Time-Aware Prior Fitted Networks for Zero-Shot Forecasting with Exogenous Variables

Andres Potapczynski; Ravi Kiran Selvam; Tatiana Konstantinova; Malcolm Wolff; Kin G. Olivares; Ruijun Ma; Michael W. Mahoney; Andrew Gordon Wilson

cs.LGMar 16, 2026

Time-Aware Prior Fitted Networks for Zero-Shot Forecasting with Exogenous Variables

Authors: Andres Potapczynski, Ravi Kiran Selvam, Tatiana Konstantinova, Malcolm Wolff, Kin G. Olivares, Ruijun Ma, Michael W. Mahoney, Andrew Gordon Wilson,

Organizations: Amazon Science, New York University · Amazon Science

Abstract

In many time series forecasting settings, the target time series is accompanied by exogenous covariates, such as promotions and prices in retail demand; temperature in energy load; calendar and holiday indicators for traffic or sales; and grid load or fuel costs in electricity pricing. Ignoring these exogenous signals can substantially degrade forecasting accuracy, particularly when they drive spikes, discontinuities, or regime and phase changes in the target series. Most current time series foundation models (e.g., Chronos, Sundial, TimesFM, TimeMoE, TimeLLM, and LagLlama) ignore exogenous covariates and make forecasts solely from the numerical time series history, thereby limiting their performance. In this paper, we develop ApolloPFN, a prior-data fitted network (PFN) that is time-aware (unlike prior PFNs) and that natively incorporates exogenous covariates (unlike prior univariate forecasters). Our design introduces two major advances: (i) a synthetic data generation framework that injects realistic temporal patterns, structural changes, and exogenous dependencies into the PFN prior; and (ii) time-aware architectural modifications that embed inductive biases needed to exploit temporal context. We demonstrate that ApolloPFN outperforms existing baselines across several forecasting benchmarks with exogenous information, including M5, electric price forecasting, UCI Air Quality, and Solar Energy datasets.

Figures & tables

Panel of Figure 1 — Figure 1: (a) Not using exogenous information can lead to catastrophic forecasting errors. We compare the predictions of ApolloPFN with and without exogenous information for the weekly sales of a real product from the M5 benchmark. Ignoring the price information leads the forecaster to predict a decreased demand via context parroting (brown), whereas the tracking of price dynamics helps the model focus on most up-to-date dynamics (pink). (b) Prior-data fitted networks such as TabPFN-TS fail to capture ordered patterns. We compare the prediction of TabPFN-TS and ApolloPFN for a synthetic time series that has a recurrent pattern of a ramp-up period before a promotion, a spike on the promotion, a ramp-down period, and then a subsequent decrease in demand. The exogenous promotion information is encoded as a binary indicator. Training data is to the left of the black line, and forecasts are to the right.

Panel of Figure 2 — Figure 2: Limitations of TabPFN-TS for time series forecasting and corresponding improvements in ApolloPFN . We provide illustrative examples of each of the identified limitations using real-world time series from different datasets: Tourism Monthly for (a), Tourism Yearly for (b), M5 Weekly for (c), and M1 Monthly for (d). In all plots, the in-context historical observations are shown to the left of the black vertical line, while the forecasting horizon lies to the right. (a) When TabPFN-TS is not provided with frequency-related features, its predictions tend to regress toward the marginal mean of the observed history (green line). Even when frequency features are available, the model may capture dominant periodic patterns while failing to represent higher-amplitude or less frequent deviations, such as pronounced spikes. (b) TabPFN-TS exhibits difficulties in extrapolating long-term trends, particularly in short-context settings where limited historical information is available. (c) In some cases, the predictions of TabPFN-TS collapse toward the most frequent observed value in the context, leading to overly conservative forecasts that fail to reflect the recent change in dynamics. (d) The predicted 90% prediction intervals of TabPFN-TS may expand substantially, reflecting uncertainty by covering previously observed values rather than accurately capturing uncertainty in future trend evolution.

