Spatiotemporal Reconstruction of Global Radiation from Sparse Sensor Sequences

Kazuma Kobayashi; Tapas Tripura; Jay Phil Yoo; Samrendra Roy; Diab Abueidda; Seid Koric; Souvik Chakraborty; Syed Bahauddin Alam

Figure 1: Schematic overview of the effective-dose reconstruction framework. a Primary cosmic-ray flux

\Phi

drives atmospheric cascades that determine effective dose

X_{j}=g_{j}(\cdot)

and neutron-monitor counts

\mathbf{Y}_{s}=f_{s}(\cdot)

(

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

\mathbf{Y}

from

S

stations and query coordinates

\mathbf{r}

to predicted dose

\hat{\mathbf{X}}

at

P

points;

\mathcal{H}_{\mathrm{phys}}

denotes the physics-based forward mapping. d Single-branch encoding of all stations versus separate station encoders followed by fusion;

\mathbf{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 (

\odot

), summed and combined with bias

\beta

to produce

\hat{z}_{j}

. Rescaling gives

\hat{X}_{j}=m_{j}+(M_{j}-m_{j})\hat{z}_{j}

, using training-period reference extrema

m_{j}

and

M_{j}

. The map shows the corresponding S-LSTM prediction (single-branch long short-term memory encoder;

T=30

days; one trained model; relative

L_{2}

error

0.096\%

). Both maps share a color scale of

0.027

–

0.036\,\mu\mathrm{Sv}\,\mathrm{h}^{-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^{\circ}

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,

(\mathrm{prediction}-\mathrm{reference})/\mathrm{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;

\odot

, 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

\rho

between 128 latent features, with the mean

|\rho|

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

\rho=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^{\circ}

EXPACS reference fields; one trained model is evaluated throughout. Errors are calculated against the EXPACS reference at the corresponding resolution. a Daily global relative

L_{2}

errors on the

1^{\circ}

and

0.25^{\circ}

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,

(\mathrm{prediction}-\mathrm{reference})/\mathrm{reference}

, on the representative day over the North America sector (

140^{\circ}

–

40^{\circ}

W,

0^{\circ}

–

60^{\circ}

N). c Relative

L_{2}

errors over the global domain and the regional sector shown in b . Light gray bars (

1^{\circ}

output, interpolated), the normalized network output bilinearly interpolated from

1^{\circ}

to

0.25^{\circ}

. For both finer-grid evaluations, the per-point output-scaling parameters

(m_{j},M_{j})

are fitted using the

0.25^{\circ}

EXPACS reference from 2001–2021. Bars show means over

n=365

test days; error bars denote

\pm 1

s.d. Regional SSIM values are

0.9985\pm 0.0006

at

1^{\circ}

and

0.9997\pm 0.0001

at

0.25^{\circ}

(mean

\pm

s.d. over test days).

Figure 7

Figure 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.	$R_{\mathrm{c}}$ (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.

R_{\mathrm{c}}

: vertical geomagnetic cut-off rigidity from the NMDB station list [ 26 ] . JUNG is the IGY-type monitor at Jungfraujoch.

Data Set	Sequence Length ( $T$ )
Data Set	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^{\prime}(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 $L_{2}$ Error (%)	RMSE	MAE	SSIM
7	Static-branch DeepONet	$1.87\times 10^{-1}$	$6.26\times 10^{-5}$	$5.05\times 10^{-5}$	$0.998934$
	S-GRU	$1.00\times 10^{-1}$	$3.37\times 10^{-5}$	$2.84\times 10^{-5}$	$0.999546$
	S-LSTM	$1.12\times 10^{-1}$	$3.78\times 10^{-5}$	$3.11\times 10^{-5}$	$0.999370$
	M-GRU	$8.98\times 10^{-1}$	$3.01\times 10^{-4}$	$2.69\times 10^{-4}$	$0.998256$
	M-LSTM	$6.24\times 10^{-1}$	$2.09\times 10^{-4}$	$1.88\times 10^{-4}$	$0.998120$
	GNO	$1.86$	$6.24\times 10^{-4}$	$4.86\times 10^{-4}$	$0.738379$

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\times 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\times 128

); the output at the last time step (128 values) is the branch output. b Multi-branch TRON: each neutron monitor’s sequence (

T\times 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 (

\odot

) with the trunk network’s spatial embedding and summed, and a learnable scalar bias

\beta

is added, giving a normalized dose; this is rescaled with the per-point output scaling

(m_{j},M_{j})

to the effective dose rate at query point

j

.

