OutageDiT: A Generative Foundation Model for Power Outage Forecasting and Scenario Simulation

# OutageDiT: A Generative Foundation Model for Power Outage Forecasting and Scenario Simulation
Source: [https://arxiv.org/html/2609.01896](https://arxiv.org/html/2609.01896)
Yunqin ZhuAffiliation:H\. Milton Stewart School of Industrial and Systems EngineeringGeorgia Institute of TechnologyEmail:[yao\.xie@isye\.gatech\.edu](mailto:)Yao XieAffiliation:H\. Milton Stewart School of Industrial and Systems EngineeringGeorgia Institute of Technology

###### Abstract

Power\-outage planning requires scenarios before an event occurs\. These scenarios must represent uncertainty in magnitude, timing, and duration while preserving temporal dependence\. However, severe events are rare, and data from any single region contain few examples of extreme outage and restoration patterns\. To address this challenge, we introduce OutageDiT, a foundation model for generating seven\-day outage trajectories at quarter\-hour resolution, trained on outage and weather records across the United States\. Specifically, a condition encoder processes the historical context and known future covariates once per forecast, and a shallow flow decoder reuses the resulting horizon\-aligned states to generate complete trajectories\. The resulting samples support point forecasting, uncertainty quantification, and conditional event simulation within one deep generative model\. Across outage forecasting benchmarks, OutageDiT improves forecast accuracy and scenario quality over strong baselines and supports zero\-shot transfer to held\-out regions\. Together, these results position conditional outage simulation as a bridge from outage forecasting to operational planning under uncertainty\.

## 1Introduction

Power\-outage response decisions are made before an event’s magnitude and duration are fully known\. Examples include crew staging and resource allocation, which require forecasts of how outage counts may evolve over the planning horizon under uncertainty\. Scenario\-based planning further requires complete trajectories that preserve temporal dependence and plausible patterns of escalation, peak severity, and restoration\[[8](https://arxiv.org/html/2609.01896#bib.bib19),[2](https://arxiv.org/html/2609.01896#bib.bib18)\]\. We formulate this task as estimating

p\(y1:T∣h−L\+1:0,w−L\+1:T,a−L\+1:T\),p\\\!\\left\(y\_\{1:T\}\\mid h\_\{\-L\+1:0\},w\_\{\-L\+1:T\},a\_\{\-L\+1:T\}\\right\),\(1\)whereh−L\+1:0h\_\{\-L\+1:0\}denotes the 14\-day outage and tracked\-customer history,w−L\+1:Tw\_\{\-L\+1:T\}anda−L\+1:Ta\_\{\-L\+1:T\}denote weather and calendar covariates over the historical and forecast windows, andy1:Ty\_\{1:T\}denotes the complete seven\-day outage trajectory\. Learning this distribution is difficult because severe outage and restoration trajectories are rare and heterogeneous\. Existing regional models incorporate weather, infrastructure, and outage history\[[6](https://arxiv.org/html/2609.01896#bib.bib1),[7](https://arxiv.org/html/2609.01896#bib.bib2),[20](https://arxiv.org/html/2609.01896#bib.bib3),[19](https://arxiv.org/html/2609.01896#bib.bib4),[17](https://arxiv.org/html/2609.01896#bib.bib5)\], but models trained on data from one region may encounter few such events\. The scarcity of severe trajectories and the need to preserve local context motivate a foundation model approach\. Large\-scale pretraining across regions broadens the range of outage and restoration patterns seen during training, while conditioning on local outage history, weather, and calendar covariates preserves local context\. This combination supports transfer to regions with little or no local training data\.

Inspired by the Diffusion Transformer \(DiT\) architecture for image synthesis\[[12](https://arxiv.org/html/2609.01896#bib.bib14),[11](https://arxiv.org/html/2609.01896#bib.bib15)\], we develop OutageDiT, a generative foundation model for power outage trajectories\. A condition encoder maps 14 days of history and covariates available at forecast time to horizon\-aligned states\. A shallow flow decoder then reuses these states to model dependence across the forecast horizon\. Trained with conditional flow matching\[[10](https://arxiv.org/html/2609.01896#bib.bib13)\], the decoder generates outage trajectories through ODE integration\. An analytic count representation guarantees nonnegative integer outage counts\. The model is pretrained on outage and weather records across the United States\. Across national seven\-day forecasting and held\-out Michigan benchmarks, OutageDiT improves point and probabilistic forecast accuracy over supervised and fine\-tuned foundation\-model baselines, better represents temporal dependence and forecast uncertainty in generated scenarios, and transfers directly to a state excluded from training\.

## 2Method

#### Count representation\.

Outage and tracked\-customer counts are nonnegative integers with a wide dynamic range \(up to10710^\{7\}in our dataset\)\. A direct decimal encoding is discontinuous at carry boundaries, where an increment of one can change several digit coordinates\. For a countnnwith zero\-padded decimal digits, we therefore reflect each lower digit when the next higher digit is odd, obtaining carry\-aware coordinatesr6,…,r0r\_\{6\},\\ldots,r\_\{0\}that keep consecutive counts adjacent across carries\. We further append normalized log magnitude to encode the overall count scale and determine valid decimal positions during analytic decoding\. The resulting representation is

g⁡\(n\)=\[2​log10⁡\(1\+n\)7−1,\{2​rj−910\}j=60\]∈ℝ8\.g\(n\)=\\left\[2\\frac\{\\log\_\{10\}\(1\+n\)\}\{7\}\-1,\\left\\\{\\frac\{2r\_\{j\}\-9\}\{10\}\\right\\\}\_\{j=6\}^\{0\}\\right\]\\in\\mathbb\{R\}^\{8\}\.\(2\)

