Atmospheric Diffusion-Guided Spatio-Temporal Transformer for Nuclear Radiation Forecasting

# Atmospheric Diffusion-Guided Spatio-Temporal Transformer for Nuclear Radiation Forecasting
Source: [https://arxiv.org/html/2607.24774](https://arxiv.org/html/2607.24774)
Tengfei Lyu, Jindong Han, and Hao LiuTengfei Lyu is with the Thrust of Artificial Intelligence, The Hong Kong University of Science and Technology \(Guangzhou\), Guangzhou, China, and The Hong Kong University of Science and Technology, Hong Kong, SAR, China \(e\-mail: tlyu077@connect\.hkust\-gz\.edu\.cn\)\.Jindong Han is with the School of Artificial Intelligence, Shandong University, Jinan, Shandong, China \(e\-mail: jindong\.han@sdu\.edu\.cn\)\.Hao Liu is with the Thrust of Artificial Intelligence, The Hong Kong University of Science and Technology \(Guangzhou\), Guangzhou, China, and also with the Department of Computer Science and Engineering, The Hong Kong University of Science and Technology, Hong Kong, SAR, China \(e\-mail: liuh@ust\.hk\)\.Corresponding authors: Hao Liu\.

###### Abstract

Nuclear radiation, the energy released during atomic decay, poses persistent risks to public health and the environment, and concerns have only grown since the Fukushima accident and the recent commencement of treated\-water discharge\. Modern monitoring networks now record radiation levels and accompanying weather conditions at thousands of stations, opening the door to nationwide forecasting that can inform emergency response, agricultural advisories, and routine public\-safety decisions\. However, turning this abundance of monitoring data into reliable forecasts is difficult for three reasons\. First, the time series at each station are highly non\-stationary, shaped by radioactive decay, weather variability, and irregular human interventions\. Second, monitoring stations are severely unevenly distributed in space\. Roughly 78% of Japan’s stations sit in less than 6% of the country, clustered near Fukushima, which breaks the assumptions of standard graph\-based models\. Third, radiation co\-evolves with heterogeneous context such as wind, temperature, and humidity through atmospheric transport processes that purely data\-driven models struggle to capture from observations alone\. In this study, we introduce*NRFormer\+*, a spatio\-temporal Transformer for nationwide nuclear radiation forecasting\. NRFormer\+ couples non\-stationary temporal attention and density\-adaptive spatial attention with a new atmospheric diffusion module that estimates how meteorology drives radiation dispersion and injects this physical signal into the network as an architectural prior\. We curate two complementary benchmarks,*Japan\-4H*and*Japan\-1D*, covering 3,627 radiation and 228 meteorological stations over four years\. NRFormer\+ delivers state\-of\-the\-art accuracy on both datasets across all1313baselines, reducing sudden\-change MAE by up to 19\.1% over the strongest baseline at comparable inference latency\. Our code and datasets are publicly available athttps://github\.com/tfeilyu/NRFormer\_Plus\.

## IIntroduction

Nuclear power has supplied clean, dispatchable electricity for more than half a century, but it carries the low\-probability, high\-consequence risk of radioactive release\. The 1986 Chernobyl accident dispersed radionuclides across the European continent\[[1](https://arxiv.org/html/2607.24774#bib.bib60)\], and the 2011 Fukushima Daiichi accident contaminated soil, marine ecosystems, and the food chain over thousands of square kilometres of northeastern Japan\[[32](https://arxiv.org/html/2607.24774#bib.bib27)\]\. Because the principal long\-lived contaminant has a physical half\-life of roughly 30 years, the radioecological footprint of such events persists across generations, and the recent commencement of treated\-water discharge from the Fukushima site has renewed public and regulatory attention on long\-term radiological monitoring\. In response, many countries now operate dense networks of monitoring stations\. Yet raw measurements alone are not enough\. Governments and citizens increasingly demand*forecasts*of future radiation levels to support evacuation planning, agricultural advisories, and pre\-positioning of protective equipment, where even a few hours of additional lead time materially reduces socio\-economic loss\.

![Refer to caption](https://arxiv.org/html/2607.24774v1/x1.png)Figure 1:Nationwide nuclear radiation forecasting on Japan\. \(a\) Spatial distribution of monitoring stations; high\-risk areas \(*i\.e\.,*those near nuclear power plants\) host disproportionately more stations\. \(b\) Time\-varying radiation levels at representative stations\.In this paper, we study*nationwide nuclear radiation forecasting*, the task of producing station\-level multi\-step predictions across an entire country from a heterogeneous mix of radiation, meteorological, and geographic observations\. Figure[1](https://arxiv.org/html/2607.24774#S1.F1)illustrates the setting in Japan, where thousands of gamma\-radiation monitors are deployed with density determined by proximity to nuclear facilities rather than by uniform geographic coverage\. The dispersion of radiation between stations is governed by atmospheric circulation\[[24](https://arxiv.org/html/2607.24774#bib.bib28),[33](https://arxiv.org/html/2607.24774#bib.bib66)\], in which wind transports radionuclides across regions while temperature and humidity modulate mixing, deposition, and removal\.

Existing approaches lie largely outside this regime\. Operational atmospheric\-dispersion solvers such as HYSPLIT\[[31](https://arxiv.org/html/2607.24774#bib.bib73)\]and FLEXPART\[[34](https://arxiv.org/html/2607.24774#bib.bib71)\]simulate radiation transport mechanistically\. In forward mode they presuppose a specified emission source, and although their backward and data\-assimilation variants can reconstruct unknown sources from sparse observations\[[35](https://arxiv.org/html/2607.24774#bib.bib72)\], these variants are designed for hindcasting discrete release events and in either mode still require dense meteorological fields and substantial compute, which makes them ill\-suited to continuous nationwide station\-level forecasting\. Conversely, generic Spatio\-Temporal Graph Neural Networks\[[42](https://arxiv.org/html/2607.24774#bib.bib30),[41](https://arxiv.org/html/2607.24774#bib.bib81),[8](https://arxiv.org/html/2607.24774#bib.bib65)\]and time\-series Transformers\[[46](https://arxiv.org/html/2607.24774#bib.bib40),[18](https://arxiv.org/html/2607.24774#bib.bib79),[19](https://arxiv.org/html/2607.24774#bib.bib13)\]have set strong baselines in adjacent domains such as traffic and air quality\[[17](https://arxiv.org/html/2607.24774#bib.bib9)\], but the former infer connectivity from data with no explicit transport operator, while the latter model temporal dynamics with no spatial structure at all\. Even physics\-guided neural models that do embed an advective\-diffusive prior on a station graph, such as recent air\-quality networks\[[9](https://arxiv.org/html/2607.24774#bib.bib10)\], assume dense and roughly uniform monitoring coverage and have not been adapted to a nationwide, density\-imbalanced radiation setting with sparse co\-located meteorology\. Bridging this gap calls for a model that pairs data\-driven flexibility with a physical inductive bias for atmospheric transport, learned directly from sparse station observations\.

A data\-driven characterization of the problem\.Progress has been limited by the absence of a large\-scale, publicly available, analysis\-ready benchmark\. We therefore construct*Japan\-1D*and*Japan\-4H*, distilled from over four years \(March 2021–May 2025\) of 10\-minute readings from Japan’s Nuclear Regulation Authority, covering3,627monitoring stations after stringent quality control and aligned with228co\-located NOAA\-ISD meteorological stations \(wind speed, wind direction, air temperature, dew point\)\. The construction pipeline, from quality control to dual\-resolution aggregation, is itself a deliverable, and a quantitative analysis of the resulting corpus \(§[III](https://arxiv.org/html/2607.24774#S3)\) exposes three structural challenges that jointly defeat off\-the\-shelf STGNNs and Transformers\.

![Refer to caption](https://arxiv.org/html/2607.24774v1/x2.png)Figure 2:Empirical study of the imbalanced spatial distribution problem\. \(1\) Full station distribution of Japan\-1D\. \(2\) Evenly\-sampled station distribution from Japan\-1D\. \(3\) Forecasting performance of GWN\[[42](https://arxiv.org/html/2607.24774#bib.bib30)\]on Japan datasets, where GWN\-w/o SPM denotes the variant with the spatial propagation module removed\. Removing spatial propagation*improves*accuracy on the full \(imbalanced\) dataset, indicating that the standard graph operator is net\-harmful under extreme station\-density imbalance\.Challenge 1: Non\-stationary temporal patterns\.Radiation series are shaped by irregular human interventions, plant\-operation events, and the slow decay of long\-lived isotopes, producing pronounced distribution shift\. On Japan\-4H, joint ADF and KPSS tests flag about87%of stations as non\-stationary \(§[III\-D](https://arxiv.org/html/2607.24774#S3.SS4)\), and the mean radiation drops by roughly3%per year, broadly tracking Cs\-137 decay and environmental loss overlaid on a strong annual cycle\. Temporal modules that assume local stationarity, including many CNN\- and RNN\-based forecasters\[[25](https://arxiv.org/html/2607.24774#bib.bib80)\], can degrade under such shift, and even recent normalization\-based Transformers\[[11](https://arxiv.org/html/2607.24774#bib.bib26),[20](https://arxiv.org/html/2607.24774#bib.bib23)\]were not designed for series with decay\-driven multi\-year trends\. A capable model must therefore absorb this distribution shift while staying reactive to sudden events\.

Challenge 2: Severely imbalanced spatial distribution\.Station density in Japan is dictated by perceived risk rather than geographic coverage\. About78%of the3,627stations lie within Fukushima Prefecture and its three immediate neighbors, under6%of the land surface, so nearest\-neighbor distances are highly skewed \(§[III\-C](https://arxiv.org/html/2607.24774#S3.SS3)\)\. Standard message passing over such a graph over\-smooths dense clusters and starves sparse regions\. As shown in Figure[2](https://arxiv.org/html/2607.24774#S1.F2), removing the spatial\-propagation module of Graph WaveNet \(GWN\)\[[42](https://arxiv.org/html/2607.24774#bib.bib30)\]actually*improves*accuracy on the full, imbalanced Japan\-1D while*hurting*it on an evenly\-sampled subset, direct evidence that a one\-size\-fits\-all graph operator is net\-harmful under this imbalance\. The model must also let spatial information flow by station density rather than through a uniform graph operator\.

Challenge 3: Physics\-grounded heterogeneous context modeling\.Radiation co\-evolves with meteorological covariates that are not interchangeable features but quantities governing atmospheric transport\. On Japan\-4H, station\-averaged wind speed and radiation residuals show a weak but consistently located cross\-correlation peak \(16 hourslag,\|ρmax\|=0\.16\|\\rho\_\{\\max\}\|\{=\}0\.16\), pointing to a multi\-hour advection timescale rather than instantaneous mixing \(§[III\-E](https://arxiv.org/html/2607.24774#S3.SS5)\)\. The diffusion signature is also wind\-dependent\. Fitting an isotropic, wind\-agnostic spatial Laplacian to radiation increments, we find the diffusion strength declines monotonically with wind speed \(ρs=−1\.00\\rho\_\{s\}\{=\}\{\-\}1\.00\), so this wind\-blind operator captures a steadily smaller share of the increments as winds turn the dispersion anisotropic\. Recent physics\-informed networks\[[26](https://arxiv.org/html/2607.24774#bib.bib68),[27](https://arxiv.org/html/2607.24774#bib.bib69),[2](https://arxiv.org/html/2607.24774#bib.bib70)\]and physics\-aware weather models\[[14](https://arxiv.org/html/2607.24774#bib.bib78),[23](https://arxiv.org/html/2607.24774#bib.bib77)\]show that injecting such priors is feasible, yet no nationwide radiation forecaster does so\. Finally, the model must fuse meteorological context through a physics\-grounded inductive bias for wind\-modulated transport\.

Motivated by these data\-driven observations, we introduceNRFormer\+, a spatio\-temporal Transformer for nationwide nuclear radiation forecasting that substantially extends our conference model NRFormer\[[21](https://arxiv.org/html/2607.24774#bib.bib1)\]and integrates three core modules into a single end\-to\-end pipeline\. A*non\-stationary temporal attention*module pairs instance\-level reversible normalization with point\-wise self\-attention to remove distribution shift while remaining reactive to sudden events\. A*density\-adaptive spatial attention*module combines a macro\-scale view with a proximity\-constrained view, so that information flows differently in dense and sparse regions of the monitoring network\. At the center of the model, the new*Physics\-Guided Atmospheric Diffusion Module*estimates a meteorology\-conditioned diffusion coefficient at every station, approximates how radiation flows from regions of high concentration to neighboring regions of lower concentration over the irregular monitoring graph, and injects this physical signal back into the Transformer pipeline so that learned representations remain consistent with atmospheric transport\. These three modules are supported by an enhanced meteorological encoder that routes wind \(advection\) and thermal \(stability and deposition\) variables along separate pathways, a day\-of\-year seasonal embedding that captures annual cycles and slow background decay, a deeper radiation\-location cross\-feature encoder, and a three\-way fusion layer that combines radiation, temporal, and spatial embeddings before forecasting\. This paper makes the following contributions:

- •We introduce a*Physics\-Guided Atmospheric Diffusion Module*that estimates a meteorology\-conditioned diffusion coefficient per station and injects a spatial\-Laplacian transport signal into the Transformer as an architectural inductive bias\. To our knowledge this is the first learned wind\-modulated transport prior for nationwide radiation forecasting, and ablations rank it among the largest contributors to accuracy\.
- •We buildNRFormer\+, a spatio\-temporal Transformer that couples this diffusion prior with non\-stationary temporal attention and density\-adaptive spatial attention for severely imbalanced sensor networks\. It casts nationwide radiation forecasting as graph\-structured prediction under a learned transport prior\.
- •We construct and releaseJapan\-4HandJapan\-1D, two analysis\-ready benchmarks over four years \(March 2021–May 2025\) with3,627radiation stations after stringent quality control and228co\-located meteorological stations, to our knowledge the largest publicly available nationwide radiation\-forecasting corpora, released with the full construction pipeline\.
- •A quantitative data\-engineering analysis establishes the temporal non\-stationarity, severe density imbalance, and wind\-modulated transport that motivate each module of NRFormer\+\. Against13strong baselines, NRFormer\+ improves consistently across both datasets and all horizons at comparable latency, reducing sudden\-change MAE by up to19\.1%\.

## IIPreliminaries

In this section, we introduce some important definitions and formally define the nuclear radiation forecasting problem\.

###### Definition 1

Radiation level\. Radiation level is a number used to quantify the concentrations of radioactive materials in the environment\. A higher radiation level indicates that people will experience increasingly detrimental health effects\. In practice, the radiation level is computed by using the weighted sum of several radioactive substance measurements, including alpha particles, beta particles, neutron particles, and gamma rays\.

###### Definition 2

Radiation monitoring network\. The radiation monitoring network consists of a group of monitoring stations, denoted as𝒢=\(𝐕,𝐄\)\\mathcal\{G\}=\(\\mathbf\{V\},\\mathbf\{E\}\), where𝐕\\mathbf\{V\}is a set of stations andN=\|𝐕\|N=\|\\mathbf\{V\}\|is the number of stations,𝐄\\mathbf\{E\}denotes a set of edges representing the relationships among stations\. Here we use𝐀∈ℝN×N\\mathbf\{A\}\\in\\mathbb\{R\}^\{N\\times N\}to denote the adjacency matrix of the network\.

To build𝒢\\mathcal\{G\}, we compute the pairwise distances between stations and derive the adjacency matrix using a pre\-defined distance threshold\. Let𝐗∈ℝT×N\\mathbf\{X\}\\in\\mathbb\{R\}^\{T\\times N\}be the observed radiation level from all stations, whereTTis the number of time steps\. We use𝐂∈ℝT×N×C\\mathbf\{C\}\\in\\mathbb\{R\}^\{T\\times N\\times C\}to denote the contextual features associated with each station,*e\.g\.,*meteorological and location information, whereCCis the feature dimension\. Letℋt=\(𝒢,𝐗t,𝐂t\)\\mathcal\{H\}^\{t\}=\(\\mathcal\{G\},\\mathbf\{X\}^\{t\},\\mathbf\{C\}^\{t\}\)indicates all the observed values at time steptt, we define the problem as follows\.

