of the surface vector field while RHMP tracks the ground truth.

Panel of Figure 13 — Figure 13: Additional qualitative samples on Maxwell magnetostatics. Heatmaps show $\|\mathbf{B}\|$ and streamlines show $\mathbf{B}$ ; the divergence-free constraint $d^{2}\!=\!0$ is encoded structurally in RHMP, whose flux lines close through the core and the air gaps as in the ground truth.

Panel of Figure 14 — Figure 14: Additional qualitative samples on $U(1)$ Wilson-loop curvature. The localized $F=d\theta$ peaks are recovered with the correct amplitude by RHMP; GEM-CNN and GaugeEquivCNN leave small residuals, while SchNet and CW-Net add spurious structure away from the peaks.

Panel of Figure 15 — Figure 15: Additional qualitative samples on $SU(2)$ Yang–Mills curvature. Rows display the three Lie-algebra components $F^{1},F^{2},F^{3}$ of $F=dA+[A,A]$ ; in these examples only RHMP captures both the differential $dA$ structure and the algebraic $[A,A]$ coupling at the correct amplitude.

Panel of Figure 16 — Figure 16: Additional qualitative samples on AirfRANS pressure. Each test simulation is shown on its 1.9K-node mesh (training size) and on its 6.9K-node mesh without retraining; rows display the inputs, the pressure, and the prediction error.

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