Spatiotemporal Reconstruction of Global Radiation from Sparse Sensor Sequences
Authors: Kazuma Kobayashi, Tapas Tripura, Jay Phil Yoo, Samrendra Roy, Diab Abueidda, Seid Koric, Souvik Chakraborty, Syed Bahauddin Alam
Organizations: Nuclear, Plasma and Radiological Engineering, The Grainger College of Engineering, University of Illinois Urbana-Champaign, IL, USA. · Department of Applied Mechanics, Indian Institute of Technology Delhi, New Delhi, India. · Civil and Urban Engineering Department, New York University Abu Dhabi, Abu Dhabi, UAE. · National Center for Supercomputing Applications, 1205 W Clark St,, Urbana, 61801, IL, USA. · Mechanical Science and Engineering, The Grainger College of Engineering, University of Illinois Urbana-Champaign, IL, USA. · Yardi School of Artificial Intelligence, Indian Institute of Technology Delhi, New Delhi, India.
Much of the physical world is inaccessible to direct measurement and can be reached only through the sparse traces it leaves in other observable variables. Here we introduce the Temporal Radiation Operator Network (TRON), a spatiotemporal neural operator that learns proxy-to-field mappings, reconstructing otherwise unobserved physical fields from sparse, indirect sensor histories. For cosmic-radiation mapping, TRON reconstructs sea-level effective dose at 65,341 locations from histories of 12 neutron monitors, achieving 0.096% mean relative L2 error against EXPACS-generated reference fields in a chronologically held-out year; batched GPU inference evaluates a global field in approximately ∼1ms, compared with ∼57.6s for EXPACS on one CPU thread. On measured weather networks, TRON reconstructs temperature at held-out stations from thermometer or barometer histories, outperforming interpolation, calendar-based and hypsometric baselines. These results establish proxy-to-field reconstruction as a general route to recovering spatially distributed physical information beyond the direct reach of sparse sensor networks.
Figures & tables
Figure 1: Schematic overview of the effective-dose reconstruction framework. a Primary cosmic-ray flux Φ drives atmospheric cascades that determine effective dose Xj=gj(⋅) and neutron-monitor counts Ys=fs(⋅) ( A , atmospheric conditions; G , geomagnetic field; h , altitude; counts schematic). Light-blue arrows denote the forward process (dashed, proxy relationship); the salmon arrow denotes TRON reconstruction. The map shows the EXPACS reference on 10 September 2023, the median-error test day. b Twelve input stations, colored by latitude; codes and coordinates are given in Supplementary Table S1. c TRON maps T -day count histories Y from S stations and query coordinates r to predicted dose X^ at P points; Hphys denotes the physics-based forward mapping. d Single-branch encoding of all stations versus separate station encoders followed by fusion; y(t) contains the S counts on day t . e Spatial trunk evaluated at latitude–longitude queries. f Branch and trunk features are multiplied element-wise ( ⊙ ), summed and combined with bias β to produce z^j . Rescaling gives X^j=mj+(Mj−mj)z^j , using training-period reference extrema mj and Mj . The map shows the corresponding S-LSTM prediction (single-branch long short-term memory encoder; T=30 days; one trained model; relative L2 error 0.096% ). Both maps share a color scale of 0.027 – 0.036μSvh−1 .
Figure 2
Figure 3: Location-specific dose reconstruction across twelve globally distributed cities. Results are for S-LSTM with a 30-day input window, evaluated over the 2023 test year using one trained model. Each city is represented by the nearest node of the 1∘ grid, whose 2001–2021 values were training targets. a City locations, colored by latitude. b Left, predicted versus EXPACS reference dose rates for all city–day pairs ( n=4,380 ), colored as in a ; dashed line, identity. Right, distribution of absolute relative errors over the same pairs; solid line, median; dashed line, 95th percentile. c Daily signed relative errors, (prediction−reference)/reference , for each city ( n=365 days), ordered from south to north and colored as in a . Center lines indicate medians, boxes the interquartile ranges, whiskers 1.5 times the interquartile range, and points observations beyond the whiskers; dashed line, zero. d Daily mean signed relative error across the twelve cities (line), with the 5th–95th percentile range across cities (band); dashed line, zero.
Figure 4: Correlation structure of the learned latent representations. a Single-branch and multi-branch architectures. Blue cells, input neutron counts; green, branch networks; pink, trunk network; ⊙ , element-wise multiplication. Labels (b)–(d) identify the branch-side tensors analyzed in the corresponding panels, before multiplication with the trunk output; the dashed ring marks the station-specific outputs pooled in c . b – d Pearson correlations ρ between 128 latent features, with the mean ∣ρ∣ over off-diagonal pairs: b single-branch S-LSTM, 0.814 , calculated over n=365 test days; c multi-branch M-LSTM before fusion, 0.246 , calculated by pooling the twelve station outputs over n=4,380 station–days; d M-LSTM after fusion, 0.522 , calculated over n=365 test days, with the 16 features constant over the year assigned ρ=0 . Both models use 30-day input windows; one trained model is analyzed per architecture.