Panel of Figure 3 — Figure 3: Overall synthetic data generation workflow of ApolloPFN. A stochastic root signal is propagated through a nonlinear structural causal graph to generate multivariate node trajectories, from which covariates and forecasting targets are selected to construct the final dataset $(X_{t},y_{t})$

Level	Model	M5(D-B)	M5(W-B)	M5(M-B)	M5(D-S)	M5(W-S)	M5(M-S)
State	AutoARIMAX	0.900	1.374	2.218	0.840	1.555	3.006
	Prophet	0.525	1.685	3.223	0.859	1.945	4.593
	Sundial-Base (0)	0.572	2.052	2.398	0.825	1.614	3.042
	Chronos-Large (0)	0.565	1.223	2.572	0.894	1.853	2.989
	Moirai-Large ${}^{(\dagger\text{x})}$	0.794	1.533	2.336	0.879	1.681	2.780
	TabPFN-TS ${}^{(0\text{x})}$	0.527	1.251	2.599	0.883	1.522	2.864

Table 1: M5 Competition. RMSSE results on M5 at state and store levels across different temporal aggregations. Lower is better. (D/W/M) denote Daily, Weekly, Monthly frequencies; (B/S) denote Category and SKU-level data. Best results are bold , second-best are underlined .

sCRPS	DE(24)	NP(24)	FR(24)	BE(24)	PJM(24)	DE(48)	NP(48)	FR(48)	BE(48)	PJM(48)
AutoARIMAX	$0.156$	$0.053$	$0.120$	$0.117$	$0.200$	$0.194$	$0.065$	$0.128$	$0.149$	$0.171$
Prophet	$0.064$	$0.032$	$0.073$	$0.080$	$0.129$	$0.069$	$0.040$	$0.075$	$0.099$	$0.088$
Sundial-Base (0)	$0.063$	$0.022$	$0.054$	$0.053$	$0.121$	$0.081$	$0.032$	$0.053$	$0.074$	$0.057$
Chronos-Large (0)	$0.059$	$0.022$	$0.048$	$\underline{0.049}$	$0.108$	$0.077$	$0.030$	0.047	$\underline{0.070}$	$0.055$
Moirai-Large ${}^{(\dagger\text{x})}$	$0.102$	$0.035$	$0.081$	$0.077$	$0.140$	$0.128$	$0.049$	$0.094$	$0.105$	$0.101$
TabPFN-TS ${}^{(0\text{x})}$	$\underline{0.043}$	0.018	$\underline{0.043}$	$0.054$	0.086	0.048	0.023	$\underline{0.048}$	0.069	0.049

Table 2: Electricity Price Forecasting. ApolloPFN achieves best or second best sCRPS results across most of the datasets and prediction horizons (24, 48). Lower values are better. The notation

{}^{(0\text{x})}

denotes zero-shot forecasters with exogenous variables;

{}^{(\dagger\text{x})}

denotes models trained with exposure to the datasets; and (0) denotes zero-shot univariate models without exogenous inputs. Best results are bold , second-best are underlined .

sCRPS	Air Quality	Solar Weather
AutoARIMAX	$0.160$	$0.712$
Prophet	$0.121$	$0.295$
Sundial-Base (0)	$0.118$	$0.142$
Chronos-Large (0)	$0.121$	$0.157$
Moirai-Large ${}^{(\dagger\text{x})}$	$0.121$	$0.346$
TabPFN-TS ${}^{(0\text{x})}$	$\underline{0.110}$	$\underline{0.135}$

Table 3: Multivariate datasets from other domains. ApolloPFN beats other neural forecasters for series from other domains as well. sCRPS results for Air Quality dataset and Solar with Weather dataset. Lower values are better. The notation

{}^{(0\text{x})}

denotes zero-shot forecasters that leverage exogenous information;

{}^{(\dagger\text{x})}

denotes forecasters that leverage exogenous information but were exposed to the data during training; and (0) denotes zero-shot univariate forecasters that do not use exogenous information. Best results for each dataset are bold , and second best are underlined .

sCRPS	M1(M)	M1(Y)	M3(M)	M3(O)	M4(D)	M4(M)	M4(Y)	Tou(M)	Tou(Y)
Sundial-Base	$0.157$	$0.183$	$0.121$	$0.047$	$\underline{0.026}$	$0.116$	$0.160$	$\underline{0.126}$	$0.174$
Chronos-Large	$0.173$	0.119	$0.113$	$0.036$	$0.028$	$0.108$	0.106	$0.155$	0.103
Moirai-Large	0.135	$0.210$	0.093	$0.035$	$0.033$	$0.117$	$0.187$	$0.275$	$0.275$
TabPFN-TS	$0.169$	$\underline{0.123}$	$0.106$	$\underline{0.035}$	$0.027$	$\underline{0.096}$	$0.115$	$0.203$	$0.146$
ApolloPFN (ours)	$\underline{0.152}$	$0.142$	$\underline{0.094}$	0.034	0.023	0.092	$\underline{0.113}$	0.084	$\underline{0.137}$

Table 4: ApolloPFN performance on the classical univariate benchmarks. Lower values are better. Best results for each dataset are bold , and second best are underlined .

