Wave Physics as Representation: Tensor-Structured Neural Reconstruction for Ocean Acoustic Fields

Yifan Sun; Panqi Chen; Weifeng Shi; Xiao Fu; Lei Cheng

cs.LGNov 1, 2026

Wave Physics as Representation: Tensor-Structured Neural Reconstruction for Ocean Acoustic Fields

Authors: Yifan Sun, Panqi Chen, Weifeng Shi, Xiao Fu, Lei Cheng

Organizations: College of Information Science and Electronic Engineering, Zhejiang University · Department of ECE, University of Minnesota - Twin Cities

Abstract

Reconstructing three-dimensional (3D) complex ocean acoustic fields from sparse measurements is challenging due to highly oscillatory wave structures and restricted sensing patterns/routes. Learning-based approaches fall into two categories. First, purely data-driven models can learn complex field patterns, but do not explicitly exploit wave physics that may enhance interpretability and reduce training costs. Second, physics-informed methods impose a specified physical equation as a reconstruction constraint, but often require environment-dependent parameters that are difficult to acquire. As a result, practical 3D acoustic field reconstruction that is both physically grounded and robust to uncertain environmental information remains largely unexplored. This work takes an alternative route: instead of solving a fully specified wave equation, it extracts a structured representation from wave physics and learns its components from data. From the Helmholtz equation and normal mode propagation, we derive a structured spatial-angular tensor representation, termed the S/c representation. Under a low-rank specialization, it takes the form of a rank-(L,L,1) block-term tensor decomposition, which connects structured sampling patterns/routes used in ocean sensing to sufficient conditions for field recovery. Building on this tensor representation, we propose TIDAL, a neural reconstructor that uses the S/c representation as an inductive bias for structured decoding and aligns information aggregation with the sampling geometry. The neural learner removes the need for known model parameters required by conventional physics-based recovery methods. Experiments across three structured sampling geometries show that TIDAL achieves the best reconstruction performance and fastest inference among baselines, with benefits from the physics-derived representation under limited data and model capacity.

Figures & tables

Figure 1: Overview of the proposed framework. Starting from the Helmholtz equation, we derive the physics-derived $S/c$ tensor representation; under a low-rank specialization, it connects H/V observations to LL1 recoverability theory (top). Building on this representation, TIDAL uses the $S/c$ structure as an inductive bias through data-adaptive decoding, while the H/V observation geometry guides information aggregation through sampling-aligned axial attention (bottom).

Figure 2: Qualitative reconstruction under three sampling geometries. Columns show the ground truth, sparse observations, and reconstructions and absolute complex errors of MxT and TIDAL for the same test field and angular slice. Field magnitudes and errors use shared scales across all cases.

Method	Metric	Equispaced		Shifted		Jittered		Time (s) $\downarrow$
Method	Metric	Full	Unobs.	Full	Unobs.	Full	Unobs.	Time (s) $\downarrow$
LaMO	RMSE $\downarrow$	$0.712\pm 0.068$	$0.749\pm 0.073$	$0.704\pm 0.068$	$0.743\pm 0.072$	$0.717\pm 0.067$	$0.757\pm 0.071$	$0.91$
LaMO	MAC $\uparrow$	$0.488\pm 0.098$	$0.434\pm 0.110$	$0.501\pm 0.097$	$0.444\pm 0.108$	$0.483\pm 0.096$	$0.423\pm 0.108$	$0.91$
SCPNet	RMSE $\downarrow$	$0.504\pm 0.056$	$0.541\pm 0.060$	$0.486\pm 0.055$	$0.526\pm 0.059$	$0.542\pm 0.052$	$0.586\pm 0.056$	$1.71$
SCPNet	MAC $\uparrow$	$0.743\pm 0.056$	$0.704\pm 0.065$	$0.761\pm 0.053$	$0.720\pm 0.062$	$0.704\pm 0.056$	$0.653\pm 0.066$	$1.71$
MxT	RMSE $\downarrow$	$0.278\pm 0.073$	$0.298\pm 0.078$	$0.281\pm 0.070$	$0.303\pm 0.076$	$0.293\pm 0.072$	$0.317\pm 0.078$	$0.72$

Table 1: Reconstruction performance and inference efficiency under three sampling geometries. Results are reported as mean

\pm

standard deviation over all test fields. “Time” denotes per-field inference time. Best results are shown in bold.