#### Condition encoder\.

Each forecast uses 14 days of transformed quarter\-hour outage and tracked\-customer counts together with weather and calendar covariates\. Four consecutive observations are patched into an hourly token\. The condition encoder applies transformer blocks\[[18](https://arxiv.org/html/2609.01896#bib.bib11)\]to 336 history tokens, an overlapping 24\-token recent outage stream, and 168 future tokens formed from weather and calendar variables known over the seven\-day horizon\. The streams share a signed hourly time axis represented with rotary position embeddings \(RoPE\)\[[15](https://arxiv.org/html/2609.01896#bib.bib12)\]\. History remains a read\-only key–value bank; recent and future tokens use one joint attention softmax, and the final encoder block returns 168 horizon\-aligned conditioning statesQ=\(q1,…,q168\)Q=\(q\_\{1\},\\ldots,q\_\{168\}\)\.

#### Flow decoder\.

The decoder patchifies the noisy flow stateYτ∈ℝ672×8Y\_\{\\tau\}\\in\\mathbb\{R\}^\{672\\times 8\}into 168 hourly tokens\. Each layer applies one joint attention softmax: future\-token queries attend to condition keys and values projected fromQQtogether with keys and values from the noisy future tokens themselves\. Flow timeτ\\taumodulates the attention and feed\-forward residuals through adaptive layer normalization \(adaLN\)\[[12](https://arxiv.org/html/2609.01896#bib.bib14)\]\. Independent noiseϵj∼𝒰⁡\(−0\.05,0\.05\)\\epsilon\_\{j\}\\sim\\mathcal\{U\}\(\-0\.05,0\.05\)makes each future digit coordinate continuous without crossing quantization\-bin boundaries\. For the resulting trajectoryYYandZ∼𝒩⁡\(0,I\)Z\\sim\\mathcal\{N\}\(0,I\), conditional flow matching\[[10](https://arxiv.org/html/2609.01896#bib.bib13)\]samplesτ∼𝒰⁡\(0,1\)\\tau\\sim\\mathcal\{U\}\(0,1\)alongYτ=\(1−τ\)​Y\+τ​ZY\_\{\\tau\}=\(1\-\\tau\)Y\+\\tau Zand targets velocityZ−YZ\-Y\. We further use an auxiliary head to predictlog10⁡\(1\+y\)\\log\_\{10\}\(1\+y\)fromQQat quarter\-hour resolution\. The full objective is

ℒ=𝔼⁡\[18​T​‖vθ​\(Yτ,τ,Q\)−\(Z−Y\)‖F2\]\+10−3​ℒaux,\\mathcal\{L\}=\\mathbb\{E\}\\\!\\left\[\\frac\{1\}\{8T\}\\\|v\_\{\\theta\}\(Y\_\{\\tau\},\\tau,Q\)\-\(Z\-Y\)\\\|\_\{F\}^\{2\}\\right\]\+10^\{\-3\}\\mathcal\{L\}\_\{\\mathrm\{aux\}\},\(3\)whereℒaux\\mathcal\{L\}\_\{\\mathrm\{aux\}\}denotes this auxiliary log\-count MSE\. Sampling initializesY1=ZY\_\{1\}=Zand integrates the flow ODE fromτ=1\\tau=1toτ=0\\tau=0\. At inference, the condition encoder runs once; its layer\-wise key–value projections are cached across Monte Carlo samples and ODE solver steps\. Only the noisy future tokens and their projections update during sampling\. Hourly outputs unfold into four quarter\-hour velocities, andg−1g^\{\-1\}maps the integrated coordinates to integer counts\. Inference requires one encoder pass and one decoder evaluation per MC sample and ODE solver step\. Appendix[B](https://arxiv.org/html/2609.01896#A2)provides the complete architecture and optimization details\.

## 3Experiments

#### Experimental setup\.

Training uses U\.S\. county\-level windows outside Michigan ending before 2025\. Validation covers January\-\-February 2025 and test covers March\-\-May 2025\. Each split contains 30 normal windows and 30 event windows drawn from ten distinct events\. An event must exceed both 800 outages and 4% of tracked customers for at least six hours\. Inputs combine historical outage and customer counts, calendar variables, and 84 hourly meteorological covariates derived from NOAA/NCEP High\-Resolution Rapid Refresh \(HRRR\) fields111[https://emc\.ncep\.noaa\.gov/emc/pages/numerical\_forecast\_systems/hrrr\.php](https://emc.ncep.noaa.gov/emc/pages/numerical_forecast_systems/hrrr.php); weather is observed through the forecast horizon\. All supervised models use the same county/origin sampling distribution, and validation MSE selects checkpoints\. We report MSE and weighted quantile loss \(WQL\) inlog10⁡\(1\+y\)\\log\_\{10\}\(1\+y\)space; lower values are better\. Appendices[A](https://arxiv.org/html/2609.01896#A1),[B](https://arxiv.org/html/2609.01896#A2),[C\.1](https://arxiv.org/html/2609.01896#A3.SS1), and[C\.2](https://arxiv.org/html/2609.01896#A3.SS2)provide the data, training, and evaluation details\.