Panel of Figure 4 — Figure 4: SRNGN graph generation algorithm used by ApolloPFN accelerates learning. We compare the test benchmark performance of our ApolloPFN model trained with the random growing network (RGN) algorithm and our Single Node Growing Network (SRNGN) algorithm at different training steps. With SRNGN, we achieve better performance at 20K iterations than at 80K with RGN.

Panel of Figure 5 — Figure 5: Ablation of data generation components in ApolloPFN . We evaluate the impact of incorporating time series root node excitation and SRNGN graph generation algorithm into the prior data generating process. Results show the relative change over the TabPFN-TS baseline across multiple forecasting benchmarks.

Figure 6: Architectural modifications in ApolloPFN for time series forecasting. Ablation on the use of RoPE positional encoding and full attention. We evaluate the effect of progressively introducing these architectural components in ApolloPFN and report relative change over the TabPFN-TS baseline across multiple forecasting benchmarks.

Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

Panel of Figure 7 — Figure 7: Example graphs from distinct graph generation algorithms.

Dataset	Source
M1	Makridakis and Hibon (1979)
M3	Makridakis and Hibon (2000)
M4	Makridakis et al. (2020)
Tourism	Hyndman et al. (2008)
M5	Makridakis et al. (2022)
Electricity Price	Lago et al. (2021b)

Appendix

Table 5: Data sources used for benchmarking.

Model	Hyperparameter Details	Code Repo
AutoARIMAX	Per-series fitting via statsforecast.AutoARIMA . Search: $p_{\max}{=}5$ , $q_{\max}{=}5$ , $d_{\max}{=}2$ , $P_{\max}{=}3$ , $Q_{\max}{=}3$ , $D_{\max}{=}1$ , season length based on frequency grain, stepwise with approximation. Covariates: exogenous regressors (NaN imputed with mean per covariate) + sin/cos seasonality features. Quantiles from prediction intervals.	statsforecast
Prophet	Per-series fitting with default hyperparameters. Interval width $=0.8$ . Covariates: all available features as additional regressors (NaN imputed with mean per covariate). Quantiles from MCMC posterior samples.	prophet
Sundial-base	Pre-trained thuml/sundial-base-128m (128M params). 100 sample trajectories via autoregressive generation. Quantiles computed empirically from samples. Univariate only (no covariates).	HuggingFace
Chronos-large	Pre-trained amazon/chronos-t5-large (710M params). T5 relative position encoding. Quantiles predicted directly via predict_quantiles() . Univariate only (no covariates).	chronos-forecasting
Moirai-large	Pre-trained Moirai-1.1-R-large (311M params). Patch size 16, 100 Monte Carlo samples. Quantiles computed empirically from samples. Covariates: Exogenous regressors.	uni2ts
TabPFN-TS	Pre-trained checkpoint tabpfn-v2-regressor-v2_default.ckpt from HuggingFace (11M params). 12-layer Transformer ( $d{=}192$ , 6 heads, FFN 768, no positional encoding, no dropout). Output: Bar Distribution (5000 bins). $z$ -normalization per series. Covariates: Fourier features + exogenous regressors. Preprocessing: QuantileTransformer, StandardScaler, SVD.	Internal 2 2 2 We implement the TabPFN-v2 architecture following the publicly available design specifications and initialize models directly from the released pre-trained checkpoints. We validate our implementation against the official tabpfn-time-series library, confirming agreement in outputs within numerical tolerance.

