TRACE: Retrospective Streaming Generation of Physical Fields under Sparse Structured Sensing

Xinyu Zhang; Lihao Chen; Panqi Chen; Lei Cheng; Ting Zhang; Jianlong Li; Shikai Fang

stat.MLAug 26, 2026

TRACE: Retrospective Streaming Generation of Physical Fields under Sparse Structured Sensing

Authors: Xinyu Zhang, Lihao Chen, Panqi Chen, Lei Cheng, Ting Zhang, Jianlong Li, Shikai Fang

Organizations: College of Information Science and Electronic Engineering, Zhejiang University

Abstract

Reconstructing continuous physical fields from sparse measurements is central to scientific monitoring, inverse modeling, and digital-twin construction. Generative reconstruction has recently emerged as a promising paradigm for this task by learning data-driven physical priors that complete plausible full fields from limited observations. However, existing methods largely assume fixed, batch conditioning, whereas real sensing systems often produce structured streams: probes scan local regions, instruments observe moving fields of view, and communication constraints may leave entire frames missing. We propose TRACE, a retrospective streaming generative reconstruction framework for physical fields under structured sensing. TRACE performs approximate Bayesian inference in a learned continuous-coordinate latent space, converting sparse off-grid measurements into generative latent evidence, fusing it with a state-space temporal prior through Kalman-style filtering, and refining under-observed past frames via retrospective smoothing. Experiments on active matter, ocean sound-speed fields, and supernova simulations show that TRACE matches or surpasses frame-wise generative reconstructors, offline spatiotemporal methods, and streaming data-assimilation baselines in reconstruction quality under temporally sparse and spatially localized sensing protocols.

Figures & tables

Figure 1: Overview of TRACE. Top: offline pretraining of the continuous-coordinate field representation and latent diffusion prior. Bottom: online streaming inference, which fuses per-frame generative evidence with a state-space prior via Kalman filtering and refines it via retrospective smoothing.

(a) Temporally Sparse Observations
	Active Matter			Ocean			Supernova
	Miss	Blk-5	Blk-10	Miss	Blk-5	Blk-10	Miss	Blk-5	Blk-10
LRTFR	0.413	0.258	0.461	0.205	0.163	0.177	0.561	0.430	0.503
MMGN	0.313	0.217	0.360	0.097	0.080	0.092	0.318	0.318	0.320
SDIFT	0.490	0.391	0.530	0.158	0.127	0.145	0.436	0.398	0.424
DBF	0.960	0.814	0.873	0.809	0.627	0.709	0.855	0.809	0.845

Table 1: Field-domain RMSE under structured sensing (

\rho=1\%

). (a) Temporally sparse: Miss (gap

n{=}3

, one active frame in four) and Blk-

L

(contiguous blackout of

L\in\{5,10\}

frames). (b) Spatially localized: moving window at local density

\rho_{\text{loc}}=15\%

, with S-curve single-pass and circular sweeps of one and three loops. Lower is better; bold is best per column; “—” marks cells a per-frame method cannot reconstruct.

Figure 2: Moving-window trajectories on Active Matter ( $256\times 256$ domain, $96\times 96$ window). The centre path (colour-coded by time) and shaded squares mark representative window positions; circular revisits accumulate coverage over time.

Figure 3: Qualitative reconstruction on Active Matter under a one-lap circular moving window.

Figure 4: Regime-structured analysis of TRACE.

Appendix figures & tables26 assets

Supplementary material from the paper’s appendix.

Appendix

	AM	Ocean	SN
Tucker rank $R$	$(1,48,48)$	$(5,20,20)$	$(32,32,32)$
Latent dim. $d$	$2304$	$2000$	$32768$
SIREN $\omega$	$20$	$10$	$20$

Appendix

Table 2: Per-dataset architecture configuration: Tucker multilinear rank

R

, vectorised-core latent dimension

d=\prod_{k}R_{k}

, and SIREN basis frequency

\omega

Dataset	Regime	$\zeta$	$(\alpha,\ell,\sigma_{f})$
AM	Control	$0.001$	$(0.5,1.0,2.0)$
	Temporally Sparse	$0.001$	$(0.5,5.0,1.0)$
	Spatially Localized	$0.01$	$(0.5,5.0,1.0)$
Ocean	Control	$0.06$	$(0.5,2.0,1.0)$
	Temporally Sparse	$0.06$	$(0.5,5.0,2.0)$
	Spatially Localized	$0.2$	$(0.5,5.0,2.0)$

Appendix

Table 3: Inference hyperparameters per dataset and observation regime. Shared across all runs:

S{=}20

N{=}100

\sigma_{\text{obs}}{=}0.05

\beta{=}1

, diagonal fusion.