Figure 5: Computational cost of global dose-field reconstruction. a Time per global field for EXPACS on one CPU thread (AMD Ryzen 9 7950X3D; gray points, three runs) and the trained networks on an NVIDIA A100 GPU in batches of 16, shown on a logarithmic scale. Colored points represent observation windows of 7, 30, 60 and 90 days, from left to right. The arrow indicates the speed-up range across the four TRON variants, including differences in hardware and parallel execution. b Neural-network inference time as a function of observation-window length. S- and M-, single- and multi-branch TRON; GRU, gated recurrent unit; LSTM, long short-term memory; DeepONet, deep operator network with a static branch. GPU times are the total inference time divided by the number of fields evaluated over the 2023 windows, measured in one timed pass; no error bars are shown. These are average times per field under batching, not independently measured single-field latencies. Numerical values are provided in Supplementary Table S5.
Figure 6: Evaluation on a finer spatial grid without retraining the network. The S-LSTM model uses a 30-day observation window and was trained on 1∘ EXPACS reference fields; one trained model is evaluated throughout. Errors are calculated against the EXPACS reference at the corresponding resolution. a Daily global relative L2 errors on the 1∘ and 0.25∘ grids over the 2023 test year. The star marks the median-error day, 10 September 2023. Right, comparison of errors at the two resolutions, with one point per day ( n=365 ); dashed line, identity; r , Pearson correlation. b Signed relative errors, (prediction−reference)/reference , on the representative day over the North America sector ( 140∘ – 40∘ W, 0∘ – 60∘ N). c Relative L2 errors over the global domain and the regional sector shown in b . Light gray bars ( 1∘ output, interpolated), the normalized network output bilinearly interpolated from 1∘ to 0.25∘ . For both finer-grid evaluations, the per-point output-scaling parameters (mj,Mj) are fitted using the 0.25∘ EXPACS reference from 2001–2021. Bars show means over n=365 test days; error bars denote ±1 s.d. Regional SSIM values are 0.9985±0.0006 at 1∘ and 0.9997±0.0001 at 0.25∘ (mean ± s.d. over test days).
Figure 7Figure 8
Model
Type
# Branches
Input Size
Hidden Dim.
# layers
d
# parameters
S-GRU
GRU
1
12
128
4
128
401,921
S-LSTM
LSTM
1
12
128
4
128
519,169
M-GRU
GRU
12
1 (per sensor)
128
4
128
4,404,865
M-LSTM
LSTM
12
1 (per sensor)
128
4
128
5,794,945
Table 1: Architectural specifications of TRON variants. S-GRU and S-LSTM denote TRON models with single-branch GRU and LSTM encoders, respectively. M-GRU and M-LSTM denote TRON models with multi-branch GRU and LSTM encoders, where each sensor is processed independently. d , latent dimension shared by the branch and trunk outputs.
Station Code
Location
Country
Lat.
Lon.
Rc (GV)
References
ATHN
Athens
Greece
37.97
23.78
8.53
[ 40 , 41 , 42 ]
JUNG
Jungfraujoch
Switzerland
46.55
7.98
4.49
[ 43 ]
NEWK
Newark
USA
39.68
−75.75
2.40
[ 44 ]
KERG
Kerguelen
France
−49.35
70.25
1.14
[ 45 ]
OULU
Oulu
Finland
65.05
25.47
0.81
[ 46 ]
APTY
Apatity
Russia
67.57
33.40
0.65
[ 47 ]
Table S1: List of neutron monitoring stations used in this study. Lat., latitude (positive north); Lon., longitude (positive east), in degrees. Rc : vertical geomagnetic cut-off rigidity from the NMDB station list [ 26 ] . JUNG is the IGY-type monitor at Jungfraujoch.
Data Set
Sequence Length ( T )
7
30
60
90
Training
7664
7641
7611
7581
Validation
365
365
365
365
Test
365
365
365
365
Table S2: Number of sequences for each dataset and sequence length after applying the sliding window approach. Each sequence corresponds to a contiguous temporal segment of daily neutron counts across 12 monitoring stations, with target fields of size 65,341 grid points.
Figure S1: Climatological annual cycle of the simulated grid-point mean effective dose over 2001–2023. a Daily grid-point mean dose obtained by averaging the effective dose field over all 65,341 grid points and then averaging by calendar day (day-of-year). b Corresponding anomaly after removal of a 365-day running mean, with the dashed line indicating zero.
Figure S2: Autocorrelation function of the grid-point mean effective dose anomaly D′(t) for the period 2001–2023, shown for lags up to 800 days. The shaded band denotes the 95% confidence interval. The autocorrelation first enters the confidence bounds at a lag of 32 days and exhibits no secondary peak near a one-year lag, indicating short-to-medium-term temporal correlations and the absence of a strong annual seasonal cycle.