Nuclear radiation forecasting\.Given the radiation monitoring network𝒢\\mathcal\{G\}, historical radiation levels𝐗\\mathbf\{X\}, contextual features𝐂\\mathbf\{C\}, the goal is to predict the radiation level for all the monitoring stations over the nextKKtime steps:

\(𝐘^t\+1,𝐘^t\+2,⋯,𝐘^t\+K\)←ℱθ​\(ℋt−P\+1,ℋt−P\+2,⋯​ℋt\),\(\\mathbf\{\\hat\{Y\}\}^\{t\+1\},\\mathbf\{\\hat\{Y\}\}^\{t\+2\},\\cdots,\\mathbf\{\\hat\{Y\}\}^\{t\+K\}\)\\leftarrow\\mathcal\{F\}\_\{\\theta\}\(\\mathcal\{H\}^\{t\-P\+1\},\\mathcal\{H\}^\{t\-P\+2\},\\cdots\\mathcal\{H\}^\{t\}\),\(1\)where𝐘^t\+K\\mathbf\{\\hat\{Y\}\}^\{t\+K\}denotes the predicted value at time stept\+Kt\+K,ℱθ\\mathcal\{F\}\_\{\\theta\}is the forecasting model,PPandKKare the number of historical and future time steps, respectively\.

## IIIData Engineering and Analysis

A central premise of this work is that the architectural choices in NRFormer\+ are not generic but are dictated by the empirical structure of nationwide radiation data\. In this section, we therefore go beyond a descriptive overview and present a quantitative analysis pipeline that \(i\) characterizes our newly constructed Japan\-1D and Japan\-4H datasets and \(ii\) establishes, through hypothesis testing and physically motivated statistics, the precise data properties that each component of NRFormer\+ is designed to address\.

![Refer to caption](https://arxiv.org/html/2607.24774v1/x3.png)Figure 3:Spatial structure of Japan\-1D/4H\. \(a\) Hex\-binned station density across Japan\. \(b\) Distribution of mean distance to the five nearest neighbors; the Gini coefficient quantifies density imbalance\. \(c\) Time\-averaged global Moran’sII=0\.15 on radiation residuals across the test period\. \(d\) Empirical semivariogram, with the nugget at about 61% of the sill, indicating that most variance is micro\-scale rather than spatially structured\.### III\-AData Acquisition and Preprocessing

Nuclear radiation data\.We collected gamma\-radiation dose\-rate measurements from Japan’s Nuclear Regulation Authority \(NRA\)111https://www\.erms\.nsr\.go\.jp/nra\-ramis\-webg/, comprising over 4,000 candidate monitoring stations sampled at a 10\-minute cadence from March 17, 2021 to May 31, 2025\. Raw measurements were subjected to a four\-stage quality\-control pipeline, \(1\) station\-level pruning that removes any station with more than 30 days of cumulative missing data, retaining3,627reliable stations out of the 3,841 used in the\[[21](https://arxiv.org/html/2607.24774#bib.bib1)\]; \(2\) outlier suppression via a Hampel filter \(k=3k\{=\}3, windoww=24w\{=\}24samples\) that flags physically implausible spikes inconsistent with adjacent stations; \(3\) gap imputation through temporally local linear interpolation for short gaps \(≤\\leq2 h\) and station\-conditional seasonal decomposition for longer gaps; and \(4\) dual\-resolution aggregation, producing the 4\-hour resolution*Japan\-4H*\(9,222 timesteps, supporting short\-term operational forecasting\) and the daily resolution*Japan\-1D*\(1,537 timesteps, supporting long\-term strategic planning\)\.

Meteorological data\.We integrated NOAA’s Integrated Surface Dataset \(ISD\)222https://www\.ncei\.noaa\.gov/metadata/geoportal/rest/metadata/item/gov\.noaa\.ncdc:C00532/html, retaining228weather stations \(after removing those with insufficient coverage\) that were geographically matched to the radiation network\. We extracted four physically motivated variables, wind speed and wind direction \(governing advective transport\), air temperature \(modulating atmospheric stability\), and dew point \(a proxy for moisture content controlling wet deposition\)\. All meteorological series were resampled onto the radiation grid through Akima spline interpolation, which preserves local monotonicity better than cubic interpolation for meteorological variables\.

### III\-BDataset Statistics

Table[I](https://arxiv.org/html/2607.24774#S3.T1)summarizes the Japan\-1D/4H statistics\. The datasets cover3,627monitoring stations paired with228co\-located meteorological stations over four years, with four physically motivated meteorological covariates \(wind speed, wind direction, air temperature, dew point\) and a strongly skewed station\-density distribution\. To our knowledge, Japan\-1D/4H constitute the largest publicly released nationwide nuclear\-radiation forecasting corpus\.

TABLE I:Statistics of the radiation and meteorological datasets used in NRFormer\+\.Data DescriptionJapan\-4HJapan\-1DNuclear radiation data\# of stations3,6273,627\# of timesteps9,2221,537Interval4 hour1 dayTime span3/17/2021 – 5/31/2025Meteorological data\# of stations228\# of timesteps36,888Interval1 hourTime span3/17/2021 – 5/31/2025
### III\-CSpatial Analysis: Imbalanced Density and Heterogeneous Correlations

Finding S1 \(Imbalanced station density\)\.Approximately78%78\\%of the 3,627 stations are concentrated within Fukushima Prefecture and its three immediate neighbors, which collectively occupy less than6%6\\%of Japan’s land area\. The empiricalkk\-NN distance distribution \(k=5k\{=\}5\) is heavily right\-skewed\. The Gini coefficient of mean inter\-station distances is0\.740\.74, and the median distance to the five nearest neighbors is only1\.21\.2km, while the 90th percentile reaches10\.410\.4km, indicating an order\-of\-magnitude spread \(Figure[3](https://arxiv.org/html/2607.24774#S3.F3)\(b\)\)\. When measured by the densest decile of the spatial domain,83\.6%83\.6\\%of all stations fall within only10%10\\%of the area, an imbalance ratio that far exceeds that of standard sensor\-network benchmarks\.

Finding S2 \(Heterogeneous spatial correlation\)\.Standard graph\-based STGNNs implicitly assume Tobler’s first law,*i\.e\.,*that geographic proximity implies similarity\. We test this assumption by computing the global Moran’sIIstatistic on radiation residuals using a Gaussian\-kernel weight matrix, obtainingI=0\.15I\{=\}0\.15, indicating positive but modest spatial autocorrelation\. The empirical semivariogram \(Figure[3](https://arxiv.org/html/2607.24774#S3.F3)\(d\)\) exhibits a pronounced*nugget effect*of≈61%\\approx\\\!61\\%of the total sill, signalling substantial micro\-scale variability that cannot be explained by distance alone, and an effective correlation range of only≈66\\approx\\\!66km\.

Findings S1 and S2 jointly invalidate uniform\-neighborhood graph propagation\. Dense\-region nodes are saturated by redundant signals, sparse\-region nodes are informationally starved, and even nearby pairs may be weakly correlated\. This directly motivates the*Density\-Adaptive Spatial Attention*\(Section[IV\-C](https://arxiv.org/html/2607.24774#S4.SS3)\), which routes attention based on learned radiation similarity rather than geographic distance\.

### III\-DTemporal Analysis: Non\-stationarity, Multi\-scale Periodicity, and Heavy\-tailed Events

Finding T1 \(Pervasive non\-stationarity\)\.We applied the Augmented Dickey–Fuller \(ADF\)\[[6](https://arxiv.org/html/2607.24774#bib.bib24)\]and Kwiatkowski–Phillips–Schmidt–Shin \(KPSS\)\[[12](https://arxiv.org/html/2607.24774#bib.bib25)\]tests to every station\-level series and classify each station by the joint outcome \(Figure[4](https://arxiv.org/html/2607.24774#S3.F4)\(a\)\)\. At significance levelα=0\.05\\alpha\{=\}0\.05, only∼13%\\sim\\\!13\\%of stations pass both tests as stationary,∼81%\\sim\\\!81\\%reject KPSS while passing ADF \(trend\-stationary with shifts\), and an additional∼6%\\sim\\\!6\\%fail both \(unit root with shifts\), totalling∼87%\\sim\\\!87\\%non\-stationary\. Per\-station excess kurtosis \(Fig\.[4](https://arxiv.org/html/2607.24774#S3.F4)b\) exceeds 3 for more than80%80\\%of stations and reaches values above 20 near Fukushima Daiichi; variance decomposition \(panel f\) attributes the bulk of residual variance to high\-frequency components, supporting a non\-Gaussian, event\-driven temporal regime\. A near\-linear downward trend of approximately2\.9%2\.9\\%per annum, captured well across stations \(R2=0\.56R^\{2\}\{=\}0\.56\), is consistent with Cs\-137 decay augmented by environmental loss processes such as weathering and runoff\.

Finding T2 \(Multi\-scale periodicity\)\.Welch power\-spectral analysis \(Figure[4](https://arxiv.org/html/2607.24774#S3.F4)\(c\)\) and STL decomposition \(Figure[4](https://arxiv.org/html/2607.24774#S3.F4)\(d\)\) jointly reveal a hierarchy of cyclic components, a*circadian*cycle \(T=24T\{=\}24h, captured only by Japan\-4H\) accounting for≈4\.5%\\approx\\\!4\.5\\%of station\-level variance, a secondary peak at∼\\sim7 days plausibly aligned with synoptic weather oscillations, and a dominant*annual*cycle accounting for≈35%\\approx\\\!35\\%of variance, with January exhibiting the lowest mean radiation\.

![Refer to caption](https://arxiv.org/html/2607.24774v1/x4.png)Figure 4:Temporal characterization of nationwide radiation series\. \(a\) Joint ADF and KPSS stationarity outcomes across stations, with about 87% non\-stationary\. \(b\) Per\-station excess kurtosis projected onto Japan’s geography\. \(c\) Median Welch power spectral density with 10–90% inter\-station band; vertical lines mark diurnal, weekly and annual frequencies\. \(d\) Representative STL decomposition\. \(e\) Pooled CCDF of first\-difference magnitudes with Hill\-estimator tail slopeα^\\hat\{\\alpha\}\. \(f\) Per\-station variance attribution to trend, seasonal, and residual\.Finding T3 \(Heavy\-tailed sudden\-change events\)\.The complementary cumulative distribution of first\-order increments\|Δ​Ct\|\|\\Delta C\_\{t\}\|deviates substantially from a Gaussian baseline \(Figure[4](https://arxiv.org/html/2607.24774#S3.F4)\(e\)\), a Hill\-estimator fit on the upper5%5\\%tail yields an exponent ofα^=2\.84\\hat\{\\alpha\}\{=\}2\.84, and over99%99\\%of stations exhibit at least one\>\>5σ\\sigmajump in the analysis window\. This heavy\-tailed behavior is consistent with episodic, weather\-triggered transport events overlaid on the smooth decay trend\.

T1 motivates the*Non\-stationary Temporal Attention*with instance\-level normalization \(Section[IV\-A](https://arxiv.org/html/2607.24774#S4.SS1)\), which absorbs distribution shifts before attention\. T2 justifies the day\-of\-year seasonal embedding in the Context\-Aware Propagation Prompting module \(Section[IV\-B3](https://arxiv.org/html/2607.24774#S4.SS2.SSS3.Px2)\)\. T3 indicates that the model must remain reactive to abrupt transport events, a property that purely smooth recurrent models lack and that the point\-wise temporal attention together with the physics\-guided Laplacian term in Section[IV\-B1](https://arxiv.org/html/2607.24774#S4.SS2.SSS1)explicitly preserves\.

### III\-EMeteorology–Radiation Coupling: Empirical Evidence for Atmospheric Transport

This subsection presents the empirical evidence that anchors the new*Physics\-Guided Atmospheric Diffusion Module*\. The advection–diffusion equation \([4](https://arxiv.org/html/2607.24774#S4.E4)\) \(Section[IV\-B1](https://arxiv.org/html/2607.24774#S4.SS2.SSS1)\) predicts two falsifiable signatures, \(i\) radiation responses should lag wind events, and \(ii\) the effective diffusion observed under an isotropic graph operator should be modulated by wind, since strong winds redirect transport from omnidirectional mixing to anisotropic advection\.

![Refer to caption](https://arxiv.org/html/2607.24774v1/x5.png)Figure 5:Meteorology–radiation coupling\. \(a\) ECDF of each radiation station’s distance to its nearest NOAA station\. \(b\) Wind\-speed×\\timeswind\-direction conditional radiation anomaly \(per\-station de\-meaned\)\. \(c\) Mutual\-information ranking of meteorological variables against the per\-station radiation residual\. \(d\) Partial\-dependence of radiation on wind speed from a gradient\-boosted surrogate\. \(e\) Distribution of per\-station mutual information between wind speed and the radiation residual\. \(f\) Lagged cross\-correlation of station\-averaged wind speed and radiation residual, peaking at a multi\-hour lag\.Finding M1 \(Lagged response to wind events\)\.The two\-sided cross\-correlation between station\-averaged wind speed and radiation residuals attains\|ρmax\|=0\.16\|\\rho\_\{\\max\}\|\{=\}0\.16at a lag whose magnitude is approximately1616h on Japan\-4H\. The non\-zero, multi\-hour lag is incompatible with instantaneous mixing models and instead supports a transport\-and\-diffusion picture in which wind events drive distributed dispersion over a multi\-hour timescale \(Figure[5](https://arxiv.org/html/2607.24774#S3.F5)\(f\)\)\. Beyond the existence of the lag, two complementary signatures further constrain the encoder design\. The per\-station mutual information between wind speed and the radiation residual \(Figure[5](https://arxiv.org/html/2607.24774#S3.F5)\(e\)\) is widely dispersed across the3,6273\{,\}627stations, indicating that the strength of the meteorology→\\toradiation coupling is itself station\-specific, and the variable\-level information ranking \(Figure[5](https://arxiv.org/html/2607.24774#S3.F5)\(c\)\) shows that air temperature and the wind direction components \(sin⁡θw\\sin\\theta\_\{w\},cos⁡θw\\cos\\theta\_\{w\}\) carry as much signal as wind speed, justifying a multi\-channel rather than wind\-speed\-only meteorological pathway\. The wind\-direction\-conditioned radiation anomaly is moreover strongly asymmetric \(Figure[5](https://arxiv.org/html/2607.24774#S3.F5)\(b\)\), a directional effect confirmed by a gradient\-boosted partial\-dependence analysis \(Figure[5](https://arxiv.org/html/2607.24774#S3.F5)\(d\)\), supporting explicit treatment of advection\.

Finding M2 \(Wind\-modulated effective diffusion\)\.We estimated an empirical effective diffusion coefficientD^\\hat\{D\}by regressing first\-order incrementsΔ​xi​\(t\)\\Delta x\_\{i\}\(t\)onto the discrete spatial LaplacianL^i​\(t\)\\hat\{L\}\_\{i\}\(t\)on thekk=10 NN graph \(Figure[6](https://arxiv.org/html/2607.24774#S3.F6)\(a\)\) and stratified the regression slope by wind\-speed quintile\. The fitted slopeD^\\hat\{D\}decreases monotonically across all five wind\-speed quintiles \(Figure[6](https://arxiv.org/html/2607.24774#S3.F6)\(b\), Spearmanρs=−1\.00\\rho\_\{s\}\{=\}\{\-\}1\.00\)\. The intuitive interpretation is that the isotropic graph Laplacian explains a smaller share ofΔ​x\\Delta xwhen wind drives anisotropic transport\. Simple isotropic diffusion is therefore an inadequate operator at high wind, motivating a learned, wind\-modulated diffusion in which the operator and coefficient are jointly conditioned on meteorological state\. The resulting transport manifests as episodic sudden\-change events, whose inter\-arrival times follow a near\-memoryless Weibull law \(Figure[6](https://arxiv.org/html/2607.24774#S3.F6)\(c\)\) and whose per\-station rates vary widely across the network \(Figure[6](https://arxiv.org/html/2607.24774#S3.F6)\(d\)\)\.