Full-availability RMSE	Ocean	Active Matter	Supernova
DBF (matched $\rho=3\%$ )	$0.346$	$0.473$	$0.681$
TRACE-Smoother (ours)	$0.040$	$0.066$	$0.298$
ratio	$8.8\times$	$7.1\times$	$2.3\times$

Appendix

Table 4: DBF at its matched-density upper bound (

\rho=3\%

for both training and testing, no missing frames) versus our framework evaluated at the same

\rho=3\%

	Active Matter		Ocean		Supernova
	$1\%$	$3\%$	$1\%$	$3\%$	$1\%$	$3\%$
LRTFR	0.156	0.090	0.161	0.114	0.406	0.418
MMGN	0.162	0.148	0.070	0.047	0.318	0.313
SDIFT	0.313	0.220	0.112	0.078	0.379	0.342
DBF	0.747	0.473	0.527	0.346	0.771	0.681
TRACE-Frame	0.078	0.066	0.045	0.040	0.322	0.309

Appendix

Table 5: Control regime (frame-rich: every frame observes the full domain at density

\rho

), field-domain RMSE. Lower is better; bold is best per column.

	Active Matter		Ocean		Supernova
	$1\%$	$3\%$	$1\%$	$3\%$	$1\%$	$3\%$
LRTFR	0.235	0.178	0.175	0.136	0.460	0.424
MMGN	0.187	0.178	0.073	0.055	0.317	0.313
SDIFT	0.416	0.403	0.154	0.147	0.409	0.370
DBF	0.907	0.755	0.713	0.587	0.829	0.771
TRACE-Frame	—	—	—	—	—	—

Appendix

Table 6: Missing pattern with gap

n{=}1

(every other frame active), field-domain RMSE at

\rho\in\{1\%,3\%\}

. Bold is best per column.

	Active Matter			Ocean			Supernova
	Miss	Blk-5	Blk-10	Miss	Blk-5	Blk-10	Miss	Blk-5	Blk-10
LRTFR	0.377	0.221	0.442	0.177	0.128	0.142	0.450	0.425	0.445
MMGN	0.307	0.205	0.352	0.084	0.060	0.075	0.314	0.313	0.316
SDIFT	0.478	0.366	0.526	0.148	0.110	0.139	0.399	0.356	0.397
DBF	0.901	0.617	0.742	0.747	0.502	0.627	0.829	0.747	0.812
TRACE-Frame	—	—	—	—	—	—	—	—	—

Appendix

Table 7: Temporally sparse observations at

\rho=3\%

, field-domain RMSE. Columns follow main-text Table 1 ( Miss : gap

n{=}3

; Blk-

L

: blackout length

L

). Bold is best per column.

	Active Matter	Ocean	Supernova
DPS ( $S{=}1$ )	$0.091$	$0.055$	$0.365$
TRACE-Frame	$0.078$	$0.045$	$0.322$
TRACE-Smoother	$\mathbf{0.078}$	$\mathbf{0.044}$	$\mathbf{0.310}$

Appendix

Table 8: Effect of the transition prior under full observation (Control regime,

\rho=1\%

), field-domain RMSE. “DPS (

S{=}1

)” is vanilla single-sample DPS.

Fusion	Time/frame	Outcome
Diagonal (ours)	$1.7$ ms	RMSE $0.078$
Full-covariance	$\gtrsim\!20$ s	OOM at $d{>}5000$

Appendix

Table 9: Per-frame fusion time (Active Matter, Control regime,

\rho{=}1\%

d{=}2304

; CPU, excluding DPS sampling).

$S$	$1$	$5$	$20$	$50$
RMSE	$0.091$	$0.083$	$\mathbf{0.081}$	$0.080$

Appendix

Table 10: Sample-count

S

ablation for TRACE-Frame on Active Matter (Control regime,

\rho=1\%

10

trajectories), field-domain RMSE.

Panel of Figure 5 — Figure 5: Hyperparameter sensitivity on Active Matter ( $\rho=1\%$ , full TRACE-Smoother), one curve per Miss level $n\in\{0,1,3,5\}$ . (a) RMSE is strongly $\ell$ -sensitive and the optimal $\ell$ shifts with $n$ , saturating at $\ell\!\approx\!5$ for all sparse regimes. (b) RMSE is nearly $\alpha$ -insensitive (note the compressed vertical range); the dotted line is the untempered Kalman update ( $\alpha\!=\!1$ ). Other hyperparameters held fixed ( $\sigma_{f}\!=\!1$ , diagonal fusion).

$\sigma$	TRACE-Frame	TRACE-Filter	TRACE-Smoother
$0.0$	$0.078$	$0.078$	$\mathbf{0.078}$
$0.1$	$0.097$	$0.097$	$0.098$
$0.3$	$0.195$	$0.192$	$\mathbf{0.184}$

Appendix

Table 11: Observation-noise robustness (Active Matter, Control regime,

\rho=1\%

), field-domain RMSE.

\sigma

is the additive Gaussian sensor noise.

\sigma{=}0

28

trajectories;

\sigma{>}0

3

trajectories.

