Question:
In the ablation experiments of Time-R1, which component plays the most decisive role in ensuring stable convergence and enabling the full advantage of reinforcement learning?
A. The reinforcement learning (RL) stage, which suppresses flawed reasoning paths and discovers higher-reward trajectories.
B. The supervised fine-tuning (SFT) warmup, which stabilizes output formatting and provides essential reasoning patterns for RL to build upon.
C. The structural similarity reward, which improves interpretability but has limited influence on accuracy.
D. The length reward, which ensures full-sequence generation and convergence stability but contributes minimally to forecasting quality.

Answer: A

arXiv:2506.10630v1  [cs.LG]  12 Jun 2025
Time Series Forecasting as Reasoning:
A Slow-Thinking Approach with Reinforced LLMs
Yucong Luo , Yitong Zhou, Mingyue Cheng*,
Jiahao Wang, Daoyu Wang, Tingyue Pan, Jintao Zhang
State Key Laboratory of Cognitive Intelligence
University of Science and Technology of China, Hefei, China
{prime666, yitong.zhou, jiahao.wang, wdy030428}@mail.ustc.edu.cn
{pty12345, zjttt}@mail.ustc.edu.cn, mycheng@ustc.edu.cn
Abstract
To advance time series forecasting (TSF), various methods have been proposed to
improve prediction accuracy, evolving from statistical techniques to data-driven
deep learning architectures. Despite their effectiveness, most existing methods still
adhere to a fast thinking paradigm-relying on extracting historical patterns and
mapping them to future values as their core modeling philosophy, lacking an explicit
thinking process that incorporates intermediate time series reasoning. Meanwhile,
emerging slow-thinking LLMs (e.g., OpenAI-o1) have shown remarkable multi-
step reasoning capabilities, offering an alternative way to overcome these issues.
However, prompt engineering alone presents several limitations—including high
computational cost, privacy risks, and limited capacity for in-depth domain-specific
time series reasoning. To address these limitations, a more promising approach is
to train LLMs to develop slow thinking capabilities and acquire strong time series
reasoning skills. For this purpose, we propose Time-R1, a two-stage reinforcement
fine-tuning framework designed to enhance multi-step reasoning ability of LLMs
for time series forecasting. Specifically, the first stage conducts supervised fine-
tuning for warmup adaptation, while the second stage employs reinforcement
learning to improve the model’s generalization ability. Particularly, we design a
fine-grained multi-objective reward specifically for time series forecasting, and then
introduce GRIP (group-based relative importance for policy optimization), which
leverages non-uniform sampling to further encourage and optimize the model’s
exploration of effective reasoning paths. Experiments demonstrate that Time-R1
significantly improves forecast performance across diverse datasets 1.
1
Introduction
Time series forecasting (TSF) [1, 2] plays a key role in data-driven decision-making across critical
domains, including financial market analysis, energy demand planning, and traffic flow management.
Over the years, numerous research efforts have been proposed to advance this field. Classical
statistical methods such as ARIMA [3], ETS [4], and Theta [5] have long been used to predict future
data by leveraging the statistical properties of single samples. Machine learning methods such as
XGBoost[6] and LightGBM[7] remain highly robust due to their interpretability and ability to model
nonlinear relationships. With the arrival of computing power era, deep learning-based approaches
[8, 9] have since gained prominence due to their ability to capture complex temporal patterns and
adapt to non-stationary real-world data. Methodological analysis covers pioneering architectures such
as sequence dependency modeling of RNNs [10, 11], TCNs [12, 13] and transformer-based models
[14], which improve generalization through shared representations across multiple time series.
1The corresponding author is Mingyue Cheng. The code is at https://github.com/lqzxt/Time-R1.
Preprint. Under review.
Time-Series
Forecasting
(I)
Statistical and Machine Learning Methods:
ARIMA, XGBoost,…
(II) Deep Learning Methods: PatchTST, iTransformer…
(III) Foundation Models: Moirai, Timer…
(IV) Time-R1
Black-box
Interpretability
High Generalization
Low Generalization
𝑥1:𝐿
𝑥𝐿+1:𝐿+𝐻
<Knowledge>
<Time Series Data>
<Explanation>
<Forecasted Time Series Data>
Statistical methods or machine
learning models
Deep Neural Networks: RNNs or
Transformer and their variants
Large foundation models,
trained on large amounts of data
Inference under a reinforced-
tuned language model, with
text serving as input/output.