Figure 3: Effect of spatial-rank constraints. (a) Cumulative singular-value energy of the learned dense spatial components. (b) Representation error versus prescribed spatial rank

L

. (c) End-to-end reconstruction with factorized and dense-

S

decoders. Parameter counts are shown above each bar.

Figure 4: Data and model efficiency of

S/c

decoding. Unobserved-region RMSE versus (a) training-set size and (b) model size, comparing S/c and direct-output decoders.

Appendix figures & tables5 assets

Supplementary material from the paper’s appendix.

Appendix

Budget	Method	Metric	Equispaced		Shifted		Jittered
Budget	Method	Metric	Full	Unobs.	Full	Unobs.	Full	Unobs.
H10/V10	LaMO	RMSE $\downarrow$	$0.792\pm 0.060$	$0.820\pm 0.063$	$0.770\pm 0.067$	$0.800\pm 0.069$	$0.781\pm 0.061$	$0.812\pm 0.064$
	LaMO	MAC $\uparrow$	$0.370\pm 0.096$	$0.325\pm 0.104$	$0.404\pm 0.103$	$0.356\pm 0.111$	$0.387\pm 0.096$	$0.337\pm 0.104$
	SCPNet	RMSE $\downarrow$	$0.693\pm 0.044$	$0.724\pm 0.046$	$0.659\pm 0.047$	$0.694\pm 0.049$	$0.676\pm 0.045$	$0.712\pm 0.047$
	SCPNet	MAC $\uparrow$	$0.518\pm 0.061$	$0.474\pm 0.067$	$0.564\pm 0.062$	$0.517\pm 0.068$	$0.541\pm 0.060$	$0.491\pm 0.066$
	MxT	RMSE $\downarrow$	$0.329\pm 0.082$	$0.343\pm 0.086$	$0.329\pm 0.079$	$0.346\pm 0.084$	$0.343\pm 0.080$	$0.362\pm 0.084$

Appendix

Table 2: Reconstruction performance under H10/V10 and H20/V20 sampling. Results are reported as mean

\pm

standard deviation over 80 test fields. Best results are shown in bold.

Representation	$L$	Mean	Median	P10	P90
Factorized $S=AB^{\top}$	2	0.6020	0.6014	0.4201	0.7084
Factorized $S=AB^{\top}$	4	0.4616	0.4514	0.2524	0.6014
Factorized $S=AB^{\top}$	8	0.3379	0.3326	0.1188	0.5145
Factorized $S=AB^{\top}$	16	0.2518	0.2545	0.0647	0.4130
Factorized $S=AB^{\top}$	32	0.1855	0.1815	0.0344	0.3335
Factorized $S=AB^{\top}$	64	0.1355	0.1254	0.0256	0.2641

Appendix

Table 3: Oracle representation error across different prescribed spatial ranks

L

. Results are full-field relative RMSE over 20 test fields.

		Equispaced		Shifted		Jittered
Decoder	Params	Full $\downarrow$	Unobs. $\downarrow$	Full $\downarrow$	Unobs. $\downarrow$	Full $\downarrow$	Unobs. $\downarrow$
Factorized, $L=32$	21.30M	$0.6523$	$0.6668$	$0.6585$	$0.6752$	$0.6682$	$0.6855$
Factorized, $L=64$	23.27M	$0.6452$	$0.6600$	$0.6535$	$0.6707$	$0.6643$	$0.6823$
Dense $S$	19.36M	$\mathbf{0.2561}$	$\mathbf{0.2692}$	$\mathbf{0.2594}$	$\mathbf{0.2744}$	$\mathbf{0.2780}$	$\mathbf{0.2950}$

Appendix

Table 4: End-to-end reconstruction with factorized and dense spatial decoders under the H15/V15 protocol. Results are mean relative RMSE over 80 test fields. “Full” and “Unobs.” denote full-field and unobserved-region relative RMSE, respectively.