#### National forecasting\.

Table[1](https://arxiv.org/html/2609.01896#S3.T1)\(a\) compares OutageDiT with supervised probabilistic forecasters\[[13](https://arxiv.org/html/2609.01896#bib.bib8),[9](https://arxiv.org/html/2609.01896#bib.bib9),[16](https://arxiv.org/html/2609.01896#bib.bib10)\]and fine\-tuned time\-series foundation models\[[5](https://arxiv.org/html/2609.01896#bib.bib6),[1](https://arxiv.org/html/2609.01896#bib.bib7)\]\. OutageDiT performs best on every split and metric\. The advantage holds on normal and event windows, so the aggregate gain is not driven by a single regime\. Table[1](https://arxiv.org/html/2609.01896#S3.T1)\(b\) shows that every conditioning branch contributes\. Future conditioning has the largest effect, while the history and recent ablations confirm that both temporal scales remain useful\. Digit coordinates and the auxiliary log\-count loss provide the largest gains among the remaining components; uniform digit noise yields a smaller but consistent improvement\. Figure[1](https://arxiv.org/html/2609.01896#S3.F1)illustrates complementary high\-impact dynamics: the forecast follows restoration and secondary fluctuations in St\. Louis, MO, while the Douglas, NE forecast captures event onset one day before the ground\-truth peak\.

Table 1:National seven\-day forecasting performance and ablations\. All metrics evaluate outage count inlog10⁡\(1\+y\)\\log\_\{10\}\(1\+y\)space\. Ablations use the same training budget and evaluation protocol\.\(a\) Overall performance

\(b\) Ablations of OutageDiT

#### Scenario simulation\.

For models returning complete trajectories, we report the variogram score \(VS\) and the coverage and width of the central 90% prediction interval\. VS measures temporal dependence; coverage and width assess marginal calibration and sharpness\. Table[2](https://arxiv.org/html/2609.01896#S3.T2)\(a\) shows that OutageDiT has the lowest VS and highest coverage with interval width comparable to competing methods\. Lower coverage on event windows identifies extreme\-event calibration as the primary limitation\. Figure[2](https://arxiv.org/html/2609.01896#S3.F2)further shows that digit coordinates bring synthetic one\-day trajectories closer to the real joint distribution, consistent with their gains beyond pointwise forecast error\.

#### Geographic transfer\.

Michigan provides a geographically held\-out evaluation: OutageDiT is trained without Michigan data and applied directly to the state\. The INFORMS DMDA Workshop Data Challenge reports average 24/48\-hour RMSE222[https://sites\.google\.com/view/dmdaworkshop2025/data\-challenge](https://sites.google.com/view/dmdaworkshop2025/data-challenge); SARIMAX is its best\-performing forecasting method\[[21](https://arxiv.org/html/2609.01896#bib.bib17)\]\. The IISE Energy Analytics Challenge reports normal and tail RMSE, nonzero F1, and 95% interval Winkler score333[https://www\.iise\.org/Details\.aspx?id=54646](https://www.iise.org/Details.aspx?id=54646)\. We use the official scores of its winner, the Regime\-Switching Time Series Foundation Model \(RSFM\)\. Table[2](https://arxiv.org/html/2609.01896#S3.T2)\(b\) shows that OutageDiT leads every reported transfer metric without Michigan training\. The gains span average, normal, and tail errors, nonzero detection, and interval quality, supporting transfer across outage regimes and forecast summaries\.

Table 2:Scenario quality and zero\-shot Michigan transfer\. National coverage targets 0\.90; Michigan metrics follow the DMDA and IISE challenge protocols\.\(a\) National seven\-day forecasting

\(b\) Michigan competition benchmarks

Figure 1:National test forecasts for \(a\) restoration in St\. Louis, MO and \(b\) event onset in Douglas, NE\. Dashed vertical lines mark forecast origins; bands show central 50% and 90% intervals\. Values use thelog10⁡\(1\+y\)\\log\_\{10\}\(1\+y\)scale; dates are in 2025\.Figure 2:Random one\-day outage trajectories\. Each panel shows 64 trajectories over a 24\-hour forecast horizon\. Joint MMD is the empirical RBF\-kernel discrepancy over complete trajectories, with bandwidth set to the median pairwise RMS distance among real trajectories\.

## 4Conclusion

This work introduced OutageDiT, a generative foundation model for conditional seven\-day outage trajectories learned from county\-level records across the United States\. On national benchmarks, OutageDiT improves point and probabilistic forecast accuracy and better preserves scenario\-level temporal dependence; ablations attribute these gains to the full conditioning structure and count representation\. On Michigan, which is excluded from training, OutageDiT outperforms the reported DMDA and IISE competition baselines across all metrics, demonstrating zero\-shot geographic transfer\. Extreme\-event undercoverage remains the primary limitation\. Future work should evaluate these scenarios within crew\-staging and resource\-allocation decisions, account for weather\-forecast error, and extend nominal forecasts to worst\-case planning under distribution shift\[[4](https://arxiv.org/html/2609.01896#bib.bib22),[22](https://arxiv.org/html/2609.01896#bib.bib20),[3](https://arxiv.org/html/2609.01896#bib.bib21)\]\.

## Acknowledgment

The project is partially supported by the U\.S\. Department of Energy Advanced Grid Modeling Program under Grant DE\-OE 0000875\.