Figure 1: Overview of evolution of TSF methods. Time-R1 is a novel, general forecasting paradigm.
Although the specific techniques vary, most existing TSF methods follow a similar "fast thinking"
paradigm [9, 15, 16]. Specifically focusing on single-step prediction accuracy [17], these methods
typically employ sequential models to encode historical values and use one-step decoding to directly
map past observations to future values [13]. Although effective in benchmarks, their underlying
logic is largely based on pattern recognition [8] and trend prediction, lacking an explicit reasoning
process. However, in real-world scenarios, time series often reflect more complex temporal logic,
which should not merely be ’fitted’—they should be understood and reasoned.
To address this issue, a growing body of research [18, 19] has explored leveraging the reasoning
capabilities of large language models (LLMs) to analyze temporal dynamics and generate high-quality
representations, thereby enhancing lightweight TSF models [20, 21]. These approaches benefit from
the ability of LLMs to incorporate contextual information such as textual metadata [22], offering
stronger generalization performance across diverse domains [23, 24], and produce interpretable
explanations that support forecasting decisions [25, 8]. However, despite their potential, current
LLM-based TSF methods face three key limitations: First, a partial misalignment of time series
domain knowledge, and limited reasoning capabilities. General linguistic knowledge in LLM often
mismatches the temporal patterns and causal mechanisms required for time series tasks, leading
to suboptimal performance [26, 27]. Second, a lack of generalization from experiential learning.
While effective in supervised memorization, they struggle with understanding dynamics or adapting
to new, unseen scenarios, which limits their out-of-distribution performance. Third, absence of
progressive reasoning. These models map history to future directly without detecting regime changes
or performing step-by-step inference, resembling fast (not deliberate) thinking for time series. These
issues lead to a central question: Can we improve time series forecasting performance by training
LLMs to acquire time series reasoning capabilities?
Motivated by the above question, we propose Time-R1, a novel LLM-based time series forecasting
framework that trains large language models to acquire slow-thinking reasoning capabilities for
forecasting tasks. At its core, Time-R1 leverages LLMs as the time series reasoning backbone and
introduces a two-stage reinforcement fine-tuning (RFT) optimization framework: First, we begin
with warm-up supervised fine-tuning. The model is fine-tuned for memorization, learning both
effective reasoning patterns and accurate output formatting using synthetic reasoning trajectories
that demonstrate step-by-step temporal analysis. Second, the model is refined through reinforcement
learning for generalization, using fine-grained, multi-objective rewards specifically designed for
forecasting tasks, improving temporal coherence and multi-horizon accuracy. Notably, we propose
GRIP (Group-based Relative Importance for Policy Optimization), which optimizes LLM reasoning
paths in TSF through a uniform sampling strategy and adaptive weighting. Extensive experiments are
conducted on real-world datasets, showing that Time-R1 effectively enhances forecast performance
through the slow thinking paradigm. Our main contributions are as follows:
• We introduce time series reasoning by training LLMs to adopt a slow-thinking paradigm
that generates reasoning processes supporting final forecasting.
• We design a two-stage RFT framework (SFT for memorization, and RL for generalization)
that enhances the reasoning ability of LLMs. We introduce a fine-grained, multi-objective
reward specifically for TSF, along with a novel sampling strategy for RL optimization.
• Extensive experiments demonstrate the effectiveness of Time-R1, showing it enhances LLM
reasoning and improves generalization and explainability via deliberate slow thinking.
2
Preliminaries
2.1
Problem Definition
Let D = {(Xi, yi)}n
i=1 be a temporal dataset, where each Xi ∈Rt×m is a multivariate time series
with t steps and m channels, and yi ∈Rh×d contains d-dimensional targets over h future steps. The
2
The ETTh1 dataset is about
electricity power system…
Domain Knowledge
Date            HUFL
2017-10-28 00:00
12.25
2017-10-28 01:00
11.45
Please give me complete data
for next … dates.