Train fields	Decoder	Full $\downarrow$	Unobs. $\downarrow$	Full $\downarrow$	Unobs. $\downarrow$	Full $\downarrow$	Unobs. $\downarrow$	Avg. Unobs. $\downarrow$
		Equispaced		Shifted		Jittered
50	Direct output	$0.906\pm 0.021$	$0.973\pm 0.023$	$0.900\pm 0.021$	$0.972\pm 0.023$	$0.902\pm 0.020$	$0.975\pm 0.022$	0.973
	$S/c$	$\mathbf{0.876\pm 0.028}$	$\mathbf{0.928\pm 0.026}$	$\mathbf{0.871\pm 0.025}$	$\mathbf{0.928\pm 0.022}$	$\mathbf{0.872\pm 0.026}$	$\mathbf{0.930\pm 0.024}$	0.929
100	Direct output	$0.704\pm 0.076$	$0.754\pm 0.082$	$0.698\pm 0.074$	$0.754\pm 0.080$	$0.714\pm 0.076$	$0.771\pm 0.082$	0.760
	$S/c$	$\mathbf{0.641\pm 0.077}$	$\mathbf{0.677\pm 0.078}$	$\mathbf{0.632\pm 0.078}$	$\mathbf{0.671\pm 0.080}$	$\mathbf{0.650\pm 0.075}$	$\mathbf{0.692\pm 0.076}$	0.680
200	Direct output	$0.503\pm 0.096$	$0.539\pm 0.103$	$0.500\pm 0.094$	$0.539\pm 0.102$	$0.525\pm 0.094$	$0.566\pm 0.102$	0.548

Appendix

Table 5: Reconstruction performance under different training-set sizes. Results are mean

\pm

standard deviation over 80 test fields. “Full” and “Unobs.” denote full-field and unobserved-region relative RMSE, respectively. “Avg. Unobs.” is the average unobserved-region RMSE across the three sampling geometries. Lower is better.

Width	Decoder	Params	Full $\downarrow$	Unobs. $\downarrow$	Full $\downarrow$	Unobs. $\downarrow$	Full $\downarrow$	Unobs. $\downarrow$	Avg. Unobs. $\downarrow$
			Equispaced		Shifted		Jittered
128	Direct output	2.17M	$0.809\pm 0.062$	$0.867\pm 0.067$	$0.808\pm 0.064$	$0.872\pm 0.069$	$0.814\pm 0.061$	$0.879\pm 0.066$	0.873
	$S/c$	2.19M	$\mathbf{0.662\pm 0.081}$	$\mathbf{0.706\pm 0.085}$	$\mathbf{0.662\pm 0.079}$	$\mathbf{0.710\pm 0.084}$	$\mathbf{0.679\pm 0.075}$	$\mathbf{0.729\pm 0.080}$	0.715
192	Direct output	4.85M	$0.634\pm 0.103$	$0.680\pm 0.111$	$0.633\pm 0.101$	$0.683\pm 0.110$	$0.652\pm 0.099$	$0.704\pm 0.107$	0.689
	$S/c$	4.89M	$\mathbf{0.481\pm 0.106}$	$\mathbf{0.511\pm 0.111}$	$\mathbf{0.480\pm 0.103}$	$\mathbf{0.512\pm 0.110}$	$\mathbf{0.510\pm 0.105}$	$\mathbf{0.545\pm 0.111}$	0.523
256	Direct output	8.57M	$0.453\pm 0.104$	$0.485\pm 0.112$	$0.453\pm 0.101$	$0.489\pm 0.110$	$0.477\pm 0.103$	$0.515\pm 0.111$	0.497

Appendix

Table 6: Reconstruction performance under different model capacities. Results are mean

\pm

standard deviation over 80 test fields. “Params” denotes the total number of trainable parameters. Lower is better.

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Reconstructing oscillatory wave fields from scattered sensors is a severely underdetermined inverse problem. Beyond the challenges of general physical-field reconstruction, wave responses are complex-valued, frequency-sensitive, and highly oscillatory, while costly simulation and sensing often leave only extreme-sparse observations. Existing low-rank, operator, and diffusion approaches are largely designed for real-valued, smoother fields; dense pixel-space diffusion is particularly inefficient for oscillatory complex fields and difficult to scale to 3D. We propose HarmoCore, which places a generative prior in a compact, continuous, and structured wave-field latent. HarmoCore represents joint real--imaginary channels with Functional Tucker cores over shared continuous spatial bases, learns a frequency-conditioned core diffusion prior, and performs Diffusion Posterior Sampling directly in core space. At fixed sensor coordinates, the multilinear decoder induces an explicit likelihood guidance operator, avoiding dense pixel-space correction. Optional target-equation residual guidance further promotes physical consistency. Experiments on 2D Helmholtz, 2D synthetic wave fields, and 3D Helmholtz show substantial gains under 1%--2% sensing while remaining practical in three dimensions.

Abstract

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