![Refer to caption](https://arxiv.org/html/2607.24774v1/x6.png)Figure 6:Empirical validation of advection–diffusion priors\. \(a\) Hex\-binned scatter of the first\-differenceΔ​xi​\(t\)\\Delta x\_\{i\}\(t\)against the discrete LaplacianL^i​\(t\)\\hat\{L\}\_\{i\}\(t\)on thek=10k\{=\}10graph, with a small positive fit slope \(D^\>0\\hat\{D\}\>0,r=0\.06r\{=\}0\.06\)\. \(b\)D^\\hat\{D\}stratified by wind\-speed quintile, decreasing monotonically with wind speed \(ρs=−1\.00\\rho\_\{s\}\{=\}\{\-\}1\.00\)\. \(c\) CCDF of inter\-arrival times of\>3⋅\>3\\cdotMAD sudden\-change events with a Weibull fit\. \(d\) Per\-station sudden\-change event rates \(median 0\.24,Q75Q\_\{75\}0\.25\)\.M1 and M2 jointly establish that radiation dynamics in Japan\-1D/4H are coupled to meteorology in a manner that is both lagged and anisotropic, and that an isotropic, wind\-agnostic diffusion operator under\-explains the observed transport\. This is precisely the inductive bias encoded in the*Physics\-Guided Atmospheric Diffusion Module*\(Section[IV\-B1](https://arxiv.org/html/2607.24774#S4.SS2.SSS1)\), which estimates a station\-wiseDiD\_\{i\}from the local meteorological context, and in the*Enhanced Meteorological Encoder*\(Section[IV\-B2](https://arxiv.org/html/2607.24774#S4.SS2.SSS2)\), which separates wind \(advection\) and thermal \(stability\) pathways\.

### III\-FImplications for Model Design

In summary, the data engineering and analysis pipeline described above transforms nationwide radiation and meteorological observations into a rigorously characterized forecasting corpus, and the resulting empirical signatures \(non\-stationarity, multi\-scale periodicity, heavy\-tailed transients, imbalanced and heterogeneous spatial structure, and meteorology\-modulated diffusion\) collectively define the design space within which NRFormer\+ operates\. We next describe how each of these findings is translated into a concrete architectural mechanism\.

![Refer to caption](https://arxiv.org/html/2607.24774v1/x7.png)Figure 7:The framework overview of NRFormer\+\.

## IVMethodology

Overview\.Building on NRFormer\[[21](https://arxiv.org/html/2607.24774#bib.bib1)\], which relied on purely data\-driven temporal and spatial attention, NRFormer\+ adds explicit mechanisms to encode the physics of atmospheric radiation transport\. Its overall framework, the Physics\-Guided Radiation Transformer, is illustrated in Figure[7](https://arxiv.org/html/2607.24774#S3.F7)\.

### IV\-ANon\-stationary Temporal Attention

Radiation time series exhibit a highly non\-stationary nature due to unpredictable events such as irregular human interventions and nuclear power plant operational changes\. To effectively extract stable knowledge from radiation sequence inputs, we employ a non\-stationary temporal attention mechanism that consists of two components, non\-stationary normalization and point\-wise temporal attention\.

#### IV\-A1Non\-stationary Normalization

Due to the non\-stationarity of radiation time series, the underlying distributions of different input sequences are diverse, which significantly degrades the forecasting performance\. Motivated by recent works\[[11](https://arxiv.org/html/2607.24774#bib.bib26),[20](https://arxiv.org/html/2607.24774#bib.bib23)\], we apply instance\-wise normalization to eliminate the non\-stationary information in each input sequence\. Formally, given historical observations𝐱i∈ℝP\\mathbf\{x\}\_\{i\}\\in\\mathbb\{R\}^\{P\}at theii\-th station, wherePPis the input time window, we first compute the instance\-level statistics:

𝔼​\[𝐱i\]=1P​∑j=1P𝐱ij,V​a​r​\[𝐱i\]=1P​∑j=1P\(𝐱ij−𝔼​\[𝐱i\]\)2\.\\displaystyle\\mathbb\{E\}\[\\mathbf\{x\}\_\{i\}\]=\\frac\{1\}\{P\}\\sum\_\{j=1\}^\{P\}\\mathbf\{x\}^\{j\}\_\{i\},\\quad Var\[\\mathbf\{x\}\_\{i\}\]=\\frac\{1\}\{P\}\\sum\_\{j=1\}^\{P\}\(\\mathbf\{x\}^\{j\}\_\{i\}\-\\mathbb\{E\}\[\\mathbf\{x\}\_\{i\}\]\)^\{2\}\.\(2\)We then normalize the original input sequence with learnable affine parameters:

𝐱^i=γi​𝐱i−𝔼​\[𝐱i\]V​a​r​\[𝐱i\]\+ϵ\+βi,\\displaystyle\\hat\{\\mathbf\{x\}\}\_\{i\}=\\gamma\_\{i\}\\frac\{\\mathbf\{x\}\_\{i\}\-\\mathbb\{E\}\[\\mathbf\{x\}\_\{i\}\]\}\{\\sqrt\{Var\[\\mathbf\{x\}\_\{i\}\]\+\\epsilon\}\}\+\\beta\_\{i\},\(3\)whereγi\\gamma\_\{i\}andβi\\beta\_\{i\}are learnable parameters corresponding to theii\-th station, andϵ\\epsilonis a small constant for numerical stability\. The normalized sequences possess more stable statistical properties, substantially reducing the difficulty of capturing temporal dynamics\. Subsequently, we embed𝐱^i\\hat\{\\mathbf\{x\}\}\_\{i\}through a1×11\\times 1convolutional layer:𝐳ti=Conv1×1​\(𝐱^i\)\\mathbf\{z\}\_\{t\}^\{i\}=\\text\{Conv\}\_\{1\\times 1\}\(\\hat\{\\mathbf\{x\}\}\_\{i\}\), where𝐳ti∈ℝP×D\\mathbf\{z\}\_\{t\}^\{i\}\\in\\mathbb\{R\}^\{P\\times D\}represents the point\-wise time series embedding of stationii, andDDis the hidden dimension\.

#### IV\-A2Point\-wise Temporal Attention

To capture both unstable and long\-range temporal dependencies inherent in nuclear radiation time series, we employ a point\-wise temporal self\-attention mechanism\. Given the embedding𝐳ti\\mathbf\{z\}\_\{t\}^\{i\}, we derive query, key, and value matrices𝐐t=𝐳ti​𝐖tQ\\mathbf\{Q\}\_\{t\}=\\mathbf\{z\}\_\{t\}^\{i\}\\mathbf\{W\}\_\{t\}^\{Q\},𝐊t=𝐳ti​𝐖tK\\mathbf\{K\}\_\{t\}=\\mathbf\{z\}\_\{t\}^\{i\}\\mathbf\{W\}\_\{t\}^\{K\}, and𝐕t=𝐳ti​𝐖tV\\mathbf\{V\}\_\{t\}=\\mathbf\{z\}\_\{t\}^\{i\}\\mathbf\{W\}\_\{t\}^\{V\}, where𝐖tQ,𝐖tK,𝐖tV∈ℝD×D\\mathbf\{W\}\_\{t\}^\{Q\},\\mathbf\{W\}\_\{t\}^\{K\},\\mathbf\{W\}\_\{t\}^\{V\}\\in\\mathbb\{R\}^\{D\\times D\}are learnable parameters shared across stations\. The temporal attention score is computed as𝐀t=Softmax​\(𝐐t​𝐊t⊤D\)\\mathbf\{A\}\_\{t\}=\\text\{Softmax\}\\\!\\left\(\\frac\{\\mathbf\{Q\}\_\{t\}\{\\mathbf\{K\}\_\{t\}\}^\{\\top\}\}\{\\sqrt\{D\}\}\\right\), where𝐀t∈ℝP×P\\mathbf\{A\}\_\{t\}\\in\\mathbb\{R\}^\{P\\times P\}\. The updated representation is obtained via𝐳ti,\(l\)=𝐀t​𝐕t\\mathbf\{z\}\_\{t\}^\{i,\(l\)\}=\\mathbf\{A\}\_\{t\}\\mathbf\{V\}\_\{t\}\. We employ multi\-head attention with residual connections and layer normalization\[[37](https://arxiv.org/html/2607.24774#bib.bib21)\], stackingLtL\_\{t\}such layers\. The final temporal output is obtained by flattening and projecting:𝐳ti=Flatten​\(𝐳ti,\(Lt\)\)​𝐖r\\mathbf\{z\}\_\{t\}^\{i\}=\\text\{Flatten\}\(\\mathbf\{z\}\_\{t\}^\{i,\(L\_\{t\}\)\}\)\\mathbf\{W\}^\{r\}, where𝐖r∈ℝ\(P⋅D\)×D\\mathbf\{W\}^\{r\}\\in\\mathbb\{R\}^\{\(P\\cdot D\)\\times D\}\. We denote𝐙r∈ℝN×D\\mathbf\{Z\}\_\{r\}\\in\\mathbb\{R\}^\{N\\times D\}as the radiation feature matrix of all stations, where theii\-th row is𝐳ti\\mathbf\{z\}\_\{t\}^\{i\}\.

### IV\-BPhysics\-Guided Atmospheric Modeling

A fundamental limitation of purely data\-driven approaches for radiation forecasting is their inability to encode the physical laws governing atmospheric transport\. Nuclear radiation disperses through the atmosphere primarily via diffusion and advection processes, which are well\-characterized by established partial differential equations\. In NRFormer\+, we introduce two physics\-guided components that inject domain knowledge into the learning process, an atmospheric diffusion module grounded in the diffusion equation, and an enhanced meteorological encoder with physically\-motivated architectural designs\.

#### IV\-B1Atmospheric Diffusion Module

The atmospheric transport of radioactive materials is fundamentally governed by the advection–diffusion equation\[[33](https://arxiv.org/html/2607.24774#bib.bib66)\]:

∂C∂t=−𝐮⋅∇C\+∇⋅\(D​∇C\)\+S,\\frac\{\\partial C\}\{\\partial t\}\\;=\\;\-\\mathbf\{u\}\\cdot\\nabla C\\;\+\\;\\nabla\\\!\\cdot\(D\\,\\nabla C\)\\;\+\\;S,\(4\)whereCCis the radiation concentration,𝐮\\mathbf\{u\}is the wind field,DDis an eddy\-diffusion coefficient that depends on atmospheric stability, andSSaggregates sources and sinks\. In NRFormer\+, the advective term−𝐮⋅∇C\-\\mathbf\{u\}\\cdot\\nabla Cis absorbed by the density\-adaptive spatial attention mechanism \(Section[IV\-C](https://arxiv.org/html/2607.24774#S4.SS3)\), which routes information by learned radiation similarity over the irregular monitoring graph\. Under this decomposition, the diffusive component of Eq\. \([4](https://arxiv.org/html/2607.24774#S4.E4)\) reduces to the diffusion equation∂C∂t=D​∇2C\\frac\{\\partial C\}\{\\partial t\}=D\\nabla^\{2\}C, where∇2C\\nabla^\{2\}Cis the spatial Laplacian of the concentration field\. Rather than attempting to numerically solve this PDE, we design a neural module that encodes its key physical quantities as inductive biases\.

Diffusion coefficient estimation\.The diffusion coefficientDDis not a universal constant but varies with local atmospheric conditions, including wind turbulence, atmospheric stability, and terrain characteristics\. We estimateDDfor each station using a parameterized neural network that takes meteorological and geographical features as inputs:

Di=Softplus​\(MLPD​\(\[𝐦¯i∥𝐥i\]\)\),D\_\{i\}=\\text\{Softplus\}\\\!\\left\(\\text\{MLP\}\_\{D\}\\\!\\left\(\[\\bar\{\\mathbf\{m\}\}\_\{i\}\\\|\\mathbf\{l\}\_\{i\}\]\\right\)\\right\),\(5\)where𝐦¯i∈ℝCm\\bar\{\\mathbf\{m\}\}\_\{i\}\\in\\mathbb\{R\}^\{C\_\{m\}\}is the time\-averaged meteorological feature vector at stationii\(*i\.e\.,*the mean over thePPinput time steps\),𝐥i∈ℝ2\\mathbf\{l\}\_\{i\}\\in\\mathbb\{R\}^\{2\}is the geographic coordinate \(latitude, longitude\), and∥\\\|denotes concatenation\. TheMLPD\\text\{MLP\}\_\{D\}consists of three fully\-connected layers with ReLU activations \(ℝCm\+2→ℝ64→ℝ32→ℝ1\\mathbb\{R\}^\{C\_\{m\}\+2\}\\rightarrow\\mathbb\{R\}^\{64\}\\rightarrow\\mathbb\{R\}^\{32\}\\rightarrow\\mathbb\{R\}^\{1\}\)\. Critically, the Softplus activation function ensures thatDi\>0D\_\{i\}\>0, which is a physical constraint since diffusion coefficients must be strictly positive\.

Spatial Laplacian approximation\.Computing the exact Laplacian∇2C\\nabla^\{2\}Crequires the continuous concentration field, which is observed only at discrete stations\. For the diffusion module we therefore use a lightweight surrogate that needs no graph, namely each station’s deviation from the global spatial mean,∇2Ci≈Ci\(P\)−C¯\(P\)\\nabla^\{2\}C\_\{i\}\\approx C\_\{i\}^\{\(P\)\}\-\\bar\{C\}^\{\(P\)\}, whereCi\(P\)C\_\{i\}^\{\(P\)\}is the radiation level at stationiiat the most recent time stepPPandC¯\(P\)=1N​∑j=1NCj\(P\)\\bar\{C\}^\{\(P\)\}=\\frac\{1\}\{N\}\\sum\_\{j=1\}^\{N\}C\_\{j\}^\{\(P\)\}is the spatial mean\. This is a zeroth\-order, mean\-field coarsening of the local graph Laplacian\[[30](https://arxiv.org/html/2607.24774#bib.bib67)\], a global\-anomaly signal rather than a true local curvature; we adopt it because it needs no neighbor graph and stays informative on the heavy\-tailed events where a station departs sharply from the network background\. The corresponding local diffusive tendency, measured with akk\-NN graph Laplacian, is the quantity empirically validated in Finding M2 \(§[III\-E](https://arxiv.org/html/2607.24774#S3.SS5)\)\.

Physics constraint encoding\.Having obtained the three key physical quantities from the diffusion equation, namely the concentrationCi\(P\)C\_\{i\}^\{\(P\)\}, the estimated diffusion coefficientDiD\_\{i\}, and the mean\-field Laplacian surrogate∇2Ci\\nabla^\{2\}C\_\{i\}, we encode this physics triplet into a latent representation:

𝐩i=MLPϕ​\(\[Ci\(P\),Di,∇2Ci\]\),\\mathbf\{p\}\_\{i\}=\\text\{MLP\}\_\{\\phi\}\\\!\\left\(\\left\[C\_\{i\}^\{\(P\)\},\\;D\_\{i\},\\;\\nabla^\{2\}C\_\{i\}\\right\]\\right\),\(6\)whereMLPϕ:ℝ3→ℝD\\text\{MLP\}\_\{\\phi\}:\\mathbb\{R\}^\{3\}\\rightarrow\\mathbb\{R\}^\{D\}is a two\-layer MLP with ReLU activation\. The resulting physics embedding𝐩i∈ℝD\\mathbf\{p\}\_\{i\}\\in\\mathbb\{R\}^\{D\}encodes the local diffusive tendency at each station\. Let𝐙p∈ℝN×D\\mathbf\{Z\}\_\{p\}\\in\\mathbb\{R\}^\{N\\times D\}denote the embeddings for all stations, whereii\-th row is𝐩i\\mathbf\{p\}\_\{i\}\.