Seq
Model
Relative L2 Error (%)
RMSE
MAE
SSIM
7
Static-branch DeepONet
1.87×10−1
6.26×10−5
5.05×10−5
0.998934
S-GRU
1.00×10−1
3.37×10−5
2.84×10−5
0.999546
S-LSTM
1.12×10−1
3.78×10−5
3.11×10−5
0.999370
M-GRU
8.98×10−1
3.01×10−4
2.69×10−4
0.998256
M-LSTM
6.24×10−1
2.09×10−4
1.88×10−4
0.998120
GNO
1.86
6.24×10−4
4.86×10−4
0.738379
Table S3: Performance comparison across sequence lengths. All TRON variants—S-GRU, S-LSTM, M-GRU, and M-LSTM—share the same TRON backbone and differ only in temporal encoding (single vs. multi-branch, GRU vs. LSTM). Linear regression baselines are included for comparison. All values are from a single training run per configuration.
Figure S3: Branch network architectures in TRON and the fusion with the trunk output. a Single-branch TRON: neutron count sequences from all monitoring stations are concatenated into a T×S tensor ( T , input window length in days; S=12 stations) and processed through four stacked GRU (gated recurrent unit) or LSTM (long short-term memory) layers of 128 units, followed by layer normalization and a linear layer applied at every time step ( T×128 ); the output at the last time step (128 values) is the branch output. b Multi-branch TRON: each neutron monitor’s sequence ( T×1 ) is processed independently by its own GRU/LSTM encoder of the same form. The resulting sensor-specific latent representations are fused via element-wise multiplication across the stations to give the branch output (last time step, 128 values). c Fusion with the trunk network: the branch output is multiplied element-wise ( ⊙ ) with the trunk network’s spatial embedding and summed, and a learnable scalar bias β is added, giving a normalized dose; this is rescaled with the per-point output scaling (mj,Mj) to the effective dose rate at query point j .
Type
Variation
# of trainable parameters
Linear Regressions
Static
849,433
Temporal
23,588,101
Neural Operators
Static-branch DeepONet
84,609
GNO
1,558,765
GINO
6,596,995
TRON
S-GRU
401,921
Table S4: Summary of number of model parameters. The temporal linear regression has (12T+1)×65,341 parameters; the value shown is for T=30 days.
Method
Simulation (s)
Inference time (ms), input window T (days)
7
30
60
90
EXPACS (one CPU thread)
57.6
-
-
-
-
DeepONet
-
0.981
1.256
0.987
1.007
Single-branch GRU
-
0.993
1.006
1.016
1.025
Single-branch LSTM
-
1.005
1.021
1.032
1.054
Multi-branch GRU
-
1.262
1.309
1.444
1.532
Table S5: Simulation and inference time per daily global field ( 65,341 points), plotted in main-text Fig. 5. Inference on an NVIDIA A100 GPU in batches of 16: total GPU time divided by the number of fields, over the input windows lying entirely within 2023, from one timed pass. The EXPACS simulation of the same field ran on one thread of an AMD Ryzen 9 7950X3D CPU (mean of three runs). No spread is given: the standard deviation of the batch times, divided by the mean batch size, is dominated by the last, smaller batch (7, 2 and 4 fields at T=7 , 60 and 90 days; none at 30 days) rather than by timing noise. Values from inference_time_corrected.log and expacs_timing/speedup.log ( code/analysis/resource/ ).
Figure S4: Comparative performance of neural operator architectures. Mean relative L2 error versus number of trainable parameters at T=30 days; each point is a single training run. Ring, S-LSTM, the main TRON configuration. TRON variants (S-, single-branch; M-, multi-branch; LSTM, long short-term memory; GRU, gated recurrent unit) and the static-branch DeepONet (deep operator network) colored as in main-text Fig. 2; yellow, GNO (graph neural operator); brown, GINO (geometry-informed neural operator); open symbols, linear regressions (static, current day only; temporal, whole window; Section S4 ). Values in Tables S3 and S4 .
Predictor
Trained for the task
Relative L2 error (%)
SSIM
Reference field of the previous day (persistence) a
no
0.291
0.9999
TRON nowcast of the previous day, carried forward
no
0.314
0.9994
Static-branch DeepONet
yes
0.546
0.9919
TRON (S-LSTM)
yes
0.805
0.9978
TRON (S-LSTM), dropout 0.2, mean of 50 samples
yes
0.864
0.9976
Table S6: One-day-ahead prediction of the 2023 reference fields (365 target days). Each trained model is a single training run; five further runs of the dropout configuration with different seeds gave relative L2 errors of 0.90–2.04% with dropout disabled at inference. SSIM as defined in the main-text Methods.