You must follow …
Time Steps and Variables
Task Guidelines
+
+
=
Prompt Building
Base Language
Model
Supervised Fine-tuning
Reinforcement Learning
G
k
Multi-objective
Rule-based RM
SFT
Model
Template
Rollout
KL
G
G
Reward
Advantages
RFT
Model
Inference and Evaluation
Generalization
Warmup
SFT
Model
RFT
Model
Slow-thinking LLM
<think>
</think>
thinking process
<answer>
</answer>
explanation
Training prompt by
applying template
Prompt
<think>
</think>
thinking process
<answer>
</answer>
explanation
MSE
MAE
In-Domain
Out-of-Domain
Figure 2: A diagram illustrating the three steps of Time-R1: (1) building a training template with
domain context, time steps, and variables, (2) collecting long-CoT data from DeepSeek-R1 using the
template to train a supervised policy, and (3) optimizing the policy via reinforcement learning with
group-based relative importance for policy optimization (GRIP) to enhance TSF reasoning capability.
forecasting task learns a mapping fθ : Rt×m →Rh×d capturing temporal dependencies in D. Under
Time-R1, the forecasting procedure using prompt template P is: T i = LLMϕ(P, Xi), where T i is
the LLM’s textual output, and ˆyi = g(T i) parses it into numerical predictions ˆyi ∈Rh×d.
2.2
Time Series Reasoning
Although time series forecasting methods have advanced rapidly, most existing approaches rely
on pattern recognition and lack genuine time series reasoning — the ability to dynamically infer
future values by integrating observed historical data with structured logic or domain knowledge.
Traditional statistical [4, 3] and classical machine learning models [7, 6] are inherently limited in
capturing high-level temporal semantics, while deep learning models [14, 12, 10], despite their
expressive power, often function as "black boxes" without offering clear reasoning paths. Recently,
LLM-based methods [19, 18, 28] have attempted to address this gap by incorporating language-like
interpretability and contextual understanding. However, these models are typically adapted from
general-purpose LLMs not specifically designed for temporal reasoning, and they heavily depend on
prompt engineering rather than training on forecasting tasks, limiting their capacity for deliberate,
step-by-step temporal inference.
With the emergence of reasoning-oriented LLMs such as OpenAI-o1/o3 [29] and DeepSeek-R1 [30],
there is a promising path toward more structured and deliberative forecasting through mechanisms
akin to "slow thinking". Nevertheless, these models are computationally intensive, not tailored
for temporal tasks, and often unsuitable for privacy-sensitive domains like healthcare. Therefore,
prompt engineering alone is insufficient to unlock the full potential of LLMs in TSF. To overcome
these limitations, there is a pressing need for a dedicated framework that explicitly trains LLMs on
structured forecasting tasks, endowing them with domain-specific slow-thinking capabilities and
robust time series reasoning skills.
3
The Proposed Time-R1
3.1
Time-R1 Overview
Time-R1 consists of a two-stage RFT framework for LLM-based time series forecasting, built
upon a structured training template that standardizes input representations and encodes task-specific
knowledge. In the first stage, we perform warm-up SFT using synthetic chain-of-thought trajectories
to teach the model effective temporal analysis and accurate output formatting. These trajectories are
generated under strict guidelines and refined iteratively to align with ground-truth forecasts. The
second stage further improves the model via RL, guided by a fine-grained, multi-objective reward
3
function tailored for time series forecasting. To optimize reasoning paths during RL, we introduce
GRIP (Group-based Relative Importance for Policy Optimization), a novel strategy that leverages
non-uniform sampling and adaptive weighting to balance accuracy, logical consistency, and temporal
coherence. An overview of the framework is provided in Figure 2.
3.2
Training Template
Our training template employs an instructional framework that standardizes inputs while encoding
task-specific knowledge through five components: (1) Task Definition establishing objectives and
problem scope; (2) Dataset Description specifying temporal characteristics and application scenarios;
(3) Channel Information delineating input signal types; (4) Testing Data providing timestamps and
historical series; and (5) Format Instruction defining output templates. This design interleaves domain
knowledge with structural constraints, reducing inference-time formatting ambiguities (see Table 1).