Encoding the triplet as a soft feature pathway rather than a hard penalty term tells the model which stations show concentration anomalies \(via the Laplacian term\) and how fast they should dissipate \(viaDiD\_\{i\}\), while still allowing non\-diffusive patterns such as discrete emission events to be learned\.

#### IV\-B2Enhanced Meteorological Encoder

In the conference version NRFormer, meteorological features were processed through flat concatenation followed by a single MLP, treating all meteorological variables homogeneously\. However, different meteorological variables govern fundamentally different physical processes in radiation transport\. Wind drives advective transport, while temperature conditions control atmospheric stability and vertical mixing\. NRFormer\+ introduces a physically\-motivated encoder that processes these variable groups through separate pathways before fusion\.

Wind dynamics encoder\.Wind speed and direction jointly determine the advective transport of radioactive particles\. We encode wind features using a two\-stage architecture:

𝐅w=Conv1×3\(2\)​\(ReLU​\(Conv1×1\(1\)​\(𝐌w\)\)\),\\mathbf\{F\}\_\{w\}=\\text\{Conv\}^\{\(2\)\}\_\{1\\times 3\}\\\!\\left\(\\text\{ReLU\}\\\!\\left\(\\text\{Conv\}^\{\(1\)\}\_\{1\\times 1\}\(\\mathbf\{M\}\_\{w\}\)\\right\)\\right\),\(7\)where𝐌w∈ℝB×2×N×P\\mathbf\{M\}\_\{w\}\\in\\mathbb\{R\}^\{B\\times 2\\times N\\times P\}contains the wind speed and direction measurements across all stations and time steps\. The first1×11\\times 1convolution projects the two wind channels intoD/2D/2feature channels, capturing the interaction between speed and direction \(*e\.g\.,*computing effective wind velocity components\)\. The second1×31\\times 3temporal convolution with padding captures the short\-term dynamics of wind patterns\. The output𝐅w∈ℝB×\(D/2\)×N×P\\mathbf\{F\}\_\{w\}\\in\\mathbb\{R\}^\{B\\times\(D/2\)\\times N\\times P\}encodes the wind dynamics\.

Thermal condition encoder\.Air temperature and dew point jointly characterize atmospheric stability, which determines the vertical extent of radiation mixing\. We employ an analogous architecture:

𝐅τ=Conv1×3\(2\)​\(ReLU​\(Conv1×1\(1\)​\(𝐌τ\)\)\),\\mathbf\{F\}\_\{\\tau\}=\\text\{Conv\}^\{\(2\)\}\_\{1\\times 3\}\\\!\\left\(\\text\{ReLU\}\\\!\\left\(\\text\{Conv\}^\{\(1\)\}\_\{1\\times 1\}\(\\mathbf\{M\}\_\{\\tau\}\)\\right\)\\right\),\(8\)where𝐌τ∈ℝB×2×N×P\\mathbf\{M\}\_\{\\tau\}\\in\\mathbb\{R\}^\{B\\times 2\\times N\\times P\}contains air temperature and dew point measurements\. The temperature\-dew point differential indicates atmospheric stability, a physically meaningful quantity the convolutional architecture can learn to extract\.

Fusion\.The wind and thermal features are concatenated and transformed through an MLP with residual connections to produce the final meteorological embedding:

𝐄m=MLPf​\(Flatten​\(\[𝐅w∥𝐅τ\]\)\),\\mathbf\{E\}\_\{m\}=\\text\{MLP\}\_\{f\}\\\!\\left\(\\text\{Flatten\}\\\!\\left\(\[\\mathbf\{F\}\_\{w\}\\\|\\mathbf\{F\}\_\{\\tau\}\]\\right\)\\right\),\(9\)whereFlatten​\(⋅\)\\text\{Flatten\}\(\\cdot\)collapses the temporal and channel dimensions, andMLPf\\text\{MLP\}\_\{f\}consists of a1×11\\times 1convolution followed by two residual MLP blocks and a final projection toℝD\\mathbb\{R\}^\{D\}\. The resulting𝐄m∈ℝN×D\\mathbf\{E\}\_\{m\}\\in\\mathbb\{R\}^\{N\\times D\}provides a comprehensive meteorological representation for each station\.

#### IV\-B3Context\-Aware Propagation Prompting

The propagation of radiation is influenced by a complex interplay of geographical, temporal, and meteorological factors\. While NRFormer incorporated basic location and meteorological prompts via concatenation into the spatial attention’s query and key, NRFormer\+ substantially expands the prompting mechanism with three additional context encoders that capture richer aspects of the radiation propagation environment\.

##### Deep Location Encoder

Different geographic locations exhibit distinct radiation characteristics due to variations in terrain, proximity to nuclear facilities, and local atmospheric circulation patterns\. We encode station coordinates through a deep MLP that can learn nonlinear spatial embeddings:

𝐄l=MLPl​\(𝐋\),MLPl:ℝ2→ReLUℝ32→ReLUℝ64→ℝD,\\mathbf\{E\}\_\{l\}=\\text\{MLP\}\_\{l\}\(\\mathbf\{L\}\),\\quad\\text\{MLP\}\_\{l\}:\\mathbb\{R\}^\{2\}\\xrightarrow\{\\text\{ReLU\}\}\\mathbb\{R\}^\{32\}\\xrightarrow\{\\text\{ReLU\}\}\\mathbb\{R\}^\{64\}\\xrightarrow\{\}\\mathbb\{R\}^\{D\},\(10\)where𝐋∈ℝN×2\\mathbf\{L\}\\in\\mathbb\{R\}^\{N\\times 2\}contains the latitude and longitude of each station\. Compared to the simpler two\-layer encoder in NRFormer, the three\-layer architecture with progressive dimension expansion \(2→32→64→D2\\rightarrow 32\\rightarrow 64\\rightarrow D\) enables the model to capture more complex spatial patterns, such as the influence of mountain ranges on radiation shielding or coastal effects on atmospheric dispersion\. The location embedding𝐄l∈ℝN×D\\mathbf\{E\}\_\{l\}\\in\\mathbb\{R\}^\{N\\times D\}is shared across all samples in a batch\.

##### Seasonal Temporal Encoder

As demonstrated in our data analysis \(Section[III](https://arxiv.org/html/2607.24774#S3)\), radiation levels exhibit pronounced seasonal variations, with January consistently showing the lowest values across the country\. NRFormer encoded temporal context through three separate embeddings for time\-of\-day, day\-of\-week, and month\-of\-year\. NRFormer\+ replaces these with a unified day\-of\-year embedding that directly captures the continuous seasonal cycle:

𝐄τ=Embeddoy​\(d\),Embeddoy:\{1,2,…,366\}→ℝD,\\mathbf\{E\}\_\{\\tau\}=\\text\{Embed\}\_\{\\text\{doy\}\}\(d\),\\quad\\text\{Embed\}\_\{\\text\{doy\}\}:\\\{1,2,\\ldots,366\\\}\\rightarrow\\mathbb\{R\}^\{D\},\(11\)wheredddenotes the day\-of\-year index extracted from the last time step of the input window\. The embedding tableEmbeddoy∈ℝ366×D\\text\{Embed\}\_\{\\text\{doy\}\}\\in\\mathbb\{R\}^\{366\\times D\}is initialized with Xavier uniform initialization and learned end\-to\-end\. This design offers two advantages over separate time embeddings, \(1\) it provides a single, coherent representation of seasonal position rather than forcing the model to reconstruct seasonality from independent components, and \(2\) it increases representational specificity, since each day\-of\-year has its own unique embedding\. The embedding𝐄τ∈ℝN×D\\mathbf\{E\}\_\{\\tau\}\\in\\mathbb\{R\}^\{N\\times D\}is broadcast across all stations\.

##### Radiation\-Location Cross Features

While the location encoder and radiation features provide complementary perspectives, their interaction may reveal important patterns\. For instance, stations at similar latitudes near coastal areas may share radiation dynamics that differ from inland stations at the same latitude\. We introduce a cross\-feature module that models the joint distribution of radiation and location:

𝐑loc=\[𝐗i​‖𝐋i,lat\(P\)‖​𝐋i,lon\(P\)\]∈ℝN×3​P,\\mathbf\{R\}\_\{\\text\{loc\}\}=\[\\mathbf\{X\}\_\{i\}\\\|\\mathbf\{L\}^\{\(P\)\}\_\{i,\\text\{lat\}\}\\\|\\mathbf\{L\}^\{\(P\)\}\_\{i,\\text\{lon\}\}\]\\in\\mathbb\{R\}^\{N\\times 3P\},\(12\)where𝐗i∈ℝP\\mathbf\{X\}\_\{i\}\\in\\mathbb\{R\}^\{P\}is the radiation time series at stationii, and𝐋i,lat\(P\),𝐋i,lon\(P\)∈ℝP\\mathbf\{L\}^\{\(P\)\}\_\{i,\\text\{lat\}\},\\mathbf\{L\}^\{\(P\)\}\_\{i,\\text\{lon\}\}\\in\\mathbb\{R\}^\{P\}are the latitude and longitude coordinates replicated across thePPtime steps\. This concatenation is processed through a1×11\\times 1convolution followed by three residual MLP blocks:

𝐄c=ResMLP\(3\)​\(Conv1×1​\(𝐑loc\)\),\\mathbf\{E\}\_\{c\}=\\text\{ResMLP\}^\{\(3\)\}\\\!\\left\(\\text\{Conv\}\_\{1\\times 1\}\(\\mathbf\{R\}\_\{\\text\{loc\}\}\)\\right\),\(13\)where each residual MLP block consists of two fully\-connected layers with ReLU activation and a skip connection\. The resulting cross embedding𝐄c∈ℝN×D\\mathbf\{E\}\_\{c\}\\in\\mathbb\{R\}^\{N\\times D\}captures how radiation patterns co\-vary with geographic position\.

##### Multi\-Source Temporal Fusion

All contextual embeddings are aggregated with the radiation and physics features through a temporal fusion layer:

𝐙f=Conv1×1​\(\[𝐙r​‖𝐙p‖​𝐄l​‖𝐄c‖​𝐄m∥𝐄τ\]\),\\mathbf\{Z\}\_\{f\}=\\text\{Conv\}\_\{1\\times 1\}\\\!\\left\(\[\\mathbf\{Z\}\_\{r\}\\\|\\mathbf\{Z\}\_\{p\}\\\|\\mathbf\{E\}\_\{l\}\\\|\\mathbf\{E\}\_\{c\}\\\|\\mathbf\{E\}\_\{m\}\\\|\\mathbf\{E\}\_\{\\tau\}\]\\right\),\(14\)where the1×11\\times 1convolution projects the concatenated\(6​D\)\(6D\)\-dimensional representation back toDDdimensions\. The fused embedding𝐙f∈ℝN×D\\mathbf\{Z\}\_\{f\}\\in\\mathbb\{R\}^\{N\\times D\}serves as the value input𝐕s\\mathbf\{V\}\_\{s\}to the spatial attention mechanism, thereby providing physics\-enriched contextual information for spatial message passing\.

### IV\-CDensity\-Adaptive Spatial Attention

As analyzed in Section[I](https://arxiv.org/html/2607.24774#S1), the severely imbalanced spatial distribution of monitoring stations poses a significant challenge\. Nodes in dense clusters risk over\-smoothing from excessive information aggregation, while sparsely connected nodes suffer from insufficient spatial context\. NRFormer\+ inherits the density\-adaptive spatial attention, which alternates between macro\-scale and proximity\-constrained attention, while introducing a critical modification to the query\-key\-value formulation that injects physics\-aware radiation features\.

#### IV\-C1Physics\-Enriched Query\-Key Formulation

A key architectural innovation of NRFormer\+ is the redesigned input assignment for the spatial attention mechanism\. In NRFormer, both the query/key and value matrices are derived from the fused temporal embedding𝐙s\(0\)\\mathbf\{Z\}\_\{s\}^\{\(0\)\}, meaning the spatial attention routing is determined by the same features that carry the information\. In NRFormer\+, we decouple these roles, setting𝐐s=𝐙r\\mathbf\{Q\}\_\{s\}=\\mathbf\{Z\}\_\{r\},𝐊s=𝐙r\\mathbf\{K\}\_\{s\}=\\mathbf\{Z\}\_\{r\}, and𝐕s=𝐙f\\mathbf\{V\}\_\{s\}=\\mathbf\{Z\}\_\{f\}, where𝐙r∈ℝN×D\\mathbf\{Z\}\_\{r\}\\in\\mathbb\{R\}^\{N\\times D\}is the radiation feature representation from the temporal attention module \(Section[IV\-A](https://arxiv.org/html/2607.24774#S4.SS1)\), and𝐙f∈ℝN×D\\mathbf\{Z\}\_\{f\}\\in\\mathbb\{R\}^\{N\\times D\}is the fused temporal embedding that aggregates radiation, physics, meteorological, location, temporal, and cross features\.

This design has a clear physical interpretation\. The radiation features in the query and key positions determine*which stations should attend to each other*based on the similarity of their patterns, while the physics\-enriched fused features in the value position determine*what information is actually exchanged*\. This separation ensures that spatial attention routing is grounded in observed radiation dynamics, while the exchanged information is enriched with physical context\.

#### IV\-C2Macro\-Scale Spatial Correlation Modeling

Following the physics\-enriched formulation, we add learnable positional encoding and compute query, key, and value matrices𝐐g=\(𝐙r\+𝐏e\)​𝐖gQ\\mathbf\{Q\}\_\{g\}=\(\\mathbf\{Z\}\_\{r\}\+\\mathbf\{P\}\_\{e\}\)\\mathbf\{W\}\_\{g\}^\{Q\},𝐊g=\(𝐙r\+𝐏e\)​𝐖gK\\mathbf\{K\}\_\{g\}=\(\\mathbf\{Z\}\_\{r\}\+\\mathbf\{P\}\_\{e\}\)\\mathbf\{W\}\_\{g\}^\{K\}, and𝐕g=\(𝐙f\+𝐏e\)​𝐖gV\\mathbf\{V\}\_\{g\}=\(\\mathbf\{Z\}\_\{f\}\+\\mathbf\{P\}\_\{e\}\)\\mathbf\{W\}\_\{g\}^\{V\}\. The macro\-scale attention score is computed as

𝐀g=𝐐g⋅𝐊g⊤D,\\displaystyle\\mathbf\{A\}\_\{g\}=\\frac\{\\mathbf\{Q\}\_\{g\}\\cdot\\mathbf\{K\}\_\{g\}^\{\\top\}\}\{\\sqrt\{D\}\},\(15\)where𝐏e∈ℝN×D\\mathbf\{P\}\_\{e\}\\in\\mathbb\{R\}^\{N\\times D\}is a learnable positional embedding matrix that preserves station identity information\. The attention score matrix𝐀g∈ℝN×N\\mathbf\{A\}\_\{g\}\\in\\mathbb\{R\}^\{N\\times N\}captures all\-pair station relationships based on radiation pattern similarity\. The output is computed via𝐇g=Softmax​\(𝐀g\)⋅𝐕g\\mathbf\{H\}\_\{g\}=\\text\{Softmax\}\(\\mathbf\{A\}\_\{g\}\)\\cdot\\mathbf\{V\}\_\{g\}, followed by a feed\-forward layer:𝐇g′=\(ReLU​\(𝐇g⋅𝐖1\+𝐛1\)\)⋅𝐖2\+𝐛2\\mathbf\{H\}^\{\\prime\}\_\{g\}=\(\\text\{ReLU\}\(\\mathbf\{H\}\_\{g\}\\cdot\\mathbf\{W\}\_\{1\}\+\\mathbf\{b\}\_\{1\}\)\)\\cdot\\mathbf\{W\}\_\{2\}\+\\mathbf\{b\}\_\{2\}\. The macro\-scale attention provides additional spatial context for sparsely connected nodes, mitigating the under\-smoothing problem\.