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## Appendix AData and Evaluation Protocol

### A\.1Data and Inputs

The training dataset contains quarter\-hour outage and tracked\-customer counts from U\.S\. counties outside Michigan\. Michigan forms a geographically held\-out transfer benchmark\. Outage records span 2018–2024; complete weather covariates are available from January 2022\. Each example contains 14 days of context and a seven\-day target, corresponding to 1,344 and 672 quarter\-hour observations\. The model receives outage and customer\-count history, 84 weather variables, and eight calendar variables\. Its conditioning path includes observed weather and calendar values over the seven\-day forecast horizon\. The hourly weather variables are derived from NOAA/NCEP HRRR fields supplied on a county\-indexed grid and span temperature, wind, pressure, moisture, cloud, precipitation, convection, and radar diagnostics\. Each weather feature is standardized using pre\-2025 training data and aligned to its four corresponding quarter\-hour observations\.

### A\.2Training and Evaluation Splits

Training windows follow the natural county–date distribution: each draw first selects a county and then a valid forecast origin in that county\. The complete 14\-day history stream is retained independently for half of the training examples, while the overlapping 24\-hour recent stream is always retained\. Weather conditioning is augmented with probabilities 0\.80 for full past and future weather, 0\.10 for past weather with future weather masked, and 0\.10 for all weather masked\.

The national validation set contains 30 event and 30 normal windows with origins from January 15 through February 20, 2025; the test set contains the same composition with origins from March 16 through May 23, 2025\. The 30 event windows in each split are formed from ten distinct events with three forecast origins per event\. An event must exceed both 800 outages and 4% of tracked customers continuously for at least six hours\. Normal windows are drawn from counties outside the selected events\. Every national score gives equal weight to each forecast case\.

## Appendix BArchitecture and Optimization

Four consecutive quarter\-hour observations are patchified into one hourly token, yielding 336 history, 24 recent, and 168 future tokens\. The condition encoder has four transformer blocks; the flow decoder has two\. Both use width 1,024, 16 attention heads, and a feed\-forward expansion factor of four\. The complete model has 183\.46 million trainable parameters\. Outage count is the prediction target, and tracked\-customer count is a historical covariate\.

### B\.1Tokenization and Position

For a quarter\-hour streamUU, each modalitymmuses the two\-layer patch map

𝒫m​\(U\)i=Wm,2​SiLU⁡\(Wm,1​vec⁡\(U4​i,…,U4​i\+3\)\+bm,1\)\+bm,2\.\\mathcal\{P\}\_\{m\}\(U\)\_\{i\}=W\_\{m,2\}\\,\\operatorname\{SiLU\}\\\!\\left\(W\_\{m,1\}\\operatorname\{vec\}\(U\_\{4i\},\\ldots,U\_\{4i\+3\}\)\+b\_\{m,1\}\\right\)\+b\_\{m,2\}\.\(4\)The three condition streams are

H\\displaystyle H=LN⁡\(𝒫c​\(UH\)\+𝒫w​\(WH\)\+𝒫a​\(AH\)\),\\displaystyle=\\operatorname\{LN\}\\\!\\left\(\\mathcal\{P\}\_\{c\}\(U\_\{H\}\)\+\\mathcal\{P\}\_\{w\}\(W\_\{H\}\)\+\\mathcal\{P\}\_\{a\}\(A\_\{H\}\)\\right\),\(5\)R\\displaystyle R=LN⁡\(𝒫c​\(UR\)\+𝒫a​\(AR\)\),\\displaystyle=\\operatorname\{LN\}\\\!\\left\(\\mathcal\{P\}\_\{c\}\(U\_\{R\}\)\+\\mathcal\{P\}\_\{a\}\(A\_\{R\}\)\\right\),F\\displaystyle F=LN⁡\(𝒫w​\(WF\)\+𝒫a​\(AF\)\),\\displaystyle=\\operatorname\{LN\}\\\!\\left\(\\mathcal\{P\}\_\{w\}\(W\_\{F\}\)\+\\mathcal\{P\}\_\{a\}\(A\_\{F\}\)\\right\),wherec,w,ac,w,adenote count, weather, and calendar modalities\. Missing count or weather patches are zeroed before summation\. For zero\-based hourly indexii, the signed positions arepiH=i\+12−336p\_\{i\}^\{H\}=i\+\\tfrac\{1\}\{2\}\-336,piR=i\+12−24p\_\{i\}^\{R\}=i\+\\tfrac\{1\}\{2\}\-24, andpiF=i\+12p\_\{i\}^\{F\}=i\+\\tfrac\{1\}\{2\}\. RoPE rotates each query/key pair by

\[\(ℛp​x\)2​k\(ℛp​x\)2​k\+1\]=\[cos⁡\(p​ωk\)−sin⁡\(p​ωk\)sin⁡\(p​ωk\)cos⁡\(p​ωk\)\]\[x2​kx2​k\+1\],ωk=10,000−k/32,k=0,…,31\.\\begin\{bmatrix\}\(\\mathcal\{R\}\_\{p\}x\)\_\{2k\}\\\\ \(\\mathcal\{R\}\_\{p\}x\)\_\{2k\+1\}\\end\{bmatrix\}=\\begin\{bmatrix\}\\cos\(p\\omega\_\{k\}\)&\-\\sin\(p\\omega\_\{k\}\)\\\\ \\sin\(p\\omega\_\{k\}\)&\\cos\(p\\omega\_\{k\}\)\\end\{bmatrix\}\\begin\{bmatrix\}x\_\{2k\}\\\\ x\_\{2k\+1\}\\end\{bmatrix\},\\qquad\\omega\_\{k\}=10\{,\}000^\{\-k/32\},\\quad k=0,\\ldots,31\.\(6\)Thus all streams share one time axis, with context before and forecast tokens after the origin\.