Appendix

Table 6: Hyperparameter and implementation details for all evaluated models. All models use input size 512 and predict quantiles at levels

\tau\in\{0.1,0.2,\ldots,0.9\}

Perturbation	Level	M5(D-B)	M5(W-B)	M5(M-B)	M5(D-S)	M5(W-S)	M5(M-S)
None	State	0.500	1.275	2.088	0.875	1.497	2.721
SNR (50%, medium)	State	0.512	1.630	2.227	0.878	1.544	2.819
SNR (50%, high)	State	0.576	1.726	2.436	0.888	1.564	2.867
Partial (50%)	State	0.657	1.795	2.455	0.896	1.566	2.868
Gaussian extra (50%)	State	0.503	1.317	2.183	0.873	1.494	2.739
AR(1) extra (50%)	State	0.511	1.312	2.006	0.874	1.507	2.777

Appendix

Table 7: Robustness of ApolloPFN to noisy and irrelevant exogenous features under controlled perturbations across various M5-state aggregation benchmarks

Dataset	20% PI		40% PI		60% PI		80% PI
	ApolloPFN	TabPFN-TS	ApolloPFN	TabPFN-TS	ApolloPFN	TabPFN-TS	ApolloPFN	TabPFN-TS
BE(24)	0.214	0.148	0.434	0.325	0.622	0.521	0.813	0.747
BE(48)	0.229	0.209	0.438	0.412	0.620	0.616	0.793	0.799
DE(24)	0.212	0.177	0.422	0.375	0.608	0.563	0.813	0.778
DE(48)	0.206	0.201	0.398	0.387	0.608	0.583	0.816	0.782
FR(24)	0.242	0.181	0.480	0.359	0.672	0.542	0.833	0.753

Appendix

Table 8: Average empirical coverage at different prediction interval (PI) levels for ApolloPFN and TabPFN-TS across EPF, Solar Energy, and UCI Air Quality datasets. Values closer to the nominal targets (0.2, 0.4, 0.6, 0.8) indicate better calibration.

Dataset	20% PI		40% PI		60% PI		80% PI
	ApolloPFN	TabPFN-TS	ApolloPFN	TabPFN-TS	ApolloPFN	TabPFN-TS	ApolloPFN	TabPFN-TS
BE(24)	0.014	0.053	0.034	0.075	0.022	0.079	0.013	0.053
BE(48)	0.029	0.009	0.038	0.012	0.020	0.016	0.008	0.001
DE(24)	0.012	0.023	0.022	0.025	0.008	0.037	0.013	0.022
DE(48)	0.006	0.001	0.002	0.013	0.008	0.017	0.016	0.018
FR(24)	0.042	0.019	0.080	0.041	0.072	0.058	0.033	0.048

Appendix

Table 9: Absolute deviation from nominal coverage, computed as

|\text{empirical coverage}-\text{nominal coverage}|

for prediction intervals at 20%, 40%, 60%, and 80% for ApolloPFN and TabPFN-TS across EPF, Solar Energy and UCI Air Quality datasets. Lower values indicate better calibration, i.e., closer alignment between predicted and target coverage levels.

Panel of Figure 8 — Figure 8: How TabPFN combines attention across features and samples. Taken from Hollmann et al. (2025) , the figure illustrates the main components of the TabPFN architecture, discussed in Equation 1 , plus the translation of the embedding into a Riemann distribution approximation of the PPD $p(y_{\text{test}}|\bm{x}_{\text{test}},\mathcal{D}_{\text{train}})$ .

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Future exogenous variables provide valuable information for forecasting endogenous time series. Existing covariate-aware methods primarily learn the direct influence of exogenous variables on endogenous variables. However, these effects can be complex and change with the pattern of the exogenous variables, making them difficult to capture. Beyond this perspective, we observe that a given exogenous pattern often co-occurs with only a small set of endogenous response patterns. These associations motivate a strategy that matches future and historical exogenous patterns and uses the corresponding endogenous patterns to enhance forecasting. However, in real-world forecasting scenarios with multiple exogenous variables, each exogenous variable provides a distinct dimension for matching, creating a dilemma for this strategy between precise matching and sufficient historical support. To bridge this gap, we propose XMatch (EXogenous MATCHing), a covariate-aware forecasting model that realizes the aforementioned strategy through a tree-structured matching process that adaptively adjusts the number of exogenous variables used as matching conditions. Specifically, we first introduce the ProtoTree Creator, which organizes historical correspondences between exogenous and endogenous patterns into a ProtoTree, whose deeper levels incorporate additional exogenous variables for matching. For forecasting, we then design the ProtoTree Matcher, which uses future exogenous variables to query the ProtoTree and adaptively determines how many exogenous variables to use for matching based on exogenous pattern similarity and historical support. Finally, the matched endogenous patterns are used as explicit historical evidence to enhance forecasting. Extensive experiments on 12 real-world datasets demonstrate that XMatch outperforms state-of-the-art baselines.

Abstract

Exogenous Time Series Forecasting

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