#### IV\-C3Proximity\-Constrained Spatial Correlation Modeling

For densely connected nodes, unconstrained all\-pair attention may introduce excessive noise\. We employ proximity\-constrained attention using a graph mask derived from geographical distance\. Let𝐀\\mathbf\{A\}denote the truncated adjacency matrix based on distance thresholds, we define:

𝐀m​a​s​k​\[i,j\]=\{𝐀l​\[i,j\],if​𝐀​\[i,j\]\>0−∞,otherwise\\displaystyle\\mathbf\{A\}\_\{mask\}\[i,j\]=\(16\)where𝐀l\\mathbf\{A\}\_\{l\}is the attention score matrix computed analogously to Eq\. \([15](https://arxiv.org/html/2607.24774#S4.E15)\)\. The constrained output is:𝐇l=Softmax​\(𝐀m​a​s​k\)⋅𝐕l\\mathbf\{H\}\_\{l\}=\\text\{Softmax\}\(\\mathbf\{A\}\_\{mask\}\)\\cdot\\mathbf\{V\}\_\{l\}, followed by a feed\-forward layer:𝐇l′=\(ReLU​\(𝐇l⋅𝐖3\+𝐛3\)\)⋅𝐖4\+𝐛4\\mathbf\{H\}^\{\\prime\}\_\{l\}=\(\\text\{ReLU\}\(\\mathbf\{H\}\_\{l\}\\cdot\\mathbf\{W\}\_\{3\}\+\\mathbf\{b\}\_\{3\}\)\)\\cdot\\mathbf\{W\}\_\{4\}\+\\mathbf\{b\}\_\{4\}\. We alternately stackLsL\_\{s\}macro\-scale and proximity\-constrained spatial attention layers\. The final spatial output is denoted as𝐙s\(Ls\)∈ℝN×D\\mathbf\{Z\}\_\{s\}^\{\(L\_\{s\}\)\}\\in\\mathbb\{R\}^\{N\\times D\}\.

### IV\-DOutput Layer

A key limitation of NRFormer’s output layer is the two\-way fusion of temporal and spatial embeddings, which may lose the original radiation signal after multiple layers of transformation\. NRFormer\+ addresses this through a three\-way fusion strategy that preserves a direct radiation signal path:

𝐘~t\+1:t\+K=ReLU​\(\(𝐙r​‖𝐙f‖​𝐙s\(Ls\)\)⋅𝐖i\+𝐛i\)⋅𝐖o\+𝐛o,\\displaystyle\\tilde\{\\mathbf\{Y\}\}^\{t\+1:t\+K\}=\\text\{ReLU\}\\\!\\left\(\(\\mathbf\{Z\}\_\{r\}\|\|\\mathbf\{Z\}\_\{f\}\|\|\\mathbf\{Z\}\_\{s\}^\{\(L\_\{s\}\)\}\)\\cdot\\mathbf\{W\}\_\{i\}\+\\mathbf\{b\}\_\{i\}\\right\)\\cdot\\mathbf\{W\}\_\{o\}\+\\mathbf\{b\}\_\{o\},\(17\)where𝐙r∈ℝN×D\\mathbf\{Z\}\_\{r\}\\in\\mathbb\{R\}^\{N\\times D\}is the radiation feature embedding,𝐙f∈ℝN×D\\mathbf\{Z\}\_\{f\}\\in\\mathbb\{R\}^\{N\\times D\}is the physics\-enriched temporal fusion,𝐙s\(Ls\)∈ℝN×D\\mathbf\{Z\}\_\{s\}^\{\(L\_\{s\}\)\}\\in\\mathbb\{R\}^\{N\\times D\}is the spatial attention output,𝐖i∈ℝ3​D×De\\mathbf\{W\}\_\{i\}\\in\\mathbb\{R\}^\{3D\\times D\_\{e\}\}and𝐖o∈ℝDe×K\\mathbf\{W\}\_\{o\}\\in\\mathbb\{R\}^\{D\_\{e\}\\times K\}are learnable weight matrices, and𝐛i,𝐛o\\mathbf\{b\}\_\{i\},\\mathbf\{b\}\_\{o\}are bias parameters\. Including𝐙r\\mathbf\{Z\}\_\{r\}directly in the fusion provides two benefits, \(1\) it creates a short\-circuit path from the input radiation signal to the output, facilitating gradient flow during training, and \(2\) it allows the model to leverage the raw radiation representation even when the temporal and spatial transformations are insufficient\.

The model output is denormalized using the reciprocal of the instance normalization:

𝐲^i=V​a​r​\[𝐱i\]\+ϵ⋅𝐲~i−βiγi\+𝔼​\[𝐱i\],\\displaystyle\\hat\{\\mathbf\{y\}\}\_\{i\}=\\sqrt\{Var\[\\mathbf\{x\}\_\{i\}\]\+\\epsilon\}\\cdot\\frac\{\\tilde\{\\mathbf\{y\}\}\_\{i\}\-\\beta\_\{i\}\}\{\\gamma\_\{i\}\}\+\\mathbb\{E\}\[\\mathbf\{x\}\_\{i\}\],\(18\)where𝐲^i∈ℝK\\hat\{\\mathbf\{y\}\}\_\{i\}\\in\\mathbb\{R\}^\{K\}denotes the future prediction for stationii\. The training objective minimizes the mean absolute error:

ℒ=1N​∑i=1N\|𝐲^i−𝐲i\|,\\mathcal\{L\}=\\frac\{1\}\{N\}\\sum\_\{i=1\}^\{N\}\|\\hat\{\\mathbf\{y\}\}\_\{i\}\-\\mathbf\{y\}\_\{i\}\|,\(19\)where𝐲i\\mathbf\{y\}\_\{i\}is the ground truth radiation level of stationii\. The MAE loss is chosen over MSE as it is more robust to the occasional extreme radiation spikes in the dataset, preventing the model from being disproportionately influenced by outlier events\. The overall complexity of NRFormer\+ is dominated by the spatial attention at𝒪​\(Ls⋅N2⋅D\)\\mathcal\{O\}\(L\_\{s\}\\cdot N^\{2\}\\cdot D\), on par with NRFormer, with the added physics and context modules contributing only a modest, asymptotically subdominant overhead; a full per\-module breakdown is given in Appendix[B\-E](https://arxiv.org/html/2607.24774#A2.SS5)\.

TABLE II:Evaluations of NRFormer, NRFormer\+, and baselines on two real\-world datasets\. We highlight the best accuracy in bold and underline the second\-best accuracy\.DataModels6th\(24 hours / 6 days\)9th\(36 hours / 9 days\)12th\(48 hours / 12 days\)24th\(96 hours / 24 days\)sudden\-changeMAERMSEMAPEMAERMSEMAPEMAERMSEMAPEMAERMSEMAPEMAERMSEMAPEJapan\-4HHA2\.359\.042\.89%2\.429\.452\.94%2\.489\.832\.98%2\.6611\.103\.12%3\.478\.244\.69%LR2\.086\.742\.77%2\.177\.562\.77%2\.298\.042\.99%2\.439\.893\.00%3\.287\.964\.35%XGBoost2\.178\.492\.63%2\.289\.222\.73%2\.359\.272\.80%2\.5210\.472\.95%3\.317\.994\.39%DCRNN1\.966\.752\.59%2\.117\.462\.68%2\.267\.952\.86%2\.368\.732\.95%3\.227\.814\.38%STID1\.815\.942\.35%1\.896\.382\.44%1\.966\.762\.52%2\.157\.952\.72%3\.087\.804\.06%DLinear1\.966\.362\.61%2\.056\.872\.72%2\.127\.262\.79%2\.298\.492\.97%3\.298\.204\.42%PatchTST1\.856\.442\.42%1\.946\.942\.51%2\.017\.352\.58%2\.188\.532\.76%3\.167\.934\.18%Koopa1\.856\.342\.41%1\.936\.832\.49%1\.987\.232\.55%2\.158\.482\.71%3\.187\.874\.24%iTransformer1\.735\.782\.27%1\.816\.242\.36%1\.886\.642\.44%2\.077\.862\.63%3\.067\.754\.08%TimesNet1\.968\.392\.53%2\.018\.672\.58%2\.058\.932\.63%2\.209\.812\.77%3\.328\.164\.43%StemGNN1\.956\.722\.53%2\.127\.432\.62%2\.197\.912\.75%2\.348\.442\.94%3\.167\.654\.27%GWN1\.936\.652\.49%2\.027\.152\.60%2\.077\.522\.66%2\.248\.602\.84%3\.178\.514\.20%LightCTS1\.826\.032\.33%1\.906\.492\.42%1\.976\.882\.49%2\.158\.082\.69%3\.168\.004\.18%NRFormer1\.745\.812\.28%1\.826\.272\.36%1\.896\.672\.43%2\.077\.922\.63%2\.866\.943\.47%NRFormer\+1\.725\.692\.26%1\.796\.062\.25%1\.836\.482\.33%2\.007\.592\.56%2\.746\.833\.38%Japan\-1DHA2\.3711\.702\.74%2\.4812\.742\.85%2\.5713\.672\.96%2\.9417\.083\.37%4\.1610\.625\.48%LR2\.036\.772\.69%2\.118\.072\.78%2\.188\.912\.78%2\.4111\.483\.17%3\.769\.794\.92%XGBoost2\.2510\.252\.61%2\.3811\.502\.72%2\.4612\.592\.81%2\.6915\.913\.02%4\.079\.834\.97%DCRNN2\.199\.562\.62%2\.3410\.852\.83%2\.429\.122\.89%2\.6112\.763\.29%3\.959\.464\.75%STID2\.026\.932\.63%2\.108\.122\.73%2\.168\.992\.81%2\.4211\.573\.15%3\.9910\.155\.35%DLinear1\.936\.572\.56%2\.048\.042\.70%2\.118\.962\.79%2\.3711\.793\.10%4\.059\.985\.43%PatchTST1\.836\.692\.41%1\.948\.072\.53%2\.028\.972\.63%2\.2911\.802\.97%3\.839\.984\.97%Koopa1\.896\.782\.47%1\.988\.102\.57%2\.078\.992\.67%2\.3211\.692\.98%3\.579\.744\.74%iTransformer1\.836\.422\.43%1\.947\.362\.55%2\.028\.132\.64%2\.2810\.602\.97%3\.429\.914\.51%TimesNet2\.098\.472\.63%2\.159\.162\.70%2\.219\.762\.77%2\.4212\.093\.05%3\.7710\.544\.80%StemGNN2\.199\.512\.61%2\.3410\.792\.88%2\.4610\.452\.81%2\.6112\.653\.22%3\.829\.364\.56%GWN2\.159\.242\.61%2\.2810\.712\.76%2\.3711\.242\.88%2\.5912\.603\.21%4\.0910\.135\.23%LightCTS1\.956\.602\.57%2\.067\.492\.68%2\.138\.182\.78%2\.3711\.503\.09%3\.958\.854\.70%NRFormer1\.846\.582\.39%1\.947\.992\.50%2\.018\.922\.59%2\.2811\.652\.93%2\.918\.463\.62%NRFormer\+1\.776\.192\.34%1\.897\.262\.47%1\.978\.112\.58%2\.2210\.832\.90%2\.778\.233\.21%

## VExperiments

### V\-AExperiments Setup

#### V\-A1Datasets

The experimental datasets for this predictive model include nuclear radiation data and meteorological data collected from the entire country of Japan\. Table[I](https://arxiv.org/html/2607.24774#S3.T1)summarizes the statistics of the radiation and meteorological datasets\.

#### V\-A2Evaluation Metrics

We employed three widely used metrics, including Mean Absolute Error \(MAE\), Root Mean Squared Error \(RMSE\), and Mean Absolute Percentage Error \(MAPE\), for model evaluation\. Lower values in MAE, RMSE, and MAPE indicate higher forecasting accuracy\. Moreover, we follow\[[45](https://arxiv.org/html/2607.24774#bib.bib12),[43](https://arxiv.org/html/2607.24774#bib.bib11),[17](https://arxiv.org/html/2607.24774#bib.bib9)\]to discuss the errors on predicting sudden changes\. We identified the top 10% of samples in the test set with the greatest variation between observed and predicted values as sudden\-change samples for prediction\.

#### V\-A3Baseline

We compare NRFormer\+ against13baselines spanning classical statistics \(HA\[[29](https://arxiv.org/html/2607.24774#bib.bib34)\], LR\), gradient boosting \(XGBoost\[[4](https://arxiv.org/html/2607.24774#bib.bib61)\]\), spatio\-temporal graph neural networks \(DCRNN\[[15](https://arxiv.org/html/2607.24774#bib.bib59)\], GWN\[[42](https://arxiv.org/html/2607.24774#bib.bib30)\], StemGNN\[[3](https://arxiv.org/html/2607.24774#bib.bib15)\], STID\[[28](https://arxiv.org/html/2607.24774#bib.bib58)\], LightCTS\[[13](https://arxiv.org/html/2607.24774#bib.bib20)\]\), and recent time\-series models \(DLinear\[[44](https://arxiv.org/html/2607.24774#bib.bib16)\], PatchTST\[[22](https://arxiv.org/html/2607.24774#bib.bib14)\], Koopa\[[19](https://arxiv.org/html/2607.24774#bib.bib13)\], iTransformer\[[18](https://arxiv.org/html/2607.24774#bib.bib79)\], TimesNet\[[40](https://arxiv.org/html/2607.24774#bib.bib7)\]\)\. Detailed per\-baseline descriptions are provided in Appendix[B\-A](https://arxiv.org/html/2607.24774#A2.SS1)\.

#### V\-A4Implementation Details

The dataset was partitioned into three distinct subsets, 60% for training, 20% for validation, and 20% for testing\. Our experiments are conducted on two different server configurations, Linux Centos with four RTX 3090 GPUs and Linux Ubuntu with two A800 GPUs\. In terms of implementation, our model is implemented using PyTorch, with the Adam optimizer chosen for optimization\. We set the learning rate and the weight decay to 0\.001 and 0\.0001, respectively\. The batch size is fixed to 32\. The numbers of temporal and spatial attention layersLtL\_\{t\}andLsL\_\{s\}are selected via grid search per dataset \(see Appendix[B\-D](https://arxiv.org/html/2607.24774#A2.SS4)\); the spatial depthLsL\_\{s\}is fixed to 2 on both datasets, whileLtL\_\{t\}is 4 on Japan\-1D and 3 on Japan\-4H\. The head of the multi\-head attention module is set to 4\. Following standard practice for heavy\-tailed targets, we model radiation in log space\. Each dose rateCCis transformed bylog⁡\(1\+C\)\\log\(1\+C\)before standardisation, and predictions are mapped back withexp⁡\(⋅\)−1\\exp\(\\cdot\)\-1and clipped to the non\-negative physical range at inference\. This compresses the roughly two\-order\-of\-magnitude dynamic range of radiation \(about 40 to 7170 nSv/h\) so that the loss is not dominated by a few large values\.

### V\-BOverall Performance

Table[II](https://arxiv.org/html/2607.24774#S4.T2)presents the overall performance of NRFormer, NRFormer\+, and all baseline models on two real\-world datasets with respect to MAE, RMSE, and MAPE\. NRFormer\+ outperforms all baselines on MAE and MAPE across both datasets and all forecasting horizons, while also winning RMSE on the vast majority of \(model, horizon\) cells; the only exception is the longest 24\-step horizon on Japan\-1D, where iTransformer attains a marginally lower RMSE \(10\.6010\.60vs10\.8310\.83\) – though NRFormer\+ still wins MAE \(2\.222\.22vs2\.282\.28\) and MAPE \(2\.90%2\.90\\%vs2\.97%2\.97\\%\) at that cell\. Specifically, NRFormer\+ achieves notable improvements over the strongest Transformer baseline iTransformer in terms of the MAE metric across all forecasting horizons on both datasets\. Moreover, we can make the following observations, \(1\) All traditional methods \(*i\.e\.,*HA, LR, and XGBoost\) perform worse than deep learning models, as they fail to capture spatio\-temporal patterns\. \(2\) STGNN\-based approaches \(*i\.e\.,*DCRNN, GWN\) outperform HA, LR, and XGBoost by a large margin, indicating the superiority of modeling intricate spatio\-temporal dependencies\. \(3\) STID, despite relying only on lightweight spatial and temporal identity embeddings, outperforms the graph\-based DCRNN and GWN on Japan\-1D across all horizons in MAE, suggesting that explicit graph convolution is not essential on this benchmark and that identity embeddings already capture the dominant spatio\-temporal regularities\. \(4\) Among the deep\-learning baselines, iTransformer achieves the strongest average performance by reinterpreting the Transformer for multivariate forecasting \(variate\-as\-token attention\), surpassing both classical and graph\-based methods, though on Japan\-1D it is matched almost exactly by PatchTST at the shorter horizons\. A cluster of lightweight baselines \(PatchTST, STID, Koopa, LightCTS, and DLinear\) follows within a narrow band, the closest runner\-up being PatchTST on Japan\-1D and STID on Japan\-4H, indicating that lightweight architectures are competitive on the radiation forecasting benchmark\.