Table 1: Training template for Time-R1. Contents enclosed in {} will be replaced with specific
information, including dataset description, channel information, and historical time-series data.
Here is the {Channel Information} data of the ETTh1 dataset. The ETTh1 dataset is {Dataset
Description}. I will now give you data for the past 96 recorded dates, and please help me
forecast the data for the next 96 recorded dates. The data is as follows: {Historical Time Series
Data}. Please give me the complete data for the next 96 recorded dates. Remember to give me
the complete data. You must first conduct reasoning inside <thinking>...</thinking>. When
you have the final answer, you can output the answer inside <answer>...</answer>.
3.3
Supervised Fine-tuning for Warmup Adaptation
To mitigate the linguistic readability degradation and slow convergence caused by direct reinforcement
learning on LLMs—particularly due to formatting inconsistencies—we first perform a warmup stage
via SFT. This warmup SFT step is designed to stabilize training, ensure proper output formatting, and
equip the model with basic reasoning capabilities, without requiring deep time series understanding.
Our SFT data construction involves three key steps. First, we leverage DeepSeek-R1 to generate
time-series predictions on the training set by feeding it historical time series data paired with strict
formatting guidelines. We then select the optimal prediction for each sample based on the Mean
Absolute Percentage Error metric [31]. Next, to derive a reasoning process aligned with ground-truth
labels, we inject both the true prediction value and the high-quality CoT generated in the previous
step into DeepSeek-R1 as prompts, guiding it to synthesize a revised CoT that logically culminates
in the correct prediction. Finally, we concatenate the refined CoT and the true prediction value,
demarcating the final answer using ‘<answer>‘ tags to create structured training data for SFT. The
full data collection procedure is detailed in Algorithm 1 in the Appendix.
After constructing the training data, we perform a single-epoch fine-tuning with a small learning
rate. This warm-up SFT phase effectively prepares the model for subsequent reinforcement learning,
ensuring stable learning dynamics and accurate output formatting. It also enables the model to
internalize reasoning patterns from synthetic trajectories, laying the foundation for more deliberate
and coherent decision-making in later RL stages.
3.4
Reinforcement Learning for Effective Reasoning Patterns
After warmup SFT, we further fine-tune the LLM using RL to generalize its reasoning and acquire
slow-thinking capabilities for time series forecasting. In the following sections, we first present the
reward design, and then describe the employed reinforcement learning algorithm GRIP.
3.4.1
Reward Design
To effectively apply RL for optimizing the proposed slow-thinking time series forecasting, we
introduce several fine-grained and multi-objective reward functions specifically designed to enhance
forecasting performance and slow thinking behavior.
Format Rewards.
To ensure syntactic validity and completeness of the generated reasoning paths,
we define two reward components to enforce both structural integrity and output completeness:
4
Format Reward: A binary penalty is imposed if the output does not follow the required structured
format (e.g., missing or malformed <answer> tags):
γFormat =
0
if valid <think>, <answer> tags and proper structure
−1
otherwise
(1)
Length Reward: To encourage full time point sequence generation and accelerate convergence, we
provide positive feedback based on how close the generated sequence length is to the ground truth:
γLength =



0.1
if len(answer) ≥len(ground_truth)
0.1 ·
len(answer)
len(ground_truth)
otherwise
(2)
Accuracy Rewards.
We define two accuracy-based reward components to assess numerical preci-
sion and temporal fidelity, encouraging accurate value prediction and modeling of dynamics:
MSE Reward: We compute the mean squared error between normalized prediction and target
sequences and map it into a bounded reward signal using a sigmoid transformation:
γMSE =

1 −
1
1 + e−0.3·MSE

· 2,
(3)
Seasonal-Trend Decomposition Reward: Both predicted and true sequences are decomposed into
seasonal (s) and trend (t) components via moving average-based methods, where si the seasonal
component and ti the long-trend component. We then compute separate MSE terms for each:
γSeasonal = 1
n
n
X
i=1

strue
i
−spred
i
2
, γTrend = 1
n
n
X
i=1

ttrue
i
−tpred
i
2
,
(4)
Structural Similarity Reward.