Figure S5: One-day-ahead prediction of the 2023 reference fields (365 target days); each trained model is a single training run. a , Global relative L2 error (centered 7-day running mean). Right: 2023 means, including the TRON forecaster without dropout and five further seeds of the dropout configuration with dropout disabled (open squares); dotted line, carried-forward nowcast. Persistence, the previous day’s reference field, is not available at inference; MC mean, mean of 50 Monte Carlo dropout samples; S-LSTM, single-branch TRON with an LSTM (long short-term memory) encoder; DeepONet, deep operator network. b , Reference dose rate and predictions at the Helsinki grid node (60 ∘ N, 25 ∘ E). Right: the weeks around the largest one-day decrease of 2023 (9 to 10 May), daily values marked; shaded gray at left.
Adaptation window (days)
Evaluation days
O (12 stations)
A (13-ch adapted)
B (MXCO masked)
7
358
0.126%
0.134%
0.134%
14
351
0.127%
0.136%
0.136%
30
335
0.128%
0.141%
0.141%
Table S7: Relative L2 error in the North American sector after adding MXCO, mean over the evaluation days (one adapted model per window).
Adaptation window (days)
Evaluation days
O (12 stations)
A (13-ch adapted)
B (MXCO masked)
7
358
0.097%
0.104%
0.104%
14
351
0.097%
0.107%
0.107%
30
335
0.098%
0.110%
0.110%
Table S8: Global relative L2 error after adding MXCO, mean over the evaluation days (one adapted model per window).
Figure S6: Heat-exchanger configuration: computational fluid dynamics mesh and solution of simulation 0 of the 1,546-simulation dataset of ref. [ 55 ] . a Speed on the axial section z=0 of the whole tube (to scale) with the boundary inputs of TRON: wall heat flux q(x)=Asin(πx/L) ( L , tube length; 100 samples, normalized by A ) and uniform inlet velocity vin and temperature Tin . Orange, boundary inputs; blue lines, target plane. b The last 91 mm upstream of the target plane, tube wall cut open: dimpled tube ( D , diameter) with wavy-tape insert, dimensions as labeled. c Reconstruction target: speed on the plane x=0.789 m (3,977 nodes), viewed from downstream; white lines, streamlines of the in-plane velocity. One color bar for all panels.
Figure S7: Vorticity ( ω ) reconstruction for the Reynolds number Re = 195 test sequence (training used Re = 180, 190 and 200). a – c Three instants; t , dimensionless time since the first stored snapshot (snapshots every 0.2). Rows: reference from unsteady Reynolds-averaged Navier–Stokes (URANS) simulation; TRON reconstruction from the current and two preceding readings of the 1,030 vorticity sensors (dark dots, top row; cylinder in gray); absolute error. Vorticity scale symmetric, clipped at ±3.55 (99.5th percentile of ∣ω∣ over the three reference fields; maximum 11.3); error scale clipped at 0.295 (99.5th percentile of the three error fields; maximum 1.26); both maxima at the cylinder surface. Single training run; the test sequences were also used for early stopping.
Component
Explained variance ratio
Cumulative
PC1
0.999789
0.999789
PC2
2.079×10−4
0.9999972
PC3
2.800×10−6
0.99999999
PC4
7×10−9
1.000000
PC5, PC6
<10−9
1.000000
Table S9: Principal component spectrum of the reference dose fields, computed exactly over the full grid. The participation-ratio effective rank, (∑iλi)2/∑iλi2 , is 1.000 .
Figure S8: Intrinsic dimensionality of the EXPACS reference data and of the station inputs. PC k , k th principal component; effective rank, participation ratio (Table S9 ). (a) Principal component spectrum of the reference dose field (centered at each grid point; full grid). (b) PC1 coefficient c1(t) against the solar modulation index W , reconstructed from the Oulu count rate (Sato 2015, Table 2; no fitted parameters); points colored by year; red line, quadratic fit. Inset: R2 of a quadratic fit of c1 on W and of a fifth-order fit of c1 on the mean of the twelve raw count rates. (c) Principal component spectrum of the twelve daily neutron count series, standardized per station. (d) Inter-station Pearson correlation r of the raw daily counts; diagonal masked. Station labels, NMDB (Neutron Monitor Database) identifiers (Table S1 ).