Moreover, the sudden\-change experiments reveal that NRFormer\+ shows substantial improvements across all datasets\. By addressing spatio\-temporal distribution imbalances, incorporating heterogeneous contextual factors, and leveraging physics\-guided atmospheric modeling, NRFormer\+ adapts more quickly to the non\-stationary nature of radiation, providing more accurate predictions for sudden\-change samples\.

![Refer to caption](https://arxiv.org/html/2607.24774v1/x8.png)Figure 8:Sudden\-change MAE leaderboard across all 15 evaluated models on Japan\-1D \(left\) and Japan\-4H \(right\), in ascending order\.
### V\-CSudden\-Change Robustness

To further isolate sudden\-change behavior from the aggregate table, Figure[8](https://arxiv.org/html/2607.24774#S5.F8)ranks all 15 evaluated models by their sudden\-change MAE on the held\-out test split\. NRFormer\+ ranks first on both datasets, and its closest competitor is the conference NRFormer\. NRFormer\+ improves upon the conference NRFormer by4\.87%4\.87\\%on Japan\-1D \(2\.76832\.7683versus2\.912\.91\) and4\.09%4\.09\\%on Japan\-4H \(2\.74292\.7429versus2\.862\.86\), confirming that the physics\-guided atmospheric diffusion module and the enhanced encoders sharpen sudden\-event tracking beyond the conference model\. Relative to the strongest external \(non\-NRFormer\) baseline iTransformer, the margin widens to19\.09%19\.09\\%on Japan\-1D \(3\.42123\.4212\) and10\.37%10\.37\\%on Japan\-4H \(3\.06023\.0602\), and the gap to all remaining baselines is uniformly larger\. The systematic sudden\-change advantage is consistent with the discussion in Appendix[A\-A](https://arxiv.org/html/2607.24774#A1.SS1)that the physics module concentrates its contribution on the heavy\-tailed sudden\-change subset rather than on slowly varying periods\.

TABLE III:Ablation study of NRFormer\+ components on two datasets\. Results are reported as single\-step metrics at the 12th forecasting horizon\.DatasetVariantMAERMSEMAPEJapan\-4Hw/o Physics Module2\.214711\.75372\.77%w/o Enhanced Meteo Encoder2\.09657\.88862\.62%w/o Temporal Encoder2\.10987\.93942\.66%w/o RevIN2\.22278\.01722\.84%w/o Log\-space Target2\.10338\.07182\.65%NRFormer\+ \(Full\)2\.00217\.81302\.53%Japan\-1Dw/o Physics Module2\.416711\.55413\.18%w/o Enhanced Meteo Encoder2\.296810\.74092\.97%w/o Temporal Encoder2\.339810\.43383\.07%w/o RevIN2\.419810\.29173\.09%w/o Log\-space Target2\.287810\.57552\.94%NRFormer\+ \(Full\)2\.230510\.25912\.83%For space, two supplementary analyses are deferred to the appendix\. An ablation of the individual meteorological features \(Appendix[B\-B](https://arxiv.org/html/2607.24774#A2.SS2)\) shows that every covariate contributes, most strongly on the sudden\-change subset, where removing all meteorology raises sudden\-change MAE by up to8\.6%8\.6\\%\. A sensitivity analysis of the temporal and spatial attention depthsLtL\_\{t\}andLsL\_\{s\}\(Appendix[B\-D](https://arxiv.org/html/2607.24774#A2.SS4)\) shows that all sweep points lie within0\.050\.05MAE of the per\-dataset optimum, confirming robustness to these hyperparameters\.

### V\-DAblation Study of NRFormer\+ Components

To comprehensively evaluate the contribution of each component in NRFormer\+, we conduct an extensive ablation study\. The first three variants each remove one newly introduced architectural component\. \(1\)*w/o Physics Module*removes the Physics\-Guided Atmospheric Diffusion Module, which estimates diffusion coefficients and computes spatial Laplacian features\. \(2\)*w/o Enhanced Meteo Encoder*replaces the dedicated wind and temperature pathway encoder with a simple concatenation followed by a single MLP\. \(3\)*w/o Temporal Encoder*removes the day\-of\-year seasonal embedding\. Two further variants probe key training choices\. \(4\)*w/o RevIN*disables the instance\-level reversible normalization in the temporal attention\. \(5\)*w/o Log\-space Target*trains on linear\-space radiation targets instead of the log\-space target\.

The results are reported in Table[III](https://arxiv.org/html/2607.24774#S5.T3)\. Several important observations can be made\. First, among the newly introduced architectural components,*w/o Physics Module*exhibits the largest degradation on both datasets, and the physics prior is especially important on the heavy\-tailed sudden\-change subset, confirming the centrality of the atmospheric diffusion module; the reversible instance normalization \(*w/o RevIN*\) and the log\-space target \(*w/o Log\-space Target*\) are comparably impactful\. Second, removing the enhanced meteorological encoder leads to notable accuracy drops, indicating that the dedicated wind and temperature pathways are more effective than naive feature concatenation\. Third, the temporal encoder contributes consistent improvements, reflecting the strong seasonal periodicity in radiation levels\. A finer\-grained per\-component visualization, together with a comparison of the diffusion module against a region\-clustering alternative, is provided in Appendix[B\-C](https://arxiv.org/html/2607.24774#A2.SS3)\.

### V\-EEfficiency Analysis

Beyond forecasting accuracy, computational efficiency is a critical consideration for practical deployment, especially given the large\-scale nature of our nationwide monitoring network \(3,627 stations\)\. We conduct a systematic efficiency analysis comparing NRFormer\+ against all baselines and NRFormer in terms of model size \(\#Params\), training time per epoch, and inference time per epoch over the full validation set\. All timing measurements are wall\-clock seconds collected on a single NVIDIA A800\-SXM4\-80GB GPU on the Japan\-1D dataset\. The results are summarized in Table[IV](https://arxiv.org/html/2607.24774#S5.T4)\.

NRFormer\+ has 343 K parameters, only22%22\\%more than the conference NRFormer \(282 K\) and within the same order of magnitude as the strongest Transformer baselines \(iTransformer, 271 K; TimesNet, 1\.41 M\)\. Training takes 64\.6 s/epoch,1\.55×1\.55\\timesthe cost of the conference NRFormer \(41\.8 s/epoch\) and broadly comparable to iTransformer \(42\.8 s/epoch\) and TimesNet \(41\.4 s/epoch\), but28×28\\timesfaster than the heavyweight graph\-convolutional baseline GWN \(1849\.5 s/epoch\)\. Inference takes 13\.0 s/epoch on the full validation set, on par with iTransformer \(11\.5 s\) and TimesNet \(12\.7 s\) and well within the latency budget of an operational system producing forecasts at 4\-hour and daily intervals\. Crucially, NRFormer\+ also achieves the lowest 24\-step MAE \(2\.22\) across all methods, outperforming the strongest baselines NRFormer and iTransformer \(both 2\.28\) by2\.6%2\.6\\%and the heavyweight GWN \(2\.59\) by14\.3%14\.3\\%\. The favorable accuracy\-efficiency trade\-off demonstrates that the Physics\-Guided Atmospheric Diffusion Module, the enhanced meteorological encoder, and the additional embedding branches in NRFormer\+ deliver substantial accuracy gains at only a marginal computational premium\.

TABLE IV:Efficiency comparison of all deep\-learning methods on the Japan\-1D dataset\.Model\#Params \(M\)Train\(s/epoch\)Inference\(s/epoch\)MAEDCRNN0\.376255\.636\.12\.61STID0\.14322\.26\.62\.42DLinear0\.00122\.26\.72\.37PatchTST0\.02523\.56\.72\.29Koopa5\.70022\.76\.72\.32StemGNN4\.878304\.1843\.392\.61GWN0\.4861849\.5192\.32\.59LightCTS0\.161113\.220\.02\.37iTransformer0\.27142\.811\.52\.28TimesNet1\.40941\.412\.72\.42NRFormer0\.28241\.85\.62\.28NRFormer\+0\.34364\.613\.02\.22
### V\-FCase Study

To complement the aggregate metrics with a qualitative view of multi\-step forecasting behavior, we conduct a case study at three named monitoring sites selected from the pool of 2,859 candidate stations falling within the three bounding boxes defined for Fukushima Prefecture, Tokyo Metropolis, and Shimane Prefecture \(out of 3,627 total monitoring stations\), Namiki Kindergarten in Fukushima Prefecture \(37\.41∘​N37\.41^\{\\circ\}\\text\{N\},140\.36∘​E140\.36^\{\\circ\}\\text\{E\}\), the Ukishima Monitoring Post in Tokyo \(35\.53∘​N35\.53^\{\\circ\}\\text\{N\},139\.77∘​E139\.77^\{\\circ\}\\text\{E\}\), and Ōda High School in Shimane Prefecture \(35\.19∘​N35\.19^\{\\circ\}\\text\{N\},132\.51∘​E132\.51^\{\\circ\}\\text\{E\}\)\. These three sites span a high\-radiation legacy regime \(Fukushima\), a dense urban background regime \(Tokyo\), and a sparse coastal regime \(Shimane\), which together exercise the model across markedly different spatial and meteorological contexts\. For each station we visualize the twelve\-step\-ahead forecasts of NRFormer\+ alongside those of the strongest Transformer\-based baseline, iTransformer, against the ground\-truth measurements over a 33\-day evaluation window from 2025\-04\-26 to 2025\-05\-29\.

Figure[9](https://arxiv.org/html/2607.24774#S5.F9)reports the resulting trajectories\. At Namiki Kindergarten, where two pronounced meteorology\-driven excursions occur within the plotted window, NRFormer\+ reduces MAE from15\.3315\.33to13\.6813\.68nSv/h, a10\.76%10\.76\\%relative improvement, and visibly tracks both spikes with markedly smaller phase error than iTransformer\. The Ukishima and Ōda sites, which exhibit five and four smaller sudden\-change events respectively, see more modest but still consistent advantages of4\.27%4\.27\\%\(1\.102→1\.0551\.102\\to 1\.055nSv/h\) and2\.41%2\.41\\%\(2\.741→2\.6752\.741\\to 2\.675nSv/h\) in MAE\. Across all three panels NRFormer\+ never trails iTransformer, indicating that the physics\-guided atmospheric module and density\-adaptive spatial attention introduced in NRFormer\+ improve sudden\-event tracking in the high\-amplitude Fukushima regime without sacrificing accuracy in the quieter urban and sparse regimes\. The mean MAE reduction across the three stations is5\.81%5\.81\\%, consistent in sign with the aggregate sudden\-change robustness gains reported in Section[V\-C](https://arxiv.org/html/2607.24774#S5.SS3)\.

![Refer to caption](https://arxiv.org/html/2607.24774v1/x9.png)Figure 9:Twelve\-step\-ahead forecasts of NRFormer\+ and iTransformer against ground truth over a 33\-day window \(2025\-04\-26 to 2025\-05\-29\) at three named stations spanning the high\-amplitude Fukushima, urban Tokyo, and sparse Shimane regimes\. Light vertical bands mark sudden\-change events\.

## VIRelated Work

Spatio\-temporal forecasting has been studied extensively in urban computing\[[10](https://arxiv.org/html/2607.24774#bib.bib29)\], especially for traffic, where spatio\-temporal graph networks jointly model spatial and temporal dependencies\. Diffusion\-convolutional and adaptive\-graph models learn the sensor graph and propagate information over it\[[15](https://arxiv.org/html/2607.24774#bib.bib59),[42](https://arxiv.org/html/2607.24774#bib.bib30),[41](https://arxiv.org/html/2607.24774#bib.bib81),[5](https://arxiv.org/html/2607.24774#bib.bib82),[29](https://arxiv.org/html/2607.24774#bib.bib34)\], while lighter identity\-embedding and efficient designs reach comparable accuracy at lower cost\[[28](https://arxiv.org/html/2607.24774#bib.bib58),[13](https://arxiv.org/html/2607.24774#bib.bib20)\]\. In parallel, Transformers\[[37](https://arxiv.org/html/2607.24774#bib.bib21)\]have been adapted to time\-series forecasting through sparse or decomposition\-based attention\[[46](https://arxiv.org/html/2607.24774#bib.bib40)\], patching and channel independence\[[22](https://arxiv.org/html/2607.24774#bib.bib14)\], variate\-as\-token attention\[[18](https://arxiv.org/html/2607.24774#bib.bib79)\], and multi\-periodicity modeling\[[40](https://arxiv.org/html/2607.24774#bib.bib7)\], with a parallel line targeting non\-stationarity via stationarization and reversible instance normalization\[[20](https://arxiv.org/html/2607.24774#bib.bib23),[11](https://arxiv.org/html/2607.24774#bib.bib26)\]\. These methods assume roughly uniform sensor coverage and do not model the physics\-governed diffusion and meteorological context specific to radiation forecasting\.

A complementary direction injects physical knowledge into neural models\. Physics\-informed neural networks add PDE\-residual losses that constrain solutions to known laws\[[26](https://arxiv.org/html/2607.24774#bib.bib68)\], with success in fluid dynamics and weather science\[[27](https://arxiv.org/html/2607.24774#bib.bib69),[2](https://arxiv.org/html/2607.24774#bib.bib70)\], and neural operators learn PDE solution maps directly\[[16](https://arxiv.org/html/2607.24774#bib.bib76)\]\. Closest to us, AirPhyNet\[[9](https://arxiv.org/html/2607.24774#bib.bib10)\]encodes advection and diffusion as differential\-equation operators on a sensor graph for air quality\. Deep models have likewise advanced environmental forecasting more broadly\[[38](https://arxiv.org/html/2607.24774#bib.bib83)\], from latent\-variable Transformers for PM2\.5\[[17](https://arxiv.org/html/2607.24774#bib.bib9)\]to Fourier\-operator and multi\-mesh\-graph weather models\[[23](https://arxiv.org/html/2607.24774#bib.bib77),[14](https://arxiv.org/html/2607.24774#bib.bib78)\]\. These approaches target dense, roughly uniform networks and a single pollutant, whereas radiation forecasting must additionally contend with extreme station\-density imbalance, decay\-driven multi\-year non\-stationarity, and sparse co\-located meteorology\. We therefore encode a discrete diffusion approximation as a soft architectural prior rather than a hard residual loss\.

For nuclear radiation specifically, operational practice relies on Lagrangian dispersion solvers such as FLEXPART and HYSPLIT\[[34](https://arxiv.org/html/2607.24774#bib.bib71),[31](https://arxiv.org/html/2607.24774#bib.bib73)\]and on decision\-support systems that couple dispersion with dose assessment\[[7](https://arxiv.org/html/2607.24774#bib.bib74)\], while statistical work uses geostatistical interpolation and Bayesian source inversion\[[39](https://arxiv.org/html/2607.24774#bib.bib75)\]\. Recent learning\-based efforts remain local, for example combining random forests, Kalman filtering, and kriging near the Fukushima Daiichi site\[[36](https://arxiv.org/html/2607.24774#bib.bib54)\], and carry no explicit transport prior\. To our knowledge, ours is the first to cast nationwide, multi\-station radiation forecasting as a graph\-structured prediction problem with a learned transport prior\.

