Physics-Informed Functional Tucker Method with RKHS Factors for Sparse Spatiotemporal Reconstruction

Zhengyang Liu; Jianguo Huang; Junxi Hu; Yucheng Mi; Haojie Cai; Wenxuan Shen; Zhikun Zhang; Renfeng Peng

cs.CVOct 31, 2026

Physics-Informed Functional Tucker Method with RKHS Factors for Sparse Spatiotemporal Reconstruction

Authors: Zhengyang Liu, Jianguo Huang, Junxi Hu, Yucheng Mi, Haojie Cai, Wenxuan Shen, Zhikun Zhang, Renfeng Peng,

Organizations: SandGold AI Research, Guangzhou, China · Nanyang Technological University, Singapore · Westlake University, Hangzhou, China · Chinese Academy of Sciences, Beijing, China ·

Abstract

Reconstructing physical dynamics from fixed sparse sensors and missing frames is a fundamental challenge in scientific research. Sparsity leaves large regions of the field unconstrained, making the inverse problem ill-posed: a model may fit the observed sensors while producing inaccurate off-sensor structures or unreliable predictions between support frames. To this end, we propose PhysFTM---a physics-informed functional Tucker model that parameterizes the Tucker mode factors in reproducing kernel Hilbert spaces (RKHSs) and incorporates PDE constraints through finite-difference residuals evaluated at off-sensor collocation points. These residuals provide additional physical information in regions without direct observational supervision. PhysFTM first reconstructs continuous spatial fields at support frames via an alternating minimization scheme, optimizing the Tucker core tensor and the RKHS-based factors separately. To enable missing-frame reconstruction, PhysFTM constructs a continuous trajectory in Tucker-core space, which is decoded by the learned spatial RKHS-FTM representation to recover the full spatiotemporal field. Experiments on Allen--Cahn, Navier--Stokes, three-dimensional Heat equations, and a three-dimensional gridded Copernicus ocean reanalysis demonstrate accurate 2D/3D reconstruction under two missing-frame protocols. Code is available at https://github.com/liusirzzz/PhysFTM.

Figures & tables

Figure 1: Overall framework of PhysFTM. PhysFTM reconstructs sparse grid-based sensor observations by learning an RKHS-regularized Functional Tucker spatial decoder constrained by observation loss and finite-difference PDE residuals. The inferred Tucker cores are then modeled as a continuous trajectory in core space and decoded back to the physical domain to obtain the reconstructed spatiotemporal field.

AC2D
Method	Random-drop-half				Temporal-interval-missing
Method	S-sup	Off-sup	S-miss	Off-miss	S-sup	Off-sup	S-miss	Off-miss
SDIFT	0.35804	0.38080	0.38030	0.38151	0.35699	0.36199	0.39071	0.39259
LRTFR	0.31170	0.32220	0.31340	0.32038	0.31006	0.31676	0.32059	0.32781
PhysFTM	0.01735	0.08710	0.05035	0.09988	0.01633	0.09397	0.03953	0.07924
RecFNO	0.01848	0.45283	0.02178	0.45523	0.01811	0.45875	0.02116	0.43680

Table 1: Reconstruction RMSE on four datasets at spatial sensor ratio

s=0.03

. The two column groups report the four disjoint evaluation subsets, defined by sensor/off-sensor locations and support/missing frames, under random-drop-half and temporal-interval-missing. Lower is better. For Heat3D and Ocean3D, the PhysFTM interval-missing results use the random-drop-half-trained temporal checkpoint. ( Bold : best, Underline : second best)

Protocol	Method	Off-miss	Divergence	Vort. RMSE	Spectrum
Random	SDIFT	0.510274	0.352407	0.916343	0.877309
	LRTFR	0.145757	0.459044	0.237923	0.049954
	PhysFTM	0.064976	0.134170	0.109304	0.033599
	RecFNO	0.104413	0.459014	0.118608	0.067430
	SIREN-MLP	0.244405	1.887604	0.623477	0.408029
	PINNs	0.340554	0.120243	0.586433	0.398297

Table 2: NS2D missing-frame reconstruction and complementary physical diagnostics at

s=0.03

. Divergence, vorticity, and spectrum errors are evaluated on full grids of missing frames; Off-miss is evaluated at off-sensor locations in missing frames. Lower is better.

Panel of Figure 2 — Figure 2: Qualitative comparisons on AC2D (top) at $s=0.03$ and the NS2D vorticity channel (bottom) at $s=0.15$ . The upper row of each block shows the sparse input and predictions from LRTFR, SDIFT, RecFNO, PINNs, and PhysFTM; the lower row shows the ground truth and the corresponding absolute-error maps.