GHCN TMAX
PeakWeather pressure → 2 m temperature
PeakWeather degree-hours
Target (unit)
daily maximum temperature, TMAX ( ∘ C)
hourly 2 m temperature ( ∘ C)
heating degree-hours below 18 ∘ C summed over the past 24, 48 or 72 h ( ∘ C h); instantaneous 2 m temperature as control
Branch input
TMAX at 12 input stations, 14-day window
station pressure (QFE) at 30 barometers, 24 h window; no thermometer
2 m temperature at 30 thermometers, 72 h window
Indirect input
no (same quantity)
yes (pressure for temperature)
no (same quantity; target is a window functional of it)
Source; license
GHCN-Daily [ 28 , 29 ] ; CC0-1.0
PeakWeather [ 30 ] ; CC-BY-4.0, attribution to MeteoSwiss
PeakWeather [ 30 ] ; CC-BY-4.0, attribution to MeteoSwiss
Stations: input / held-out / training
12 / 100 / 886
30 / 40 / 105
30 / 40 / 105
Station splits
3 random splits (0, 1, 2)
1 held-out split; 3 overlapping barometer draws (0, 1, 2)
1 held-out split
Table S10: Measured datasets used in main-text Fig. 8. Input stations feed the branch; training stations are the targets of the loss; held-out stations enter neither the loss, any scaler nor early stopping. Distances are great-circle, except the pre-registered pressure-input distances (Swiss LV95 plane). PC1 variance share: variance of the training-station field over the training period (886 GHCN stations, range over the 3 splits; 105 PeakWeather stations) explained by the first principal component (PC) after removing each station’s mean (raw) or harmonic climatology (anomaly; three annual harmonics, plus for PeakWeather three diurnal harmonics and their products). Unobserved GHCN targets are interpolated in time for this analysis only. The pressure time axis omits hours failing pressure quality control. QFE, station-level pressure; val., validation.
RMSE by split ( ∘ C)
method
clock
n
0
1
2
RMSE ( ∘ C)
pooled R2
S-LSTM
off
9
3.430
3.459
3.571
3.487 ± 0.066
0.912 ± 0.003
S-LSTM
on
9
3.423
3.434
3.489
3.449 ± 0.036
0.914 ± 0.004
S-GRU
off
9
3.383
3.437
3.562
3.461 ± 0.080
0.914 ± 0.002
S-GRU
on
9
3.394
3.386
3.518
3.433 ± 0.068
0.915 ± 0.003
flattened-window DeepONet
off
9
3.448
3.481
3.562
3.497 ± 0.058
0.912 ± 0.003
Table S11: GHCN TMAX: every branch and baseline at the 100 held-out stations. RMSE, root-mean-square error over all observed held-out station-days of the test period; pooled R2=1−SSE/SST about the single mean of all held-out test values. Network rows: n=9 runs (3 station splits × 3 training seeds); split columns, mean over seeds; last two columns, mean ± s.d. over the n runs (seed and split variation) or fits. Baseline rows: one deterministic fit per split ( n=3 ). S-LSTM (pre-declared primary) and S-GRU, TRON with a recurrent branch; flattened-window and static-branch DeepONets, feed-forward branch reading the whole window or the last step. Clock: sine and cosine of the day of year as extra inputs. IDW, inverse-distance weighting with power p . Baselines are described in Section S11 . ORACLE rows use the held-out stations’ own data; they are references, not attainable baselines.
contrast
clock
split
Δ RMSE ( ∘ C)
95% CI
stations better
outcome
S-LSTM − two-stage
off
0
− 0.119
[ − 0.270, − 0.008]
50/100
Better
S-LSTM − two-stage
off
1
− 0.052
[ − 0.113, + 0.008]
53/100
Unresolved
S-LSTM − two-stage
off
2
− 0.008
[ − 0.101, + 0.081]
43/100
Unresolved
S-LSTM − two-stage
on
0
− 0.101
[ − 0.241, + 0.001]
50/100
Unresolved
S-LSTM − two-stage
on
1
− 0.048
[ − 0.108, + 0.013]
59/100
Unresolved
S-LSTM − two-stage
on
2
− 0.082
[ − 0.185, + 0.012]
45/100
Unresolved
Table S12: GHCN TMAX: paired differences per station split. Δ RMSE, first minus second model’s held-out RMSE on the same station-days, mean over 3 training seeds ( ∘ C; negative favors the first); two-stage, the CV-tuned two-stage linear model with the same clock (Supplementary Table S11 ; CV, cross-validation). 95% CI, percentile interval of a paired bootstrap over the 100 held-out stations and calendar-week blocks (2000 replicates, seed 0). Stations better: held-out stations where the first model’s seed-mean RMSE is lower. Outcome, band δ=0.10∘ C: Better, CI upper bound <0 and Δ<−δ ; Worse, CI lower bound >0 and Δ>δ ; Tie, CI within [−δ,δ] ; Unresolved, otherwise. The rule was pre-registered for the pressure dataset; for GHCN it is applied by analogy, not pre-registered. Each split is one station partition, tested separately without correction for the 24 two-stage contrasts. Branches as in Supplementary Table S11 .