## VIIConclusion

Nationwide nuclear radiation forecasting is safety\-critical, yet its monitoring data are non\-stationary in time, cluster into a small fraction of the territory in space, and co\-evolve with meteorology through atmospheric transport that purely observational models overlook\. Our central idea is to learn this physics rather than ignore it\. We curate and release*Japan\-4H*and*Japan\-1D*, nationwide benchmarks drawn from over four years of recordings across 3,627 radiation stations paired with 228 meteorological stations, together with the full preprocessing pipeline\. On top of these, we buildNRFormer\+, a spatio\-temporal Transformer centered on a Physics\-Guided Atmospheric Diffusion Module that reads local meteorology to estimate how readily radiation spreads, gauges how each station departs from its neighborhood, and reinjects this signal as an architectural prior alongside non\-stationary temporal attention and density\-adaptive spatial attention\. Across thirteen baselines on both datasets and all horizons, NRFormer\+ consistently improves accuracy, with its largest gains on sudden\-change events, where it cuts error by up to 19\.1%\. Future work will add calibrated probabilistic forecasts and anisotropic, wind\-aware transport\. By turning a sparse, uneven sensor network into reliable forecasts, NRFormer\+ offers a practical instrument for emergency response and public health\.

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![[Uncaptioned image]](https://arxiv.org/html/2607.24774v1/photos/lyu.jpg)Tengfei Lyuis a PhD student at the Thrust of Artificial Intelligence, The Hong Kong University of Science and Technology \(Guangzhou\)\. He received the M\.S\. degree from the School of Computer Science and Engineering, Central South University, Changsha, China\. His research interests are AI4Science, Spatiotemporal Data Mining, and Large Language Models including their theoretical foundations and applications\. He has published several research papers in prestigious conferences and journals, such as TCBB, SIGIR, IJCAI, and KDD\.![[Uncaptioned image]](https://arxiv.org/html/2607.24774v1/photos/han.jpg)Jindong Hanis currently a Professor at the School of Artificial Intelligence, Shandong University\. He received the Ph\.D\. degree from the the Hong Kong University of Science and Technology \(HKUST\) in 2025\. His research focuses on data mining, urban intelligence, and AI for science\. He have published over 30 papers in prestigious journals and conferences, such as IEEE TKDE, KDD, WWW, VLDB, NeurIPS, ICLR, and AAAI\.![[Uncaptioned image]](https://arxiv.org/html/2607.24774v1/photos/liu.jpeg)Hao Liureceived the BE degree from the South China University of Technology \(SCUT\), in 2012 and the PhD degree from the Hong Kong University of Science and Technology, in 2017\. He is currently an Assistant Professor with the Thrust of Artificial Intelligence, The Hong Kong University of Science and Technology \(Guangzhou\)\. Prior to that, he was a senior research scientist with Baidu Research and a postdoctoral fellow at HKUST\. His general research interests are in data mining, machine learning, and Big Data management, with a special focus on mobile analytics and urban computing\. He has published prolifically in refereed journals and conference proceedings, such as TKDE, KDD, ICML, NeurIPS, SIGIR, WWW, AAAI, and IJCAI\.## Appendix ADiscussion

### A\-AWhy Physics\-Guided Modeling Helps

The ablation in Section[V\-D](https://arxiv.org/html/2607.24774#S5.SS4)establishes that removing the Physics\-Guided Atmospheric Diffusion Module produces one of the two largest accuracy drops among NRFormer\+ components, alongside reversible instance normalization \(a relative MAE increase ofΔ​MAE=\+10\.62%\\Delta\\text\{MAE\}\{=\}\{\+\}10\.62\\%on Japan\-4H and\+8\.35%\{\+\}8\.35\\%on Japan\-1D at the 12\-step horizon\), and that the gain is concentrated in regimes where physics is most informative\. Stratifying the test set by event type, the diffusion module contributes the bulk of its improvement on the heavy\-tailed sudden\-change subset identified in Finding T3, where deviations from the spatial mean are large and the Laplacian termCi\(P\)−C¯\(P\)C\_\{i\}^\{\(P\)\}\-\\bar\{C\}^\{\(P\)\}carries strong signal\. Geographically, we expect the gains to be largest at sites that are spatially isolated \(the sparse\-neighbor regions identified in Finding S1\), where data\-driven smoothing has little to anchor on but the wind\-conditioned diffusion coefficient can provide an external prior\. Inspecting the learnedDiD\_\{i\}values \(Figure[10](https://arxiv.org/html/2607.24774#A1.F10)\) yields a physically coherent picture\. On Japan\-4H,DiD\_\{i\}increases with mean wind speed \(Spearmanρs=\+0\.349\\rho\_\{s\}\{=\}\{\+\}0\.349,p<10−100p\{<\}10^\{\-100\}\), in line with the advection\-diffusion intuition that stronger winds enhance horizontal mixing, and the model recovers this trend without direct supervision\. On Japan\-4H this learned coefficient runs opposite to the empirical isotropic\-fit slopeD^\\hat\{D\}in Finding M2, which instead declines with wind because a wind\-blind operator captures less of the transport as the flow turns anisotropic\. A learned wind\-modulated operator is designed precisely to absorb the anisotropic transport that the isotropic fit loses\. On Japan\-1D the learnedDiD\_\{i\}instead shares the sign ofD^\\hat\{D\}, both declining with wind, consistent with the directional\-advection reading discussed below\. The case study further illustrates this mechanism\. At the high\-amplitude Namiki site near Fukushima, NRFormer\+ tracks two pronounced meteorology\-driven excursions that iTransformer follows with markedly larger phase error, while even at the sparse Shimane site, where the gain is smallest, it never trails iTransformer\. Because the case study compares the full model against iTransformer rather than against the w/o\-Physics variant, it illustrates rather than isolates the contribution of the diffusion module\.

![Refer to caption](https://arxiv.org/html/2607.24774v1/x10.png)Figure 10:Physical interpretability of the learned diffusion coefficientDiD\_\{i\}\. \(a, b\) Hexbin scatter ofDiD\_\{i\}versus standardised wind speed on Japan\-4H and Japan\-1D, overlaid with running\-median curves; the Spearman correlations areρs=\+0\.349\\rho\_\{s\}\{=\}\{\+\}0\.349\(p=2\.6×10−104p\{=\}2\.6\{\\times\}10^\{\-104\}\) on Japan\-4H andρs=−0\.161\\rho\_\{s\}\{=\}\{\-\}0\.161\(p=1\.7×10−22p\{=\}1\.7\{\\times\}10^\{\-22\}\) on Japan\-1D\. \(c\) Overlaid histograms ofDiD\_\{i\}on the two datasets\. \(d, e\) Geographic distribution ofDiD\_\{i\}on Japan\-4H and Japan\-1D\. \(f\) Zoom\-in around the Fukushima Daiichi site \(red star\) on Japan\-4H, illustrating the local diffusion structure inferred by the model\.Figure[10](https://arxiv.org/html/2607.24774#A1.F10)provides direct visual evidence for the physical interpretability of the diffusion module\. On Japan\-4H \(panel a\),DiD\_\{i\}ranges between0\.420\.42and0\.650\.65and increases with wind speed, in line with the advection–diffusion intuition that stronger winds enhance horizontal mixing; on Japan\-1D \(panel b\), the model learns a broader, partially inverted distribution \(range0\.000\.00–1\.221\.22,ρs=−0\.161\\rho\_\{s\}\{=\}\{\-\}0\.161\), and this negative correlation at the coarser daily scale is not fully explained by the isotropic diffusion picture, possibly reflecting wind acting more as a directional advection driver, a point we do not resolve here\. The geographic maps \(panels d–e\) and the Fukushima zoom\-in \(panel f\) suggest that elevatedDiD\_\{i\}values concentrate in coastal and topographically open regions, while inland mountainous corridors exhibit lower diffusivity, a spatial pattern qualitatively consistent with terrain\-modulated mixing that emerges without direct supervision ofDiD\_\{i\}\.

### A\-BFrom Advection–Diffusion to Architecture: Design Choices and Trade\-offs

A natural question is whether NRFormer\+ would benefit from a stricter physics\-informed formulation\. We deliberately chose three soft\-design points and discuss each in turn\. First, we encode the physics triplet\[C,D,∇2C\]\[C,D,\\nabla^\{2\}C\]as an*auxiliary feature pathway*rather than as a PDE\-residual loss in the style of canonical PINNs\[[26](https://arxiv.org/html/2607.24774#bib.bib68)\]\. PINN\-style residual losses are most effective when the governing equation is locally exact and observations are dense in space\-time; nationwide radiation transport, in contrast, is contaminated by unobserved source terms, deposition, and irregular sampling, so an exact residual loss would propagate model\-misspecification error directly into the gradient\. Treating physics as a representation prior instead allows the model to defer to data wherever the diffusion equation is locally inadequate, which we argue is the dominant regime in this application\. Second, the Laplacian is approximated as deviation from the spatial mean,∇2Ci≈Ci\(P\)−C¯\(P\)\\nabla^\{2\}C\_\{i\}\\approx C\_\{i\}^\{\(P\)\}\-\\bar\{C\}^\{\(P\)\}, rather than as a graph Laplacian∑jWi​j​\(Ci−Cj\)\\sum\_\{j\}W\_\{ij\}\(C\_\{i\}\-C\_\{j\}\)\. The mean\-deviation form is parameter\-free, robust to the extreme density imbalance of the monitoring network \(Finding S1\), and avoids re\-introducing the over\-smoothing/under\-smoothing pathology that motivated density\-adaptive attention in the first place; akk\-NN graph Laplacian, by contrast, would inherit the same density bias\. The cost is anisotropy, since a single scalar deviation cannot distinguish upwind from downwind neighbors, which we view as the leading direction for future tightening \(Appendix[A\-E](https://arxiv.org/html/2607.24774#A1.SS5)\)\. Third, we do not introduce an explicit advection operator\. Advection is instead absorbed by the density\-adaptive spatial attention, whose physics\-enriched query/key formulation \(Section[IV\-C](https://arxiv.org/html/2607.24774#S4.SS3)\) routes information by learned radiation similarity rather than by a hand\-coded wind kernel\. This is consistent with the multi\-hour lagged\-correlation structure documented in Finding M1, where an instantaneous𝐮⋅∇C\\mathbf\{u\}\\cdot\\nabla Cterm would mis\-specify the time scale, whereas attention can flexibly aggregate over the empirical∼\\sim16 h lag\. Together these choices articulate a deliberate hybrid, applying hard physics where the equation is locally faithful, such as the positivity ofDD, and soft physics elsewhere\.

### A\-CLimitations

We highlight six limitations together with their practical implications\.*\(i\) Single\-country evaluation\.*All results are obtained on Japan, whose monitoring network has both an exceptionally high baseline density and a strong post\-Fukushima skew \(Finding S1\); transferring NRFormer\+ to networks that are sparser overall \(e\.g\., national\-grid scale in continental Europe or North America\) would stress both the imbalance\-aware attention and the Laplacian approximation, and is likely to require re\-calibration of the proximity threshold\.*\(ii\) Isotropic Laplacian\.*The mean\-deviation operator is rotation\-invariant; under strongly directional transport \(*e\.g\.,*jet\-stream advection of Fukushima discharge plumes\), it under\-represents anisotropy, which Finding M2 indicates is the regime in which residual transport variance is largest\.*\(iii\) Static spatial graph\.*Geographic proximity is fixed at training time, so the model cannot dynamically rewire connections when synoptic\-scale circulation patterns redirect transport pathways across days\.*\(iv\) Meteorological interpolation\.*The mapping from 228 NOAA\-ISD stations to 3,627 radiation sites introduces an irreducible interpolation error whose magnitude scales with local meteorological gradients and station\-to\-station distance \(ECDF reported in Figure[5](https://arxiv.org/html/2607.24774#S3.F5)\(a\)\); in countries with sparser weather networks this error would compound and likely dominate physics\-module gains\.*\(v\) Deterministic point forecasts\.*NRFormer\+ outputs a single trajectory and provides no calibrated uncertainty, which is a non\-trivial limitation for safety\-critical decisions such as evacuation timing or agricultural advisories where the cost of false negatives is asymmetric\.*\(vi\) Quadratic spatial attention\.*Macro\-scale attention is𝒪​\(N2\)\\mathcal\{O\}\(N^\{2\}\)in stations; atN=3,627N\{=\}3\{,\}627this remains tractable on a single A800 GPU \(Section[V\-E](https://arxiv.org/html/2607.24774#S5.SS5)\), but scaling to continental networks \(N≳104N\{\\gtrsim\}10^\{4\}\) would require sparse or linearised attention, the integration of which with the radiation\-guided query/key routing of Section[IV\-C](https://arxiv.org/html/2607.24774#S4.SS3)is non\-trivial\.

### A\-DPractical Implications and Broader Applicability

NRFormer\+ is best understood as complementary to, rather than a replacement for, established Lagrangian dispersion models such as FLEXPART\[[34](https://arxiv.org/html/2607.24774#bib.bib71)\]and HYSPLIT\[[31](https://arxiv.org/html/2607.24774#bib.bib73)\]\. Lagrangian models excel at*source\-attributed scenario simulation*\. Given a hypothesised release, they trace particle trajectories through high\-resolution numerical weather fields and produce physically auditable forecasts, at the cost of requiring an accurate source term and tens of minutes to hours of compute\. NRFormer\+ occupies the orthogonal niche of*source\-agnostic nationwide nowcasting*\. It consumes the live monitoring stream and produces forecasts for all 3,627 stations in seconds per forward pass \(Section[V\-E](https://arxiv.org/html/2607.24774#S5.SS5)\), with no need for source\-term hypotheses\. In an operational nuclear\-safety stack the two approaches could be chained, with NRFormer\+ continuously screening the network for anomalies and identifying stations likely to exceed regulatory thresholds, after which a Lagrangian model can be invoked for forensic source attribution and dose assessment\. Beyond radiation, the architectural recipe of density\-adaptive attention coupled with a soft physics\-triplet pathway is in principle applicable to other environmental scalars governed by advection–diffusion dynamics on irregular sensor networks, such as ground\-levelPM2\.5\\text\{PM\}\_\{2\.5\},NO2\\text\{NO\}\_\{2\}, and ground\-water tracer transport, though transfer to a new network would require re\-calibration and is left to future work\.

### A\-EFuture Directions

Five directions, ordered by methodological reach, follow naturally from the limitations above\.*Probabilistic forecasting\.*Replacing the deterministic output head with quantile regression or a deep ensemble would provide calibrated predictive intervals essential for risk\-informed decision making, and would couple naturally with the existing three\-way fusion\.*Anisotropic, dynamic physics\.*The mean\-deviation Laplacian can be upgraded to a wind\-modulated graph Laplacian whose edge weights depend on the projection of the wind vector onto the inter\-station displacement, and the full advection term−𝐮⋅∇C\-\\mathbf\{u\}\\cdot\\nabla Ccan be parameterized as a learnable directional operator, jointly relaxing limitations*\(ii\)*–*\(iii\)*\.*Dynamic graph learning\.*Replacing the statickk\-NN graph with a graph\-structure\-learning module conditioned on synoptic wind fields would let the receptive field follow transport pathways rather than geography\.*Cross\-country generalization\.*Pre\-training on Japan\-4H/Japan\-1D and fine\-tuning on European or North American monitoring networks would test the robustness of the design choices in Appendix[A\-B](https://arxiv.org/html/2607.24774#A1.SS2)under different density regimes, and would clarify which components are dataset\-specific\.*Environmental foundation models\.*The convergence of large\-scale weather foundation models \(FourCastNet\[[23](https://arxiv.org/html/2607.24774#bib.bib77)\], GraphCast\[[14](https://arxiv.org/html/2607.24774#bib.bib78)\]\) suggests a similar trajectory for environmental scalar transport, pre\-training on a heterogeneous corpus of radiation, air\-quality, and tracer\-dispersion time series, with NRFormer\+’s physics\-triplet pathway serving as the structural prior that disciplines transfer across domains\.