We evaluate structural similarity by matching predicted and ground-
truth extrema within tolerance windows, ensuring change-point capture and interpretable patterns,
with correct matches receiving credit:
γCP =
Ncmax
Ngmax
· 0.2

+
Ncmin
Ngmin
· 0.2

,
(5)
where Ncmax and Ncmin respectively represent the counts of correctly identified local maxima and
minima within a tolerance window, Ngmax and Ngmin are the total ground-truth extrema counts.
The overall reward for RL training is the sum of the above rewards.
3.4.2
Reinforcement Learning Algorithm: GRIP
We introduce GRIP (Group-based Relative Importance for Policy Optimization), a general RL
optimization method designed to optimize entire trajectories for LLM time series forecasting reasoners.
The GRIP objective function, formalized in Equation 6, combines a non-uniform sampling strategy
with adaptive trajectory weighting within a policy gradient framework. In the following sections,
we elaborate on its core components:(1) GRIP formalization; (2) non-uniform sampling strategy to
balance exploration and exploitation; and (3) an adaptive weighting scheme that enhances gradient
signals from high-quality reasoning paths.
Formalization of the GRIP Objective.
The GRIP objective integrates the two key design compo-
nents into a unified policy gradient framework, as formalized in Equation 6:
JGRIP(θ) =E
q∼P (Q),
{oj}k·G
j=1∼πθold(o|q),
{oi}G
i=1∼Sample({oj}k·G
j=1;R(oj))
(
G
X
i=1
wU
i
1
|oi|
 |oi|
X
t=1
min
 πθ(oi,t|q, oi,<t)
πθold(oi,t|q, oi,<t)Ai,
clip
 πθ(oi,t|q, oi,<t)
πθold(oi,t|q, oi,<t), 1 −ϵ, 1 + ϵ

Ai

−βDKL[πθ||πref]
)
,
(6)
5
q
Policy
Model
Reward
Model
Non-uniform
Sampling
o1
…
o2
oG
r1
…
r2
rG
Reference
Model
KL
GRIP
Group
Computation
⨂
Weighted
Average
Trained
Frozen
G
k
o
G
…
A1
A2
AG
G
k
r
Figure 3: Overview of Group-based Relative Importance for Policy Optimization (GRIP).
where, ϵ and β are hyperparameters. πref is the reference model, typically initialized as the pre-trained
model before reinforcement learning begins. The output {oi} is selected through a sampling process
from policy πθold. The hyperparameter k controls the size of the rollout space, while G referred to as
the group size. DKL represents the KL divergence, which is incorporated into the loss function as
a regularization term during training. And Ai is the advantage computed using a group of rewards
{r1, r2, . . . , rG} corresponding to the completion trajectories within each group:
Ai = ri −mean({r1, r2, . . . , rG})
std({r1, r2, . . . , rG})
,
(7)
The weight wU
i denotes the adaptive weighting assigned to each trajectory. This objective balances
exploration and exploitation while mitigating gradient dilution. The sampling strategy and adaptive
weight will be discussed in the following section.
Non-uniform Sampling Strategy.
To bridge the gap between reasoning and forecasting in time
series modeling, recent RL methods like GRPO have shown promise. However, they often suffer from
an exploration-exploitation imbalance. To address this, GRIP introduces a non-uniform sampling
strategy that first generates k · G candidate trajectories {oj}k·G
j=1 from policy πθold (where k scales
exploration and G is group size), then selects G elite trajectories via reward-weighted sampling
Sample({oj}; R(oj)). These are replicated to form the update set {oi}G
i=1, maintaining GRPO’s up-
date scale while emphasizing high-reward regions. Mathematically equivalent to importance-sampled
policy gradient correction, this approach balances broad exploration with computational efficiency
through its dual-phase design. To further generalize, GRIP supports two sampling strategies:
(1) Local Random Sampling: For each input question q, we first generate k candidate trajectories
{oj}k
j=1 by independently sampling from the old policy πθold. The trajectory with the highest
reward o∗= arg max1≤j≤k R(oj) is selected as the elite sample. This process is repeated G
times to construct the final set {oi}G
i=1. This strategy emphasizes deterministic exploitation of the
top-performing sample at each iteration while maintaining computational efficiency.