### B\.2Condition Encoder

At encoder layerℓ\\ell, layer\-normalized stream projections produce

qSℓ\\displaystyle q\_\{S\}^\{\\ell\}=ℛpS​\(RMS⁡\(WQ,Sℓ​LN⁡\(Sℓ\)\)\),\\displaystyle=\\mathcal\{R\}\_\{p^\{S\}\}\\\!\\left\(\\operatorname\{RMS\}\(W\_\{Q,S\}^\{\\ell\}\\operatorname\{LN\}\(S^\{\\ell\}\)\)\\right\),\(7\)kSℓ\\displaystyle k\_\{S\}^\{\\ell\}=ℛpS​\(RMS⁡\(WK,Sℓ​LN⁡\(Sℓ\)\)\),\\displaystyle=\\mathcal\{R\}\_\{p^\{S\}\}\\\!\\left\(\\operatorname\{RMS\}\(W\_\{K,S\}^\{\\ell\}\\operatorname\{LN\}\(S^\{\\ell\}\)\)\\right\),vSℓ\\displaystyle v\_\{S\}^\{\\ell\}=WV,Sℓ​LN⁡\(Sℓ\)\.\\displaystyle=W\_\{V,S\}^\{\\ell\}\\operatorname\{LN\}\(S^\{\\ell\}\)\.HereqSℓq\_\{S\}^\{\\ell\}is defined forS∈\{R,F\}S\\in\\\{R,F\\\}, whilekSℓk\_\{S\}^\{\\ell\}andvSℓv\_\{S\}^\{\\ell\}are defined forS∈\{H,R,F\}S\\in\\\{H,R,F\\\};RMS\\operatorname\{RMS\}denotes per\-head RMS normalization\. History is read\-only\. For nonfinal layers,Gℓ=\[Rℓ;Fℓ\]G^\{\\ell\}=\[R^\{\\ell\};F^\{\\ell\}\]andqGℓ=\[qRℓ;qFℓ\]q\_\{G\}^\{\\ell\}=\[q\_\{R\}^\{\\ell\};q\_\{F\}^\{\\ell\}\]; in the final layer,Gℓ=FℓG^\{\\ell\}=F^\{\\ell\}andqGℓ=qFℓq\_\{G\}^\{\\ell\}=q\_\{F\}^\{\\ell\}, so only the future stream is updated\. One shared masked softmax attends to all three banks:

AGℓ=softmax⁡\(qGℓ​\[kHℓ;kRℓ;kFℓ\]⊤64\+ℳ\)​\[vHℓ;vRℓ;vFℓ\]\.A\_\{G\}^\{\\ell\}=\\operatorname\{softmax\}\\\!\\left\(\\frac\{q\_\{G\}^\{\\ell\}\[k\_\{H\}^\{\\ell\};k\_\{R\}^\{\\ell\};k\_\{F\}^\{\\ell\}\]^\{\\top\}\}\{\\sqrt\{64\}\}\+\\mathcal\{M\}\\right\)\[v\_\{H\}^\{\\ell\};v\_\{R\}^\{\\ell\};v\_\{F\}^\{\\ell\}\]\.\(8\)Here\[;\]\[\\,;\\,\]concatenates along the token axis andℳ\\mathcal\{M\}masks unavailable history or recent keys\. The mutable streams use pre\-normalized residual updates

G¯ℓ=Gℓ\+WOℓ​AGℓ,Gℓ\+1=G¯ℓ\+FFNℓ⁡\(LN⁡\(G¯ℓ\)\),\\bar\{G\}^\{\\ell\}=G^\{\\ell\}\+W\_\{O\}^\{\\ell\}A\_\{G\}^\{\\ell\},\\qquad G^\{\\ell\+1\}=\\bar\{G\}^\{\\ell\}\+\\operatorname\{FFN\}\_\{\\ell\}\(\\operatorname\{LN\}\(\\bar\{G\}^\{\\ell\}\)\),\(9\)whereFFNℓ⁡\(x\)=W2,ℓ​\[\(Wu,ℓ​x\)⊙SiLU⁡\(Wg,ℓ​x\)\]\\operatorname\{FFN\}\_\{\\ell\}\(x\)=W\_\{2,\\ell\}\[\(W\_\{u,\\ell\}x\)\\odot\\operatorname\{SiLU\}\(W\_\{g,\\ell\}x\)\]\. Nonfinal outputs are split back into recent and future streams; the encoder returnsQ=LN⁡\(F4\)Q=\\operatorname\{LN\}\(F^\{4\}\)\.