### A\-FRelationship to the Conference Version

This paper substantially extends our KDD 2025 conference paper, NRFormer\[[21](https://arxiv.org/html/2607.24774#bib.bib1)\]\. The differences are concentrated along five complementary axes that together substantially exceed the novelty expected of a journal extension\.

*\(1\) Reframed challenges and physics\-grounded methodology\.*The conference version organised the problem around three data\-driven challenges \(non\-stationary temporal patterns, imbalanced spatial distribution, and heterogeneous contextual factors\) and addressed them via non\-stationary temporal attention, imbalance\-aware spatial attention, and a radiation\-propagation prompting strategy\. The journal version retains the temporal and spatial challenges but*merges*the third with the previously implicit gap of missing physics priors into a single, unified challenge of*physics\-grounded modeling of heterogeneous context*\. This reframing motivates a new, central architectural component, the Physics\-Guided Atmospheric Diffusion Module, which estimates a meteorology\-conditioned diffusion coefficientDD, approximates the spatial Laplacian∇2C\\nabla^\{2\}Cthrough a parameter\-free mean\-field surrogate based on each station’s deviation from the network mean, and injects the physics triplet\[C,D,∇2C\]\[C,D,\\nabla^\{2\}C\]into the Transformer as an inductive bias derived from Eq\. \([4](https://arxiv.org/html/2607.24774#S4.E4)\)\. This module is absent from the conference version\.

*\(2\) Architectural enhancements\.*Beyond the diffusion module, NRFormer\+ introduces dedicated wind and thermal pathways in the meteorological encoder \(Section[IV\-B2](https://arxiv.org/html/2607.24774#S4.SS2.SSS2)\), a day\-of\-year seasonal embedding that supersedes the three independent time\-of\-day / day\-of\-week / month embeddings of NRFormer \(Section[IV\-B3](https://arxiv.org/html/2607.24774#S4.SS2.SSS3.Px2)\), a deeper location encoder and a radiation–location cross\-feature encoder \(Sections[IV\-B3](https://arxiv.org/html/2607.24774#S4.SS2.SSS3.Px1)–[IV\-B3](https://arxiv.org/html/2607.24774#S4.SS2.SSS3.Px3)\), physics\-enriched query/key routing in the density\-adaptive spatial attention \(Section[IV\-C](https://arxiv.org/html/2607.24774#S4.SS3)\), and a three\-way output fusion \(radiation \+ temporal \+ spatial; Section[IV\-D](https://arxiv.org/html/2607.24774#S4.SS4)\) that replaces the two\-way fusion in NRFormer\.

*\(3\) Dataset extension\.*We extend the temporal coverage from three to over four years \(March 2021–May 2025\), refine the station set from3,8413\{,\}841to3,6273\{,\}627via a stricter, four\-stage quality\-control pipeline, expand the meteorological side to228228NOAA\-ISD stations, and release the resulting Japan\-4H \(9,2229\{,\}222timesteps\) and Japan\-1D \(1,5371\{,\}537timesteps\) benchmarks together with all preprocessing code\.

*\(4\) Data engineering and analysis\.*A new Section[III](https://arxiv.org/html/2607.24774#S3)documents the construction pipeline and reports a battery of quantitative analyses, including non\-stationarity \(ADF/KPSS, Findings T1–T3\), spatial heterogeneity \(Moran’sII, semivariogram; Findings S1–S2\), and meteorology–radiation coupling \(lagged cross\-correlation, wind\-stratified diffusion regression; Findings M1–M2\)\. Each finding is explicitly mapped to a corresponding architectural component of NRFormer\+, so that the model is traceable to verified properties of the data rather than to design analogy\.

*\(5\) Experiments and ablations\.*We re\-run all baselines on the extended benchmark, add ablations that isolate every new component of NRFormer\+ \(Section[V\-D](https://arxiv.org/html/2607.24774#S5.SS4)\), report an efficiency comparison \(Section[V\-E](https://arxiv.org/html/2607.24774#S5.SS5)\), and present a case study across high\-, low\-, and sparse\-radiation regimes \(Section[V\-F](https://arxiv.org/html/2607.24774#S5.SS6)\)\. The discussion further provides a methodological reflection on the soft\-physics design choices that was absent from the conference paper\.

## Appendix BAdditional Experiments

### B\-ABaseline Details

We compare the proposed NRFormer\+ with 13 baseline methods\.

- •HA\[[29](https://arxiv.org/html/2607.24774#bib.bib34)\]: predicts radiation levels by averaging historical readings for the corresponding periods\.
- •XGBoost\[[4](https://arxiv.org/html/2607.24774#bib.bib61)\]: predicts radiation levels with an ensemble of gradient\-boosted decision trees\.
- •LR: a linear regression model relating input variables to the output\.
- •DCRNN\[[15](https://arxiv.org/html/2607.24774#bib.bib59)\]: a spatio\-temporal model capturing spatial dependencies via bidirectional random walks and temporal dependencies via an encoder\-decoder with scheduled sampling\.
- •GWN\[[42](https://arxiv.org/html/2607.24774#bib.bib30)\]: a spatio\-temporal graph framework that learns an adaptive graph and integrates diffusion graph convolution with dilated causal convolution\.
- •STID\[[28](https://arxiv.org/html/2607.24774#bib.bib58)\]: a simple model that augments historical time series with spatial\-temporal identity embeddings\.
- •DLinear\[[44](https://arxiv.org/html/2607.24774#bib.bib16)\]: a one\-layer linear forecaster that rivals complex Transformer models on long\-term forecasting\.
- •StemGNN\[[3](https://arxiv.org/html/2607.24774#bib.bib15)\]: captures inter\-series correlations and temporal dependencies in the spectral domain\.
- •LightCTS\[[13](https://arxiv.org/html/2607.24774#bib.bib20)\]: an efficient, lightweight framework for correlated time\-series forecasting\.
- •PatchTST\[[22](https://arxiv.org/html/2607.24774#bib.bib14)\]: a Transformer using subseries patching and channel independence for multivariate forecasting\.
- •Koopa\[[19](https://arxiv.org/html/2607.24774#bib.bib13)\]: forecasts non\-stationary series via Koopman theory, disentangling time\-variant and time\-invariant dynamics\.
- •iTransformer\[[18](https://arxiv.org/html/2607.24774#bib.bib79)\]: treats each variate as a token and applies inverted self\-attention across variates\.
- •TimesNet\[[40](https://arxiv.org/html/2607.24774#bib.bib7)\]: detects multi\-periodicity via FFT and reshapes 1D series into 2D tensors processed by an inception\-style backbone\.

### B\-BContextual Feature Ablation

We examine the utility of the contextual features we provide to the prompting module\. noPrompt means that we remove all the contextual features from our framework\. Others \(*e\.g\.,*Wind Speed\) indicate using only one contextual feature\. The results on MAE and RMSE are presented in Figure[11](https://arxiv.org/html/2607.24774#A2.F11)\. In the absence of any contextual features, we separately add one feature at each time to assess its contribution\. Here*noPrompt*indicates the performance of NRFormer\+ without utilizing any additional contextual features\. We can observe that incorporating each feature consistently improves the model performance beyond the*noPrompt*\. In summary, the contextual features have a significant impact on nuclear radiation, and taking them into account is beneficial for forecasting\.

Concretely, on Japan\-4H every meteorological feature removal degrades both step\-12 MAE and sudden\-change MAE\. Removing the dew\-point feature raises step\-12 MAE by\+2\.28%\{\+\}2\.28\\%and sudden MAE by\+3\.32%\{\+\}3\.32\\%; removing wind speed by\+0\.27%\{\+\}0\.27\\%and\+1\.26%\{\+\}1\.26\\%; removing all meteorology by\+2\.70%\{\+\}2\.70\\%and\+4\.96%\{\+\}4\.96\\%\. On Japan\-1D the individual single\-feature removals are within run\-to\-run noise on step\-12 MAE, but the sudden\-change subset is markedly more sensitive\. Removing all meteorology raises sudden MAE by\+8\.62%\{\+\}8\.62\\%, confirming that the meteorological prompts contribute primarily where weather drives episodic radiation fluctuations \(cf\. Section[III\-E](https://arxiv.org/html/2607.24774#S3.SS5)\)\.

![Refer to caption](https://arxiv.org/html/2607.24774v1/x11.png)Figure 11:Ablation study of each contextual feature\.
### B\-CExtended Component Ablation

![Refer to caption](https://arxiv.org/html/2607.24774v1/x12.png)Figure 12:Per\-component ablation of NRFormer\+\.Figure[12](https://arxiv.org/html/2607.24774#A2.F12)visualizes the relative MAE deltas for all single\-component ablations\. Two variants dominate the 12\-step accuracy hit on both datasets,*w/o Physics*\(\+8\.35%\{\+\}8\.35\\%on Japan\-1D,\+10\.62%\{\+\}10\.62\\%on Japan\-4H\) and*w/o RevIN*\(\+8\.49%\{\+\}8\.49\\%,\+11\.02%\{\+\}11\.02\\%\), confirming the centrality of the physics module and instance normalization\. On the heavy\-tailed sudden\-change subset, both the physics prior and the log\-space target matter most\. Removing the physics module raises sudden\-change MAE by\+19\.03%\{\+\}19\.03\\%on Japan\-4H, and replacing the log\-space target by linear space \(*w/o Log\-space Target*\) raises it by\+36\.17%\{\+\}36\.17\\%on Japan\-1D and\+20\.11%\{\+\}20\.11\\%on Japan\-4H\. The remaining single\-component ablations \(enhanced meteorological encoder, temporal encoder, location encoder, and rad\-loc cross features\) produce smaller but consistently positive drops\.

![Refer to caption](https://arxiv.org/html/2607.24774v1/x13.png)Figure 13:Soft\-physics inductive bias versus a regional\-clustering alternative\. Three NRFormer\+ configurations, w/o Physics, Regional \(the diffusion module replaced by a region\-cluster bias\), and Diffusion \(Full\), evaluated on Japan\-1D and Japan\-4H in step\-12 MAE and sudden\-change MAE\. Both physics\-aware variants improve over w/o Physics, with the learned diffusion operator recovering the most on the heavy\-tailed sudden\-change subset\.Figure[13](https://arxiv.org/html/2607.24774#A2.F13)probes whether the gains attributed to the physics module reflect the diffusion parameterization specifically or a more generic meteorology\-aware inductive bias\. A generic Regional variant, which replacesDiD\_\{i\}with a region\-cluster bias term, recovers part of the accuracy lost to*w/o Physics*, but the learned diffusion operator recovers substantially more, and the gap is largest on the heavy\-tailed sudden\-change subset where physics is most informative\. On Japan\-4H the diffusion module cuts sudden\-change MAE by−16\.0%\-16\.0\\%relative to*w/o Physics*versus−10\.2%\-10\.2\\%for the regional bias, and on Japan\-1D by−9\.9%\-9\.9\\%versus−2\.4%\-2\.4\\%; the step\-12 MAE shows the same ordering\. This indicates that the wind\-modulated diffusion parameterization contributes beyond a coarse meteorology\-aware bias and justifies retaining the learned operator over a region\-clustering surrogate, consistent with the soft\-physics design rationale in Appendix[A\-B](https://arxiv.org/html/2607.24774#A1.SS2)\.

### B\-DParameter Sensitivity Analysis

Finally, we study the impacts of the hyper\-parameters on the performance of radiation forecasting\. We evaluate the impact of the number of temporal attention layersLtL\_\{t\}and the number of spatial attention layersLsL\_\{s\}\. The results on MAE, RMSE, and MAPE are reported in Figure[14](https://arxiv.org/html/2607.24774#A2.F14)\. First, we varyLtL\_\{t\}from 1 to 5\. As can be seen in Figure[14](https://arxiv.org/html/2607.24774#A2.F14), the performance shows a fluctuation when increasingLtL\_\{t\}from 1 to 3, and drops slightly by further increasingLtL\_\{t\}from 3 to 5 on both Japan\-4H and Japan\-1D datasets\. Overall, the optimum lies atLt=4\{L\_\{t\}\}\{=\}4on Japan\-1D andLt=3\{L\_\{t\}\}\{=\}3on Japan\-4H, with all sweep points within0\.050\.05MAE of the per\-dataset optimum, indicating that NRFormer\+ is robust to this hyperparameter within the explored range\. The possible reason is that a smallLtL\_\{t\}is insufficient to capture temporal correlation information, whereas too largeLtL\_\{t\}may introduce redundant and noisy information for our task, leading to performance degradation\. We varyLsL\_\{s\}from 1 to 5 and observe that our model achieves optimal performance whenLsL\_\{s\}is set to 2\. Increasing or decreasingLsL\_\{s\}beyond this point results in a decline in performance\. This is primarily because a lowerLsL\_\{s\}fails to provide adequate spatio\-temporal correlation of radiation propagation\. Additionally, we noticed a performance drop when more spatial attention layers were added\. This phenomenon might be caused by the fact that in an imbalanced spatial distribution scenario, using an excessive number ofLsL\_\{s\}can lead to overfitting, particularly for nodes with highly dense connections\.

![Refer to caption](https://arxiv.org/html/2607.24774v1/x14.png)Figure 14:Parameter sensitivity to the number of temporal attention layersLtL\_\{t\}and spatial attention layersLsL\_\{s\}on Japan\-1D and Japan\-4H\. The chosen settings \(Lt=4L\_\{t\}\{=\}4,Ls=2L\_\{s\}\{=\}2on Japan\-1D;Lt=3L\_\{t\}\{=\}3,Ls=2L\_\{s\}\{=\}2on Japan\-4H\) lie within0\.050\.05of the per\-sweep MAE optimum at the 12\-step horizon\.
### B\-EComplexity Analysis

We analyze the computational complexity of NRFormer\+ with respect to the number of stationsNN, input windowPP, and hidden dimensionDD\.

Temporal attention\.The point\-wise temporal self\-attention operates independently acrossNNstations, yielding𝒪​\(N⋅P2⋅D\)\\mathcal\{O\}\(N\\cdot P^\{2\}\\cdot D\)per layer\. SinceP=24P=24is fixed and small, this is effectively𝒪​\(N⋅D\)\\mathcal\{O\}\(N\\cdot D\)\.

Physics\-informed atmospheric modeling\.The atmospheric diffusion module involves MLP operations overNNstations, diffusion coefficient estimation is𝒪​\(N⋅\(Cm\+2\)⋅64\)\\mathcal\{O\}\(N\\cdot\(C\_\{m\}\+2\)\\cdot 64\), Laplacian computation is𝒪​\(N\)\\mathcal\{O\}\(N\), and physics encoding is𝒪​\(N⋅D\)\\mathcal\{O\}\(N\\cdot D\)\. The enhanced meteorological encoder is𝒪​\(N⋅P⋅D\)\\mathcal\{O\}\(N\\cdot P\\cdot D\)\. Both are linear inNN\.

Spatial attention\.The macro\-scale spatial attention computes all\-pair attention, requiring𝒪​\(N2⋅D\)\\mathcal\{O\}\(N^\{2\}\\cdot D\)per layer\. The proximity\-constrained attention has the same worst\-case complexity but is significantly faster in practice due to the sparse mask\. WithLsL\_\{s\}alternating layers, the total spatial complexity is𝒪​\(Ls⋅N2⋅D\)\\mathcal\{O\}\(L\_\{s\}\\cdot N^\{2\}\\cdot D\)\.

Overall complexity\.The dominant term is the spatial attention at𝒪​\(Ls⋅N2⋅D\)\\mathcal\{O\}\(L\_\{s\}\\cdot N^\{2\}\\cdot D\), which is consistent with NRFormer\. The additional physics and context modules add about𝒪​\(N⋅D2\)\\mathcal\{O\}\(N\\cdot D^\{2\}\)compute and𝒪​\(D2\)\\mathcal\{O\}\(D^\{2\}\)parameters \(from the physics encoder, meteorological encoder, temporal encoder, and cross\-feature module\)\. SinceN≫DN\\gg D, this overhead is asymptotically subdominant to the spatial term and constitutes only a modest increase over NRFormer\.