Active Matter ( $\rho{=}1\%$ )
Method	Miss-3	Blk-5	Blk-10
LRTFR	$0.371/0.220/0.422$	$0.218/0.156/0.451$	$0.368/0.160/0.660$
MMGN	$0.313/0.156/0.366$	$0.217/0.164/0.418$	$0.360/0.164/0.635$
SDIFT	$0.458/0.363/0.490$	$0.352/0.280/0.626$	$0.472/0.303/0.710$
DBF	$0.930/0.826/0.965$	$0.789/0.744/0.957$	$0.843/0.749/0.975$
TRACE-Filter	$0.429/0.084/0.544$	$0.211/0.103/0.619$	$0.385/0.103/0.780$

Appendix

Table 12: Per-frame error decomposition at

\rho=1\%

, stacked by dataset (Active Matter, Ocean, Supernova). Each cell reports total / observed-frame / missing-frame RMSE under Miss-3, Blk-5, and Blk-10 (same symbols as Table 1 ). Bold marks the best total per column within each dataset block.

	nominal $0.90$	nominal $0.95$	ECE
TRACE-Smoother	$0.80$	$0.86$	$\mathbf{0.058}$
TRACE-Frame ens.	$0.56$	$0.62$	$0.220$

Appendix

Table 13: Reliability on Active Matter (Control regime,

28

trajectories): empirical coverage at two nominal credible levels, and expected calibration error (ECE) over ten levels. Closer to the nominal level is better.

$\rho$	TRACE-Filter	TRACE-Smoother
$0.5\%$	$0.104$	$\mathbf{0.099}$
$1\%$	$0.093$	$\mathbf{0.089}$
$2\%$	$0.082$	$\mathbf{0.079}$
$10\%$	$0.072$	$\mathbf{0.070}$

Appendix

Table 14: Observation-rate sweep (Active Matter, Control regime,

\sigma_{\text{obs}}=0

28

trajectories, DPS reverse steps

N{=}50

, guidance

\zeta\!\propto\!1/\rho

). Field-domain RMSE; lower is better. This sweep uses a reduced

50

-step sampler and is therefore not directly comparable to the default

100

-step results of Table 5 .

Loops	1	2	3	4	6
AM, TRACE-Smoother	0.537	0.498	0.476	0.475	0.468
AM, MMGN	0.782	0.770	0.777	0.771	0.768
Ocean, TRACE-Smoother	0.106	0.097	0.088	0.083	0.084
Ocean, MMGN	0.277	0.283	0.271	0.284	0.286
SN, TRACE-Smoother	0.334	0.320	0.318	0.314	0.311
SN, MMGN	0.321	0.321	0.321	0.321	0.321

Appendix

Table 15: Coverage accumulation: RMSE versus number of revisits (loops) under the circular moving window. “TRACE-Smoother” is the full model (RTS smoothing); “MMGN” is the strongest offline per-frame baseline.

Component	Time
DPS reverse ODE ( $N{=}100$ , $S{=}20$ )	$\approx 3.5$ s / frame
SS-GP filter (forward)	$0.64$ ms / frame
RTS smoothing (backward)	$2.35$ ms / frame
FTM decode	$1.26$ ms / frame
diagonal fusion ( $d=32768$ )	$4.1$ ms / frame

Appendix

Table 16: Per-frame inference cost on Active Matter (

d=2304

), measured on a single NVIDIA RTX 5090 GPU (the closed-form fusion step of Table 9 is timed on CPU).

Figure 6: Active Matter, Miss pattern ( $\rho=1\%$ ); five frames mixing observed and missing.

Figure 7: Ocean, Miss pattern ( $\rho=1\%$ , gap $n{=}3$ ): TRACE-Smoother recovers coherent spatial structures throughout missing intervals; TRACE-Filter drifts on gap frames; offline baselines produce temporally inconsistent transitions. Columns: ground truth, observation, TRACE-Smoother, TRACE-Filter, SDIFT, MMGN, DBF, LRTFR.

Figure 8: Supernova ( $z{=}32$ slice), Miss pattern ( $\rho=1\%$ ).

Figure 9: Active Matter, Blk- $10$ ( $\rho=1\%$ ): frames $7$ – $16$ unobserved.

Figure 10: Ocean (layer 2), Blk- $10$ ( $\rho=1\%$ ): frames $7$ – $16$ unobserved.

Figure 11: Supernova ( $z{=}32$ ), Blk- $10$ ( $\rho=1\%$ ): frames $3$ – $12$ unobserved.

Figure 12: Active Matter, moving window (circular, one lap, $\rho_{\text{loc}}=15\%$ ); nine frames span one sweep. TRACE-Smoother accumulates previously-swept regions into a coherent global field, while the per-frame baselines collapse outside the current window.

Figure 13: Ocean (layer 2), moving window (circular, one lap, $\rho_{\text{loc}}=15\%$ ).

Figure 14: Supernova ( $z{=}32$ ), moving cube (circular, one lap).

Figure 15: Failure case: Active Matter moving window where per-frame evidence is too weak for the generative prior; TRACE-Filter propagates the error and TRACE-Smoother cannot recover.

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