(2) Cluster-based Random Sampling: For each q, we generate k · G candidate trajectories {oj}k·G
j=1.
These trajectories are clustered based on their rewards (e.g., reward-binning or K-means clustering),
and G trajectories are randomly sampled across clusters to ensure diversity in the final update set. This
method balances exploration and exploitation by preserving low-reward but potentially informative
samples while still prioritizing high-reward paths.
Adaptive Weighting for Gradient Enhancement.
Traditional uniform weighting like GRPO
(1/G) across trajectories fails to account for inter-trajectory quality disparities, leading to misleading
gradients from low-quality samples and diminished signals for high-quality ones. GRIP addresses
this by assigning trajectory-specific weights via softmax:
wU
i =
exp(ˆxq,oi)
PG
j=1 exp(ˆxq,oj)
,
(8)
where the completion score ˆxq,oi can be flexibly configured. For example, when ˆxq,oi = R(oi), the
weighting amplifies the influence of high-reward trajectories. This adaptive weighting suppresses
noise from low-quality outputs and strengthens gradients from critical trajectories.
6
4
Experimental Setup
Datasets and Evaluation Metrics.
To ensure comprehensive evaluation across diverse scenarios,
we conduct experiments in nine datasets spanning multiple domains with distinct temporal char-
acteristics and data attributes (detailed in Table 4). These include: the ETT dataset [32] capturing
2016-2018 electricity load records, Exchange [33] tracking 1990-2016 foreign exchange rates, Wind
[33] with 2020-2021 wind measurements captured, AQ [34] providing four-year air quality data,
NASDAQ [35] provides complete stock market series including opening/closing prices, trading
volumes, and daily high-low values. All datasets were evaluated using Mean Squared Error (MSE)
and Mean Absolute Error (MAE) under a 96-step prediction setting, except NASDAQ which uses a
36-step configuration, with results reported as the MSE between predictions and ground truth. Data
statistics is listed in Appendix E.
Baselines.
Our baselines include competitive methods: PatchTST [36], DLinear [37], FEDformer
[38], iTransformer [39], Autoformer [40], and TimeXer [41]. We further incorporate LLMs-based
approaches, CrossTimeNet [42], GPT4TS [43], TimeLLM [27], and DeepSeek-R1 [30] for zeroshot.
Implementation Details.
Unlike traditional TSF methods that often require normalization, we
conduct experiments in the original numerical space. For Time-R1, we use Qwen2.5-7B-Instruct as
the backbone. In SFT, we train on 300 synthetic samples with a learning rate of 5e-5 for one epoch.
In RL, we implement GRIP using the Verl framework [44] with vLLM for generation. In Eq. 6,
ϵ = 0.2 and β = 0.04; group size G = 16, k = 3. The batch size is 16, learning rate is 1e-6, policy
temperature is 1, and max completion length is 3000. Both stages are run on a 4-GPU A800 cluster.
For DeepSeek-R1, we apply its prompt directly to time series prediction without training. Time-R1
is trained only on ETTh1 and generalized to other datasets without fine-tuning, whereas baseline
methods require separate models per dataset.
5
Experimental Results
5.1
Main Results
Table 2: Performance comparison of Time-R1 and baseline models with best values in bold and
second-best underlined. MSE ↓is used as the evaluation metric. Deepseek-R1 denotes zero-shot
using our reasoning template. Overall, Time-R1 achieves superior performance across dataset.