Type	Variation	# of trainable parameters
Linear Regressions	Static	849,433
Linear Regressions	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)\times 65{,}341

parameters; the value shown is for

T=30

days.

Method	Simulation (s)	Inference time (ms), input window $T$ (days)
Method	Simulation (s)	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

L_{2}

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 $L_{2}$ 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

L_{2}

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

L_{2}

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

L_{2}

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

L_{2}

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)=A\sin(\pi x/L)

(

L

, tube length; 100 samples, normalized by

A

) and uniform inlet velocity

v_{\text{in}}

and temperature

T_{\text{in}}

. 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 (

\omega

) 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

\pm 3.55

(99.5th percentile of

|\omega|

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	$2.079\times 10^{-4}$	$0.9999972$
PC3	$2.800\times 10^{-6}$	$0.99999999$
PC4	$7\times 10^{-9}$	$1.000000$
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,

(\sum_{i}\lambda_{i})^{2}/\sum_{i}\lambda_{i}^{2}

, 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

c_{1}(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:

R^{2}

of a quadratic fit of

c_{1}

on

W

and of a fifth-order fit of

c_{1}

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 $\to$ 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 $R^{2}$
S-LSTM	off	9	3.430	3.459	3.571	3.487 $\pm$ 0.066	0.912 $\pm$ 0.003
S-LSTM	on	9	3.423	3.434	3.489	3.449 $\pm$ 0.036	0.914 $\pm$ 0.004
S-GRU	off	9	3.383	3.437	3.562	3.461 $\pm$ 0.080	0.914 $\pm$ 0.002
S-GRU	on	9	3.394	3.386	3.518	3.433 $\pm$ 0.068	0.915 $\pm$ 0.003
flattened-window DeepONet	off	9	3.448	3.481	3.562	3.497 $\pm$ 0.058	0.912 $\pm$ 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

R^{2}=1-\mathrm{SSE}/\mathrm{SST}

about the single mean of all held-out test values. Network rows:

n=9

runs (3 station splits

\times

3 training seeds); split columns, mean over seeds; last two columns, mean

\pm

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	$\Delta$ 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.

\Delta

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

\delta=0.10\,^{\circ}

C: Better, CI upper bound

<0

and

\Delta<-\delta

; Worse, CI lower bound

>0

and

\Delta>\delta

; Tie, CI within

[-\delta,\delta]

; 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 $R^{2}$
S-LSTM	pressure	off	9	1.897 $\pm$ 0.034	1.853–1.964	0.949 $\pm$ 0.002
S-LSTM	pressure	on	9	1.866 $\pm$ 0.045	1.802–1.952	0.950 $\pm$ 0.002
S-GRU	pressure	off	9	1.891 $\pm$ 0.036	1.844–1.959	0.949 $\pm$ 0.002
S-GRU	pressure	on	9	1.832 $\pm$ 0.026	1.787–1.870	0.952 $\pm$ 0.001
flattened-window DeepONet	pressure	off	9	2.008 $\pm$ 0.070	1.924–2.132	0.942 $\pm$ 0.004
flattened-window DeepONet	pressure	on	9	2.104 $\pm$ 0.075	1.988–2.227	0.937 $\pm$ 0.004

Table S13: PeakWeather pressure

\to

2 m temperature: every branch, control and baseline at the 40 held-out stations. RMSE, pooled

R^{2}

and branches as in Supplementary Table S11 ; 11,086 hourly test windows. Pressure-input network rows:

n=9

runs (3 barometer draws

\times

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

\pm

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

\to

2 m temperature: pre-registered tests.

R_{T}

, RMSE of the primary cell (S-LSTM, pressure, clock on; mean of 9 runs). S1 (sanity):

R_{T}

against

0.9\times

the calendar-only linear RMSE. H1:

R_{T}

minus clock-on time-shuffled S-LSTM, paired by draw and seed. H2:

R_{T}

minus the CV-tuned two-stage linear RMSE (same inputs and clock); outcome classes as in Supplementary Table S12 ,

\delta

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.