method
input
clock
n
RMSE ( ∘ C)
range ( ∘ C)
pooled R2
S-LSTM
pressure
off
9
1.897 ± 0.034
1.853–1.964
0.949 ± 0.002
S-LSTM
pressure
on
9
1.866 ± 0.045
1.802–1.952
0.950 ± 0.002
S-GRU
pressure
off
9
1.891 ± 0.036
1.844–1.959
0.949 ± 0.002
S-GRU
pressure
on
9
1.832 ± 0.026
1.787–1.870
0.952 ± 0.001
flattened-window DeepONet
pressure
off
9
2.008 ± 0.070
1.924–2.132
0.942 ± 0.004
flattened-window DeepONet
pressure
on
9
2.104 ± 0.075
1.988–2.227
0.937 ± 0.004
Table S13: PeakWeather pressure → 2 m temperature: every branch, control and baseline at the 40 held-out stations. RMSE, pooled R2 and branches as in Supplementary Table S11 ; 11,086 hourly test windows. Pressure-input network rows: n=9 runs (3 barometer draws × 3 training seeds; one held-out station set); thermometer-input rows: n=3 seeds. Baseline rows: one deterministic fit per draw ( n=3 ) or in total ( n=1 , no s.d.). Mean ± s.d. over the n runs or fits; range, minimum–maximum. Clock: sine and cosine of hour of day and day of year (networks); annual and diurnal harmonics (linear models). Time-shuffled: station channels permuted in time, calendar aligned (pre-registered control). H2: Supplementary Table S14 . NW, Nadaraya–Watson decoder without lapse-rate detrending; last hour, last hour of the window only. Italic rows use held-out data or true temperatures and are references, not attainable baselines: ORACLE ridge, fitted to held-out targets; ORACLE two-stage, decoder tuned on them; the last row decodes the true training-station temperatures.
test
cell
runs
value ( ∘ C)
95% CI
thresh.
stations
outcome
S1
S-LSTM pressure, clock on
9
1.866
–
3.546
–
pass
H1
S-LSTM pressure, clock on
18
− 2.098
[ − 2.442, − 1.767]
0.000
–
supported
H2
S-LSTM pressure, clock on
9
− 0.040
[ − 0.151, + 0.073]
0.100
27/40
Unresolved
H2, per draw
S-LSTM pressure, clock on, draw 0
3
− 0.006
[ − 0.117, + 0.104]
0.100
–
Unresolved
H2, per draw
S-LSTM pressure, clock on, draw 1
3
− 0.041
[ − 0.157, + 0.078]
0.100
–
Unresolved
H2, per draw
S-LSTM pressure, clock on, draw 2
3
− 0.073
[ − 0.212, + 0.064]
0.100
–
Unresolved
Table S14: PeakWeather pressure → 2 m temperature: pre-registered tests. RT , RMSE of the primary cell (S-LSTM, pressure, clock on; mean of 9 runs). S1 (sanity): RT against 0.9× the calendar-only linear RMSE. H1: RT minus clock-on time-shuffled S-LSTM, paired by draw and seed. H2: RT minus the CV-tuned two-stage linear RMSE (same inputs and clock); outcome classes as in Supplementary Table S12 , δ in column thresh. (threshold). The pooled H2 row decides; per-draw rows are descriptive; H2-type rows (secondary) cover other branches and clock off. H2a: the same contrast at the 105 training stations against the ridge stage; H2 credits pressure only if H2a favors TRON. H2b: pressure minus thermometer margin over the two-stage model; H1 and H2b are supported if the CI is below 0. H3: minimum time-shuffled RMSE against a pre-set floor (pass if above). H4: thermometer- minus pressure-input mean RMSE, untested; REVERSAL, thermometer less accurate. H5: last-hour minus 24 h-window two-stage RMSE (single fits); no window claim. 95% CI as in Supplementary Table S12 (105 stations for H2a). Runs, network runs used; stations, held-out stations (of 40) where the averaged network RMSE is below the comparator’s (–, not computed). Primary-cell seed s.d. (pooled within draw): 0.048 ∘ C. Flat.-window, flattened-window DeepONet; static-branch, static-branch DeepONet; st., stations.
target
clock
S-LSTM
S-GRU
flat.-window
static-branch
RMSE, mean ± s.d.
instantaneous ( ∘ C)
off
1.770 ± 0.070
1.949 ± 0.064
1.807 ± 0.043
1.689 ± 0.070
instantaneous ( ∘ C)
on
1.854 ± 0.173
1.980 ± 0.378
1.775 ± 0.085
1.715 ± 0.038
24 h ( ∘ C h)
off
27.2 ± 1.0
29.9 ± 3.0
29.7 ± 1.5
39.5 ± 0.3
24 h ( ∘ C h)
on
28.8 ± 1.7
30.5 ± 5.9
27.9 ± 0.3
37.7 ± 0.5
48 h ( ∘ C h)
on
52.4 ± 3.1
60.4 ± 2.6
51.8 ± 2.4
87.1 ± 0.7
Table S15: PeakWeather degree-hours: every branch, horizon and clock setting at the 40 held-out stations. Heating degree-hours: hourly deficit of 2 m temperature below 18 ∘ C summed over the past 24, 48 or 72 h; instantaneous temperature is the control. One station split; n=3 training seeds per cell; mean ± s.d. over seeds. RMSE as in Supplementary Table S11 , over 9,974 hourly test windows; 1−R2 , unexplained fraction of the pooled held-out target variance (target s.d.: 8.46 ∘ C, 165.0, 325.5, 483.5 ∘ C h for instantaneous, 24, 48, 72 h). The 48 h horizon was run with clock only. Branches as in Supplementary Table S14 , clock as in Supplementary Table S13 . Held-out stations where static-branch RMSE exceeds S-LSTM’s (24 h, clock on): 37, 35 and 35 of 40 for seeds default, 11 and 22 (default-seed median ratio 1.61). A planned 120 h horizon is excluded: it exceeds the 72 h window, so its 6 runs duplicated the 72 h target (Supplementary Table S17 ). No linear baseline was fitted.