Methods
ETTh1
ETTh2
ETTm1
ETTm2
Exchange
AQWan
AQShunyi
Wind
NASDAQ
Traditional
PatchTST
9.3001
10.9735
16.3864
5.8375
0.0009
12436.8256
16693.3076
2024.8256
0.0007
DLinear
7.6954
10.4067
13.9395
7.9100
0.0014
20997.7228
20952.4161
1619.4311
0.0007
FEDFormer
15.9794
16.5863
36.0975
19.3084
0.0015
29705.7327
55307.0403
3245.5183
0.0009
iTransformer
7.5048
10.0161
12.7511
5.7713
0.0010
13482.5746
18219.6612
1591.6404
0.0008
AutoFormer
7.3876
13.9167
15.2816
7.6855
0.0012
19628.2867
23576.0196
1673.8570
0.0008
TimeXer
8.5213
11.4268
14.0023
5.7325
0.0009
14397.1884
16491.3209
1684.9856
0.0007
LLM-based
CrossTimeNet
8.3125
11.6789
16.3475
6.7924
0.0011
15120.0853
18042.1278
1931.2672
0.0012
GPT4TS
6.9928
9.7971
15.8238
5.7014
0.0009
13546.1725
16839.0718
1790.3269
0.0010
TimeLLM
6.8780
9.9814
15.8845
5.6695
0.0010
13427.4982
16665.2379
1575.8937
0.0011
DeepSeek-R1
6.7098
11.3845
14.8561
7.0063
0.0026
29653.1218
30780.9011
4047.1201
0.0021
Ours
Time-R1
5.8752
8.7093
13.1034
5.6673
0.0007
13033.1820
16150.5556
1353.9381
0.0007
We implemented the Time-R1 framework on nine datasets. The comparison with baseline models is
summarized in Table 2. A more comprehensive list of results for metrics such as MAE can be found
in Appendix Table 5. Key observations are as follows:
(1) Limitations of Traditional Methods and LLM Baselines. Traditional deep learning-based
forecasting models, such as PatchTST, DLinear, and iTransformer, achieve reasonable performance
but are limited by their one-step "fast thinking" paradigm, which struggles with complex temporal
dependencies and high-level reasoning. LLM-based methods like TimeLLM perform better by
leveraging the reasoning abilities of LLMs, especially for long-term and non-linear patterns. However,
they still treat forecasting as a direct generation task without explicit step-by-step reasoning, leading
to potentially inconsistent or logically flawed predictions. Moreover, their reliance on pre-trained
knowledge with minimal task-specific adaptation limits their explainability in the forecasting process.
7
Table 3: Ablation study on Training Strategies, Reward Design, and Template Components.
Method
ETTh1
ETTm2
Wind
MSE
MAE
MSE
MAE
MSE
MAE
Full Model
Time-R1
5.8752
1.2325
5.6673
1.3771
1353.9381
15.1095
Training
Strategies
w/o SFT
6.3558
1.4278
6.3673
1.4850
1632.6491
16.7903
w/o RL
13.2196
1.7820
12.5940
3.7759
3424.0485
28.1392
Reward
w/o Length
5.8781
1.2325
6.3210
1.4024
1358.5169
15.4612
w/o MSE
10.0948
1.5614
9.7449
2.4865
2749.2582
21.0272
w/o Seasonal Decomposition
6.0132
1.2403
6.0220
1.3859
1462.5454
15.8056
w/o Trend Decomposition
7.4775
1.3429
6.6881
1.4523
1766.9750
16.1789
w/o Structural Similarity
7.8558
1.4278
8.5940
1.7759
2316.0214
22.3412
Template
Components
w/o Timestamps
9.9146
1.5446
8.7454
2.4867
2816.0214
26.3412
w/o domain context
6.2286
1.3989
6.3517
1.5319
1639.4436
17.0182
(2) Performance Improvement and Advantage of Time-R1. Our proposed Time-R1 follows a
two-stage optimization framework. In the first stage, CoT-guided SFT is performed using synthetic
reasoning trajectories that explicitly encode time series characteristics and task constraints. This
enables the model to learn structured output formats and basic reasoning logic. In the second stage, we
further enhance the model’s reasoning capabilities through GRIP, a reinforcement learning framework
with fine-grained reward mechanisms. These include logical consistency, temporal coherence, and
multi-horizon accuracy, which serve as feedback signals to iteratively refine the model’s reasoning
paths. Experimental results show that this approach not only significantly improves overall forecasting
performance but also enhances generalization under zero-shot and out-of-distribution settings.
5.2
Ablation Study
Impact of Supervised Fine-tuning and Reinforcement Learning.