### B\.3Flow Decoder and adaLN

The noisy trajectory is patched asX\(0\)=𝒫y​\(Yτ\)X^\{\(0\)\}=\\mathcal\{P\}\_\{y\}\(Y\_\{\\tau\}\)\. A Fourier timestep embedding and MLP produce

eτ=MLPt⁡\(\[sin⁡\(τ​ν\);cos⁡\(τ​ν\)\]\),\(βa,γa,αa,βf,γf,αf\)=Wm​SiLU⁡\(eτ\),e\_\{\\tau\}=\\operatorname\{MLP\}\_\{t\}\(\[\\sin\(\\tau\\nu\);\\cos\(\\tau\\nu\)\]\),\\qquad\(\\beta\_\{a\},\\gamma\_\{a\},\\alpha\_\{a\},\\beta\_\{f\},\\gamma\_\{f\},\\alpha\_\{f\}\)=W\_\{m\}\\operatorname\{SiLU\}\(e\_\{\\tau\}\),\(10\)whereν\\nuis a fixed frequency vector, andaaandffindex attention and feed\-forward branches; each layer has its own modulation map\. For decoder layerℓ\\ell, let

X^ℓ=LN⁡\(Xℓ\)⊙\(1\+γa\)\+βa\.\\widehat\{X\}^\{\\ell\}=\\operatorname\{LN\}\(X^\{\\ell\}\)\\odot\(1\+\\gamma\_\{a\}\)\+\\beta\_\{a\}\.\(11\)Queries and future\-token keys are obtained fromX^ℓ\\widehat\{X\}^\{\\ell\}using the RMS\-normalized RoPE projections in Eq\. \([7](https://arxiv.org/html/2609.01896#A2.E7)\)\. The projected conditionQD=WC​Q\+bCQ\_\{D\}=W\_\{C\}Q\+b\_\{C\}supplies per\-layer cached keys and values\(kQℓ,vQℓ\)=KVℓ⁡\(LN⁡\(QD\)\)\(k\_\{Q\}^\{\\ell\},v\_\{Q\}^\{\\ell\}\)=\\operatorname\{KV\}\_\{\\ell\}\(\\operatorname\{LN\}\(Q\_\{D\}\)\); its keys receive the same RMS normalization and RoPE\. Decoder attention is one joint softmax,

AXℓ=softmax⁡\(qXℓ​\[kQℓ;kXℓ\]⊤64\)​\[vQℓ;vXℓ\],A\_\{X\}^\{\\ell\}=\\operatorname\{softmax\}\\\!\\left\(\\frac\{q\_\{X\}^\{\\ell\}\[k\_\{Q\}^\{\\ell\};k\_\{X\}^\{\\ell\}\]^\{\\top\}\}\{\\sqrt\{64\}\}\\right\)\[v\_\{Q\}^\{\\ell\};v\_\{X\}^\{\\ell\}\],\(12\)followed by gated residual updates

X~ℓ\\displaystyle\\widetilde\{X\}^\{\\ell\}=Xℓ\+αa⊙WOℓ​AXℓ,\\displaystyle=X^\{\\ell\}\+\\alpha\_\{a\}\\odot W\_\{O\}^\{\\ell\}A\_\{X\}^\{\\ell\},\(13\)Xℓ\+1\\displaystyle X^\{\\ell\+1\}=X~ℓ\+αf⊙FFNℓ⁡\(LN⁡\(X~ℓ\)⊙\(1\+γf\)\+βf\)\.\\displaystyle=\\widetilde\{X\}^\{\\ell\}\+\\alpha\_\{f\}\\odot\\operatorname\{FFN\}\_\{\\ell\}\\\!\\left\(\\operatorname\{LN\}\(\\widetilde\{X\}^\{\\ell\}\)\\odot\(1\+\\gamma\_\{f\}\)\+\\beta\_\{f\}\\right\)\.The output head uses its own timestep\-conditioned shift and scale,

vθ​\(Yτ,τ,Q\)=Unpatch⁡\(Wv​\[LN⁡\(X\(2\)\)⊙\(1\+γo\)\+βo\]\),v\_\{\\theta\}\(Y\_\{\\tau\},\\tau,Q\)=\\operatorname\{Unpatch\}\\\!\\left\(W\_\{v\}\[\\operatorname\{LN\}\(X^\{\(2\)\}\)\\odot\(1\+\\gamma\_\{o\}\)\+\\beta\_\{o\}\]\\right\),\(14\)where\(βo,γo\)=Wo​SiLU⁡\(eτ\)\(\\beta\_\{o\},\\gamma\_\{o\}\)=W\_\{o\}\\operatorname\{SiLU\}\(e\_\{\\tau\}\)\. The condition key–value cache is fixed across ODE solver steps and Monte Carlo samples; only the future\-token projections are recomputed\.

### B\.4Model and Training Configuration

Table 3:Configuration of the reported OutageDiT model\.Counts are transformed with Eq\. \([2](https://arxiv.org/html/2609.01896#S2.E2)\)\. For flow matching, each future digit coordinate receives independent uniform noise,ϵj∼𝒰⁡\(−0\.05,0\.05\)\\epsilon\_\{j\}\\sim\\mathcal\{U\}\(\-0\.05,0\.05\)\. Generated coordinates are decoded analytically, with the magnitude coordinate selecting valid decimal positions before inversion of the reflected digits\. The flow source is a standard isotropic Gaussian\. An auxiliary linear head mapsQQto quarter\-hourlog10⁡\(1\+y\)\\log\_\{10\}\(1\+y\)predictions; its MSE enters Eq\. \([3](https://arxiv.org/html/2609.01896#S2.E3)\) with weight10−310^\{\-3\}\. Together, the analytic count transform and its inverse provide a parameter\-free count representation\.

All linear maps are initialized with Xavier\-uniform weights and zero biases, except the timestep MLP, whose linear weights use a normal distribution with standard deviation 0\.02\. Adaptive\-normalization modulation weights in the flow decoder use a zero\-mean Gaussian with standard deviation10−310^\{\-3\}\. The final modulation and output projection are initialized to zero\.