RMSE, mean $\pm$ s.d.
target	clock	S-LSTM	S-GRU	flat.-window	static-branch
instantaneous ( ∘ C)	off	1.770 $\pm$ 0.070	1.949 $\pm$ 0.064	1.807 $\pm$ 0.043	1.689 $\pm$ 0.070
instantaneous ( ∘ C)	on	1.854 $\pm$ 0.173	1.980 $\pm$ 0.378	1.775 $\pm$ 0.085	1.715 $\pm$ 0.038
24 h ( ∘ C h)	off	27.2 $\pm$ 1.0	29.9 $\pm$ 3.0	29.7 $\pm$ 1.5	39.5 $\pm$ 0.3
24 h ( ∘ C h)	on	28.8 $\pm$ 1.7	30.5 $\pm$ 5.9	27.9 $\pm$ 0.3	37.7 $\pm$ 0.5
48 h ( ∘ C h)	on	52.4 $\pm$ 3.1	60.4 $\pm$ 2.6	51.8 $\pm$ 2.4	87.1 $\pm$ 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

\pm

s.d. over seeds. RMSE as in Supplementary Table S11 , over 9,974 hourly test windows;

1-R^{2}

, 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 $\to$ target	key numbers	runs	reason
NOAA OI SST, weekly [ 31 ]	SST at 12 grid cells $\to$ SST at withheld grid cells	random 20% of cells: held-out relative $L_{2}$ error (%) S-LSTM 5.94 $\pm$ 0.44; S-GRU 5.99 $\pm$ 0.05; flat.-window 5.55 $\pm$ 0.24; static-branch 5.74 $\pm$ 0.04; sensor-free IDW climatology 4.38 (skill against it $-$ 0.88 to $-$ 0.61). 20% in 10 ∘ blocks: held-out relative $L_{2}$ error (%) S-LSTM 6.92 $\pm$ 0.33; S-GRU 6.95 $\pm$ 0.07; flat.-window 6.54 $\pm$ 0.20; static-branch 6.79 $\pm$ 0.24; sensor-free IDW climatology 6.36 (skill against it $-$ 0.20 to $-$ 0.06)	24 runs (4 branches $\times$ 2 designs $\times$ 3 seeds)	no network beats a climatology that uses no sensors
SNOTEL [ 32 ]	snow-pillow SWE and fluxes $\to$ SWE at 40 manual snow courses	regional, flux inputs, 26-day window, 3 seeds: $R^{2}$ S-LSTM 0.237 $\pm$ 0.078; S-GRU 0.233 $\pm$ 0.055; flat.-window 0.181 $\pm$ 0.025; static-branch 0.079 $\pm$ 0.083; spatial mean of training courses 0.163 (single fit). Seed-mean $R^{2}$ over all 24 branch $\times$ configuration cells: $-$ 0.06 to 0.38	32 runs	weak reconstruction by every method
PeakWeather, instantaneous [ 30 ]	2 m temperature at 12 thermometers $\to$ 40 held-out stations (clock off, 24 h window)	held-out $R^{2}$ : S-LSTM 0.929 $\pm$ 0.009; S-GRU 0.936 $\pm$ 0.007; flat.-window 0.942 $\pm$ 0.005; static-branch 0.941 $\pm$ 0.003; per-hour lapse-rate fit 0.929 (single fit). RMSE: S-LSTM 2.16 $\pm$ 0.14; S-GRU 2.05 $\pm$ 0.11; flat.-window 1.96 $\pm$ 0.08; static-branch 1.98 $\pm$ 0.06; lapse-rate fit 2.17 ∘ C	12 runs (4 branches $\times$ 3 seeds)	little gain over a per-hour lapse-rate fit
TAO/TRITON $\to$ CMEMS SLA	mooring winds $\to$ sea-level anomaly at block-held-out 1 ∘ cells	G-T1 (planned: $\geq 10$ moorings at $\geq 70$ % valid daily wind in every split and $\geq 40$ % 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 $\geq 0.3$ 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

\pm

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

L_{2}

error, weekly

\|\hat{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-\mathrm{MSE}/\mathrm{MSE}_{\mathrm{clim}}

(MSE, mean squared error). SNOTEL: snow water equivalent (SWE) at manual snow courses from snow-pillow stations;

R^{2}

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 $\to$ 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.

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