dataset
input → target
key numbers
runs
reason
NOAA OI SST, weekly [ 31 ]
SST at 12 grid cells → SST at withheld grid cells
random 20% of cells: held-out relative L2 error (%) S-LSTM 5.94 ± 0.44; S-GRU 5.99 ± 0.05; flat.-window 5.55 ± 0.24; static-branch 5.74 ± 0.04; sensor-free IDW climatology 4.38 (skill against it − 0.88 to − 0.61). 20% in 10 ∘ blocks: held-out relative L2 error (%) S-LSTM 6.92 ± 0.33; S-GRU 6.95 ± 0.07; flat.-window 6.54 ± 0.20; static-branch 6.79 ± 0.24; sensor-free IDW climatology 6.36 (skill against it − 0.20 to − 0.06)
24 runs (4 branches × 2 designs × 3 seeds)
no network beats a climatology that uses no sensors
SNOTEL [ 32 ]
snow-pillow SWE and fluxes → SWE at 40 manual snow courses
regional, flux inputs, 26-day window, 3 seeds: R2 S-LSTM 0.237 ± 0.078; S-GRU 0.233 ± 0.055; flat.-window 0.181 ± 0.025; static-branch 0.079 ± 0.083; spatial mean of training courses 0.163 (single fit). Seed-mean R2 over all 24 branch × configuration cells: − 0.06 to 0.38
32 runs
weak reconstruction by every method
PeakWeather, instantaneous [ 30 ]
2 m temperature at 12 thermometers → 40 held-out stations (clock off, 24 h window)
mooring winds → sea-level anomaly at block-held-out 1 ∘ cells
G-T1 (planned: ≥10 moorings at ≥70 % valid daily wind in every split and ≥40 % of days jointly usable): 14 moorings qualify individually, but the best 10-mooring set is jointly usable on 0.237 of the days of its worst split (best 12-mooring set 0.159); the largest set reaching 0.40 has 7 moorings. Fail; replaced post hoc by an input-availability rule. G-T2 (planned: two-stage lagged-ridge skill ≥0.3 against a harmonic climatology, 60-day window, test years, block-held-out cells): 0.053 / 0.049 / 0.264 in partitions 0 / 1 / 2 (ORACLE ridge 0.273 / 0.212 / 0.301). Fail
no network run
dropped at gates G-T1 and G-T2
Table S16: Measured datasets attempted but not used as main-text evidence. NOAA, SNOTEL and PeakWeather instantaneous runs come from an earlier audit (Supplementary Table S17 ); TAO/TRITON stopped at pre-set gates before any network was trained. Mean ± s.d. over 3 training seeds (one station or cell partition) unless marked single fit. NOAA OI SST, optimum-interpolation sea-surface temperature, 20% of grid cells withheld; relative L2 error, weekly ∥x^−x∥2/∥x∥2 , averaged; IDW climatology, training-period week-of-year climatology interpolated to withheld cells by inverse-distance weighting (power 4), no sensors; skill, 1−MSE/MSEclim (MSE, mean squared error). SNOTEL: snow water equivalent (SWE) at manual snow courses from snow-pillow stations; R2 pooled over held-out courses. PeakWeather: lapse-rate fit, per-hour least-squares fit of temperature on elevation through the 12 inputs. TAO/TRITON: Tropical Atmosphere Ocean mooring winds as input for Copernicus Marine (CMEMS) sea-level anomaly (SLA); gates G-T1 (input availability) and G-T2 (two-stage linear skill); ORACLE, ridge fitted at held-out cells. Branches as in Supplementary Table S14 .
dataset
window
batch
cap
pat.