We optimize two\-dimensional hidden matrices in the transformer blocks with Muon and all remaining parameters with AdamW\. Training uses bfloat16 mixed precision\. An exponential moving average begins at update 1,000\. Validation is run every 500 updates\. Probabilistic and Michigan metrics are evaluated after model selection\.

### B\.5Baseline Configuration

The supervised baselines use the same county/origin sampler and available covariates\. DeepAR is a two\-layer autoregressive LSTM with a Student\-ttlikelihood; TFT uses one LSTM layer, hidden size 64, four attention heads, and nine quantiles from 0\.1 to 0\.9\. CSDI treats the future outage window as missing and retains four quarter\-hour phases in each hourly patch\. TimesFM 2\.5 is jointly fine\-tuned with one shared linear covariate head, and Chronos\-2 is fully fine\-tuned through its released interface\. All checkpoints are frozen from validation results before test and Michigan evaluation\. All supervised baselines use AdamW with 500 warmup updates followed by a constant learning rate\. Chronos\-2 uses the released interface’s default learning rate of10−610^\{\-6\}\. For the remaining baselines, we select the learning rate from\{10−4,10−3\}\\\{10^\{\-4\},10^\{\-3\}\\\}using validation MSE\.

## Appendix CMetric Definitions

### C\.1National Metrics

Final generative evaluation usesM=64M=64scrambled Sobol Gaussian trajectories and 20 Euler steps\. LetNNbe the number of cases in the evaluated subset,T=672T=672, ands⁡\(y\)=log10⁡\(1\+y\)s\(y\)=\\log\_\{10\}\(1\+y\)\. For sample models,s^n,t=M−1​∑ms⁡\(yn,t\(m\)\)\\widehat\{s\}\_\{n,t\}=M^\{\-1\}\\sum\_\{m\}s\(y^\{\(m\)\}\_\{n,t\}\); quantile\-only models use their median\. The case\-equal mean squared error is

MSE=1N​T​∑n=1N∑t=1T\[s^n,t−s⁡\(yn,t\)\]2\.\\operatorname\{MSE\}=\\frac\{1\}\{NT\}\\sum\_\{n=1\}^\{N\}\\sum\_\{t=1\}^\{T\}\\left\[\\widehat\{s\}\_\{n,t\}\-s\(y\_\{n,t\}\)\\right\]^\{2\}\.\(15\)
Weighted quantile loss uses𝒬=\{0\.1,0\.2,…,0\.9\}\\mathcal\{Q\}=\\\{0\.1,0\.2,\\ldots,0\.9\\\}:

WQL=2​∑n∑t∑q∈𝒬ρq​\(s⁡\(yn,t\)−s^n,t\(q\)\)\|𝒬\|​∑n∑t\|s⁡\(yn,t\)\|,ρq​\(e\)=max⁡\{q​e,\(q−1\)​e\}\.\\operatorname\{WQL\}=\\frac\{2\\sum\_\{n\}\\sum\_\{t\}\\sum\_\{q\\in\\mathcal\{Q\}\}\\rho\_\{q\}\\\!\\left\(s\(y\_\{n,t\}\)\-\\widehat\{s\}^\{\(q\)\}\_\{n,t\}\\right\)\}\{\|\\mathcal\{Q\}\|\\sum\_\{n\}\\sum\_\{t\}\|s\(y\_\{n,t\}\)\|\},\\quad\\rho\_\{q\}\(e\)=\\max\\\{qe,\(q\-1\)e\\\}\.\(16\)Sample\-based models use empirical quantiles of the same 64 trajectories\.

For interval level1−α1\-\\alpha, letℓn,t\\ell\_\{n,t\}andun,tu\_\{n,t\}be the empiricalα/2\\alpha/2and1−α/21\-\\alpha/2quantiles in transformed space\. We report

Coverage1−α\\displaystyle\\operatorname\{Coverage\}\_\{1\-\\alpha\}=1N​T∑n=1N∑t=1T𝟏\{ℓn,t≤s\(yn,t\)≤un,t\},\\displaystyle=\\frac\{1\}\{NT\}\\sum\_\{n=1\}^\{N\}\\sum\_\{t=1\}^\{T\}\\mathbf\{1\}\\\{\\ell\_\{n,t\}\\leq s\(y\_\{n,t\}\)\\leq u\_\{n,t\}\\\},\(17\)Width1−α\\displaystyle\\operatorname\{Width\}\_\{1\-\\alpha\}=1N​T​∑n=1N∑t=1T\(un,t−ℓn,t\),\\displaystyle=\\frac\{1\}\{NT\}\\sum\_\{n=1\}^\{N\}\\sum\_\{t=1\}^\{T\}\(u\_\{n,t\}\-\\ell\_\{n,t\}\),\(18\)withα=0\.10\\alpha=0\.10in Table[2](https://arxiv.org/html/2609.01896#S3.T2)\(a\)\.