Data	Variant	PDE	Full	Sensor	Off-sensor
AC2D	PhysFTM	0.053250	0.088078	0.015185	0.089395
	w/o physics	0.064986	0.091135	0.010852	0.092520
	w/o RKHS reg.	0.048491	0.096622	0.027551	0.097985
	solve-only	1.155543	0.171035	0.008657	0.173653
	gradient-only	$1.76{\times}10^{-5}$	0.352111	0.351441	0.352132
NS2D	PhysFTM	0.662575	0.060270	0.004189	0.061196

Table 3: Component and block-update-strategy ablation on AC2D and NS2D at

s=0.03

, without temporal missingness. RMSEs use all evaluated frames; PDE is evaluated at disjoint held-out off-sensor points and centered-stencil-valid internal times.

Dataset	$s$	Learned	Linear	Cubic
AC2D	0.01	0.180029	0.189510	0.189556
AC2D	0.03	0.085552	0.090391	0.090388
NS2D	0.01	0.130469	0.137667	0.137453
NS2D	0.03	0.055546	0.064605	0.063790

Table 4: RMSE at non-grid-aligned times within the continuous support-frame time range, using ground truth directly recorded by the PDE solver. “Learned” is the gated ODE-RNN correction to cubic Hermite interpolation in Tucker-core space.

Appendix figures & tables12 assets

Supplementary material from the paper’s appendix.

Appendix

Method	Test input	Training / fitting	Supervision / equation use	Test optimization	Spatial scope	Time interface / selection
SDIFT	Sparse spatiotemporal observations	Cross-trajectory diffusion training	Dense training trajectories; no test missing-frame targets	Posterior sampling	2D/3D	Native-grid protocols; executed checkpoint
LRTFR	Sparse coordinate–value tuples	Per-trajectory representation fitting	Sensor measurements; no equation loss	Per-trajectory fit	2D/3D	Native-grid evaluation; fitted objective
SIREN-MLP	Coordinates; no test-mask conditioning	Train once on training split	Training fields; no equation loss	None	2D/3D	Coordinate output; executed checkpoint
PINNs	Coordinates; no test-mask conditioning	Train once on training split	Training fields and implemented equation residual	None	2D/3D	Coordinate output; executed checkpoint
RecFNO	Gridded sparse field and mask	Cross-trajectory operator training	Dense training fields; no equation loss	None	2D/3D	Prescribed-grid support/missing frames; executed checkpoint
PhysFTM	Sparse support-frame sensor values	Cross-trajectory decoder/head training	Observations, missing-frame targets, and equation loss where enabled	Support-core CG solve	2D/3D	Native-grid and AC/NS in-range non-grid-aligned reconstruction; spatial: training dense RMSE; temporal: validation missing-frame relative $\ell_{2}$

Appendix

Table 5: Task-level comparison protocols. “Native” denotes prescribed dataset timestamps; “coordinate” denotes direct coordinate evaluation. Baseline budgets and checkpoint selection follow their executed run configurations.

Panel A: off-sensor reconstruction in support frames
Method	AC2D			NS2D
Method	RMSE	SSIM	PDE	RMSE	SSIM	PDE
Linear interp.	0.1275	0.6379	0.5877	0.0897	0.9322	1.1605
Cubic interp.	0.1051	0.7632	0.7174	0.0653	0.9656	0.8339
SIREN-MLP	0.2907	0.1747	1.1703	0.2155	0.5836	2.1245
MLP-FTM	0.1049	0.7515	0.2282	0.0859	0.9116	4.0426

Appendix

Table 6: Additional spatial analysis at

s=0.03

. Panel A evaluates reconstruction at off-sensor locations in support frames. Panel B varies the equation-collocation placement; the complete model uses off-sensor collocation. ( Bold : best, Underline : second best)

Diagnostic	Missing-frame/spatial reduction	Trajectory reduction
Divergence error	Global RMS over all missing frames and grid points	Included in the same global pool
Vorticity RMSE	Global RMSE over all missing frames and grid points	Included in the same global pool
Spectrum error	Per-frame relative $L_{2}$ , then missing-frame mean	Mean/std of trajectory means

Appendix

Table 7: Aggregation rules for missing-frame physical diagnostics.