LR
seeds
runs
h
hardware
GHCN TMAX
14 d
32
300
20
0.001
def., 11, 22
72
1.4
RTX PRO 4000 Blackwell; host not recorded
PeakWeather pressure → 2 m temperature
24 h
32
300
20
0.001
def., 11, 22
114
8.8
RTX PRO 4000 Blackwell (workstation)
PeakWeather degree-hours
72 h
32
300
20
0.001
def., 11, 22
109
7.7
workstation GPUs 0 and 2 (queue scripts); model not recorded
NOAA OI SST (audit)
various
–
–
10/15/20/40
–
–
290
38.4
not recorded
PeakWeather (audit)
various
–
–
10/20/30/cap30ep
–
–
312
12.5
not recorded
GHCN (audit)
various
–
–
20
–
–
116
2.6
not recorded
Table S17: Training recipe, run inventory and compute. Upper block: experiments of main-text Fig. 8 (Section S11 ); lower block: earlier audit runs behind Supplementary Table S16 and a GHCN input-count sweep. Window, input length; batch, mini-batch size; cap, maximum epochs; pat., early-stopping patience (audit: values across its run directories); LR, initial learning rate; seeds, training seeds (def., default 1234567); runs, runs with a result file (duplicate audit files counted once; cap30ep, one audit run capped at 30 epochs); h, wall-clock training time summed over runs (inference excluded; jobs could share a GPU). Degree-hours runs: 84 analyzed, 19 in cells dropped before any result was examined, 6 invalid 120 h runs (Supplementary Table S15 ). Hardware: as recorded in the run files; none records the DeltaAI GH200 system. The 295 measured-data runs took 17.9 h in total.
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.
Xinyu Zhang, Lihao Chen, Panqi Chen +4
College of Information Science and Electronic Engineering, Zhejiang University
Reconstructing spatially continuous daily temperature fields from sparse observations is important for urban climate monitoring and heat-risk analysis, but practical deployments are limited by sensor budgets and spacing constraints. This study proposes an uncertainty-aware graph neural network (GNN) framework for reconstructing daily maximum temperature fields from sparse sensors while supporting distance-constrained sensor placement and probabilistic exceedance mapping. The model predicts both the temperature field and a spatially varying predictive uncertainty field using a graph-attention-based mean-residual architecture trained with a Gaussian negative log-likelihood. Sensor placement is addressed using a Proper Orthogonal Decomposition with QR factorization (POD-QR) strategy with a 4 km minimum inter-sensor distance constraint and is compared with random feasible placement and farthest-point sampling. The framework is evaluated over a Montreal-area polygon using Daymet v4.1 daily temperature data (1 km resolution) under a strict temporal hold-out protocol (training: 2020-2023; testing: 2024). Across sensor budgets (10-40 sensors), the proposed GNN consistently outperforms inverse distance weighting and ordinary kriging in RMSE and MAE on unobserved nodes. Sensor-placement effects are most pronounced at low budgets and diminish at higher budgets, with a practical saturation regime emerging around 30 sensors under the imposed spacing constraint. Probabilistic evaluation further shows improved uncertainty calibration with increasing sensor density and a better sharpness-calibration trade-off than kriging. These results support the proposed framework as an effective tool for uncertainty-aware temperature field reconstruction and decision-oriented heat-risk mapping.
Reda Snaiki, Abdelatif Merabtine
Department of Construction Engineering, École de Technologie Supérieure, Université du Québec, Montréal, Québec, Canada
Real-time inference of inaccessible interior physical fields from sparse boundary observations is a fundamental but unresolved problem in scientific machine learning, with direct relevance to safety-critical monitoring across many engineering applications. Existing neural operators achieve high accuracy but leave deployment to embedded edge platforms unaddressed. Here we introduce VIRSO (Virtual Irregular Real-Time Sparse Operator), the first neural operator with a unique spatial-spectral architecture that explicitly addresses edge-deployment hardware. VIRSO learns a nonlinear mapping from sparse, geometrically disjoint boundary inputs to spatially continuous interior multiphysics fields on irregular unstructured meshes through a spectral-spatial decomposition explicitly aligned with hardware execution: a compute-bound graph spectral pathway and a memory-bandwidth-bound spatial-aggregation pathway, each independently characterized on datacenter and embedded accelerators. The design reduces the inference energy-delay product by 29× relative to the vanilla graph-operator baseline (206 J⋅ms → 7.0 J⋅ms on an NVIDIA H200) and enables 17.0 samples/s embedded inference on an NVIDIA Jetson Orin Nano within 7.06 W board-level power, without modification. A mesh-density-adaptive graph construction strategy (V-KNN) simultaneously improves accuracy and reduces graph edge count by 34%. Across three benchmarks with reconstruction ratios from 47:1 to 156:1, VIRSO achieves mean relative L2 errors below 1% with fewer parameters than operator baselines and delivers an inference speedup of ≈104 times over the high-fidelity reference solver. To our knowledge, this is the first demonstration of a single-digit-watt neural operator, establishing hardware co-design as a missing ingredient in operator-based inference and a tractable path to real-time deployment.
William Howes, Jason Yoo, Kazuma Kobayashi +4
Grainger College of Engineering, Nuclear, Plasma & Radiological Engineering Department, University of Illinois Urbana-Champaign, Urbana, IL, USA · Yardi School of Artificial Intelligence, Indian Institute of Technology Delhi · Department of Applied Mechanics, Indian Institute of Technology Delhi, New Delhi, India +1