Temporal dependence is measured by the variogram score of orderp=0\.5p=0\.5\[[14](https://arxiv.org/html/2609.01896#bib.bib16)\]:

VSp\(n\)=2T⁡\(T−1\)​∑t<t′\(CLOSE\\displaystyle\\operatorname\{VS\}\_\{p\}^\{\(n\)\}=\\frac\{2\}\{T\(T\-1\)\}\\sum\_\{t<t^\{\\prime\}\}\\Bigg\(\|s⁡\(yn,t\)−s⁡\(yn,t′\)\|p\\displaystyle\|s\(y\_\{n,t\}\)\-s\(y\_\{n,t^\{\\prime\}\}\)\|^\{p\}−1M∑m=1M\|s\(yn,t\(m\)\)−s\(yn,t′\(m\)\)\|p\)2,\\displaystyle\-\\frac\{1\}\{M\}\\sum\_\{m=1\}^\{M\}\|s\(y^\{\(m\)\}\_\{n,t\}\)\-s\(y^\{\(m\)\}\_\{n,t^\{\\prime\}\}\)\|^\{p\}\\Bigg\)^\{2\},\(19\)and the reported value isN−1​∑n=1NVS0\.5\(n\)N^\{\-1\}\\sum\_\{n=1\}^\{N\}\\operatorname\{VS\}\_\{0\.5\}^\{\(n\)\}\.

### C\.2Michigan Metrics

The transfer evaluation uses all held\-out Michigan counties, indexed byc=1,…,Cc=1,\\ldots,C, and the official 48\-hour target\. Michigan metrics are computed in the original count space, whereas the national metrics above uses⁡\(y\)s\(y\)\. OutageDiT uses weather over the 48\-hour forecast horizon, following our national evaluation protocol, while the reported competition entries follow their original challenge input protocols\. Lety^c,t\\widehat\{y\}\_\{c,t\}denote the county\-hour point forecast\. The county\-macro RMSE at horizonHHis

RMSEH=1C​∑c=1C1H​∑t=1H\(y^c,t−yc,t\)2\.\\operatorname\{RMSE\}\_\{H\}=\\frac\{1\}\{C\}\\sum\_\{c=1\}^\{C\}\\sqrt\{\\frac\{1\}\{H\}\\sum\_\{t=1\}^\{H\}\(\\widehat\{y\}\_\{c,t\}\-y\_\{c,t\}\)^\{2\}\}\.\(20\)The DMDA average RMSE is

AvgRMSE=12​\(RMSE24\+RMSE48\)\.\\operatorname\{AvgRMSE\}=\\tfrac\{1\}\{2\}\\left\(\\operatorname\{RMSE\}\_\{24\}\+\\operatorname\{RMSE\}\_\{48\}\\right\)\.\(21\)
For the IISE challenge metrics,κc\\kappa\_\{c\}is countycc’s 95th\-percentile training outage count\. LetAcnormal=\{t:yc,t<κc\}A\_\{c\}^\{\\mathrm\{normal\}\}=\\\{t:y\_\{c,t\}<\\kappa\_\{c\}\\\}andActail=\{t:yc,t≥κc\}A\_\{c\}^\{\\mathrm\{tail\}\}=\\\{t:y\_\{c,t\}\\geq\\kappa\_\{c\}\\\}\. Then

RMSEr=1C​∑c=1C1max⁡\{1,\|Acr\|\}​∑t∈Acr\(y^c,t−yc,t\)2,r∈\{normal,tail\}\.\\operatorname\{RMSE\}\_\{r\}=\\frac\{1\}\{C\}\\sum\_\{c=1\}^\{C\}\\sqrt\{\\frac\{1\}\{\\max\\\{1,\|A\_\{c\}^\{r\}\|\\\}\}\\sum\_\{t\\in A\_\{c\}^\{r\}\}\(\\widehat\{y\}\_\{c,t\}\-y\_\{c,t\}\)^\{2\}\},\\quad r\\in\\\{\\mathrm\{normal\},\\mathrm\{tail\}\\\}\.\(22\)
Nonzero F1 labels a county\-hour positive when its count is at least one\. After pooling all county\-hours,

Prec=TPTP\+FP,Rec=TPTP\+FN,F1=2​Prec⁡RecPrec\+Rec\.\\operatorname\{Prec\}=\\frac\{\\mathrm\{TP\}\}\{\\mathrm\{TP\}\+\\mathrm\{FP\}\},\\qquad\\operatorname\{Rec\}=\\frac\{\\mathrm\{TP\}\}\{\\mathrm\{TP\}\+\\mathrm\{FN\}\},\\qquad F\_\{1\}=\\frac\{2\\operatorname\{Prec\}\\operatorname\{Rec\}\}\{\\operatorname\{Prec\}\+\\operatorname\{Rec\}\}\.\(23\)
The 95% interval uses the empirical 2\.5th and 97\.5th percentiles of the 64 generated trajectories\. For lower and upper bounds\(ℓ,u\)\(\\ell,u\), its Winkler score is

W0\.05\(y,ℓ,u\)=\(u−ℓ\)\+40\(ℓ−y\)𝟏\{y<ℓ\}\+40\(y−u\)𝟏\{y\>u\},W\_\{0\.05\}\(y,\\ell,u\)=\(u\-\\ell\)\+40\(\\ell\-y\)\\mathbf\{1\}\\\{y<\\ell\\\}\+40\(y\-u\)\\mathbf\{1\}\\\{y\>u\\\},\(24\)averaged over allC×48C\\times 48county\-hours\.

For Table[2](https://arxiv.org/html/2609.01896#S3.T2)\(b\), the point summary for each metric is selected on validation data and frozen before Michigan evaluation\. The median is used for normal RMSE, while the 80th percentile is used for average RMSE, tail RMSE, and nonzero F1\. Winkler uses the full sample\-based interval and does not depend on the point summary\.