Protocol	$s$	Method	Off-miss	Divergence	Vort. RMSE	Spectrum
Random	0.01	SDIFT	0.509261	0.307569	0.876079	0.865337
		LRTFR	0.170679	0.475408	0.237857	0.048102
		PhysFTM	0.130659	0.274083	0.232358	0.104305
		RecFNO	0.413223	0.514580	0.396548	0.161395
		SIREN-MLP	0.244360	1.886754	0.623300	0.409266
		PINNs	0.341072	0.120243	0.586433	0.402015

Appendix

Table 8: Complete NS2D missing-frame diagnostics across spatial sensor ratios. Lower is better.( Bold : best, Underline : second best)

Dataset	Split	Stored tensor	Channels	Domain and time span
AC2D	160/20/20	$200\times 201\times 256\times 256$	scalar $u$	$[0,1)^{2}$ , $t\in[0,2]$
NS2D	640/80/80	$800\times 201\times 128\times 128\times 3$	$(u,v,\omega)$	$[0,1)^{2}$ , $t\in[0,20]$
Heat3D	160/20/20	$200\times 50\times 64\times 64\times 64\times 1$	temperature $u$	$[0,1]^{3}$ , $t\in[0,4]$
Ocean3D	80/10/10	$100\times 90\times 32\times 64\times 64\times 2$	$(u,v)$	depth–latitude–longitude, 90 days

Appendix

Table 9: Datasets used in the experiments. Shapes are trajectories

\times

time

\times

spatial axes

\times

channels.

Figure 3: Four evaluation subsets defined by sensor/off-sensor locations and support/missing frames. White dots indicate sensor measurements provided to the model.

Figure 4: Two missing-frame protocols on the original time grid: randomly withholding frames or withholding a contiguous block. Both endpoints remain support frames.

Figure 5: Internal timestamps are perturbed before being split into support and evaluation sets, while both endpoints remain fixed. Evaluation ground truth is recorded directly by the PDE solver.

Protocol	$s$	Method	S-sup	Off-sup	S-miss	Off-miss
Random-drop-half	0.01	SDIFT	0.373381	0.381700	0.378977	0.381200
		LRTFR	0.311380	0.329100	0.308521	0.328814
		PhysFTM	0.009898	0.187900	0.042055	0.191673
		RecFNO	0.020782	0.697782	0.021170	0.696267
		SIREN-MLP	0.351270	0.350988	0.349308	0.349007
		PINNs	0.354653	0.354871	0.350601	0.350840

Appendix

Table 10: Complete four-subset reconstruction RMSE on AC2D across spatial sensor ratios. Lower is better. ( Bold : best, Underline : second best)

Protocol	$s$	Method	S-sup	Off-sup	S-miss	Off-miss
Random-drop-half	0.01	SDIFT	0.518934	0.512800	0.510791	0.509261
		LRTFR	0.123841	0.172000	0.126188	0.170679
		PhysFTM	0.007849	0.137700	0.030927	0.130659
		RecFNO	0.261520	0.418079	0.262461	0.413223
		SIREN-MLP	0.244818	0.245620	0.243905	0.244360
		PINNs	0.338968	0.341335	0.339094	0.341072

Appendix

Table 11: Complete four-subset reconstruction RMSE on NS2D across spatial sensor ratios. Lower is better. ( Bold : best, Underline : second best)

Protocol	$s$	Method	S-sup	Off-sup	S-miss	Off-miss
Random-drop-half	0.01	SDIFT	0.058457	0.058594	0.061825	0.062284
		LRTFR	0.010659	0.010815	0.012014	0.012549
		PhysFTM	0.001344	0.001907	0.006379	0.006824
		RecFNO	0.339625	0.888925	0.307609	0.813064
		SIREN-MLP	0.040199	0.040240	0.042215	0.042397
		PINNs	0.040309	0.040409	0.041993	0.042295

Appendix

Table 12: Complete four-subset reconstruction RMSE on Heat3D across spatial sensor ratios. Lower is better. ( Bold : best, Underline : second best)

Protocol	$s$	Method	S-sup	Off-sup	S-miss	Off-miss
Random-drop-half	0.01	SDIFT	0.281171	0.278458	0.280994	0.280682
		LRTFR	0.113447	0.126022	0.122721	0.133850
		PhysFTM	0.006599	0.115414	0.076277	0.125325
		RecFNO	0.758963	1.005671	0.793739	1.002127
		SIREN-MLP	0.200624	0.203220	0.196868	0.200081
		PINNs	0.297127	0.301674	0.295973	0.301402

Appendix

Table 13: Complete four-subset reconstruction RMSE on Ocean across spatial sensor ratios. Lower is better. ( Bold : best, Underline : second best)

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