Next, we evaluate the necessity
of CoT-based SFT by comparing two training strategies: (i) direct RL without SFT, and (ii) SFT
followed by RL. As illustrated in Figure 4(b), the model trained without SFT suffers from slower
convergence and inferior performance, especially in early training stages. In contrast, initializing RL
with a well-aligned SFT model significantly accelerates learning and leads to better final performance.
This demonstrates that SFT provides a strong foundation for reasoning path generation, which is then
further refined through rule-augmented reinforcement learning.
Furthermore, we conducted an ablation study by completely removing RL (see Table 3). The results
demonstrate a significant degradation in TSF performance, with absolute performance drops on
the ETT and Wind dataset respectively, highlighting RL’s crucial role in optimizing SFT-initialized
reasoning paths. This finding indicates that while SFT establishes fundamental reasoning pat-
terns, RL provides indispensable optimization through the following mechanisms: (1) discovering
higher-reward reasoning trajectories through exploration, and (2) suppressing plausible-yet-incorrect
reasoning paths via reward shaping. RL proves to be a critical factor in achieving SOTA performance.
Impact of Multi-objective Reward Design.
We analyze the impact of each reward term in re-
inforcement learning by training models with partial reward components. As shown in Table 3,
removing any term degrades performance, indicating that all contribute to forecasting accuracy. The
largest drops occur when MSE or Seasonal-Trend Decomposition rewards are removed, emphasizing
the importance of point-wise precision and temporal structure. While Format and Length rewards
have smaller effects on numerical metrics, they ensure output consistency and training stability. The
Structural Similarity reward further enhances structural fidelity, especially for complex sequences.
More experiments are in Appendix F.2.
Impact of Training Template Component.
Finally, we assess how different elements of our
structured prompts affect model behavior. We consider two main components: (i) explicit timestamp
encoding, and (ii) contextual information such as seasonal period and task constraints. Table 3
show that incorporating these components consistently improves both forecasting accuracy and
generalization capability, especially under zero-shot and out-of-distribution scenarios. Models
without timestamp information struggle to capture long-range dependencies, while those lacking
contextual guidance often produce logically inconsistent outputs.
8
(a) GRIP vs. GRPO
(b) RL vs. SFT + RL
(c) Base vs. Instruct
(d) Model Scaling
Figure 4: Figure 2: (a) GRIP vs. GRPO: GRIP converges faster with slightly higher final performance.
(b) RL vs. SFT+RL: SFT+RL achieves faster initial convergence and superior final performance. (c)
Base vs. Instruct: Instruct model enables faster early reward growth, though base model achieves
higher final reward. (d) Model Scaling: Larger models show steeper reward improvement curves.
5.3
Performance Comparison w.r.t Different RL Optimization
We compare the performance of GRIP using two different sampling strategies — Local Random Sam-
pling, and Cluster-based Random Sampling — against GRPO, a commonly used policy optimization
method in reasoning-based reinforcement learning. As illustrated in Figure 4(a), the Cluster-based
Random Sampling strategy achieves the highest overall performance, slightly outperforming GRPO.
This is attributed to its ability to maintain diversity in trajectory selection by clustering samples
based on reward values, which helps preserve potentially informative yet low-reward reasoning paths
often ignored by greedy methods. In terms of convergence speed, Cluster-based Sampling also
leads, followed by Local Sampling, and finally GRPO, which converges the slowest. Although local
Sampling explores a larger search space, it tends to overfit high-reward trajectories early on, leading
to relatively poor generalization and suboptimal performance.
5.4
Performance Comparison w.r.t Different Model Types
We analyze the training dynamics of Time-R1 across model types, comparing base and instruct
models. Using two Qwen2.5 variants, Qwen2.5-7B-Base and Qwen2.5-7B-Instruct. Figure 4(c)
shows the base model converges more slowly and starts from a lower performance level. However, it
demonstrates stronger learning potential and eventually achieves slightly better results. This suggests
that while instruction tuning accelerates early learning in time series reasoning, iterative RL-based
optimization enables the base model to reach marginally superior performance.
5.5
Performance Comparison w.r.t Different Model Sizes
To evaluate the scaling behavior of Time-R1, we conduct experiments using models with 1.5B, 3B,
and 7B parameters on TSF tasks. As shown in Figure 4(d), forecasting performance consistently
improves with increasing model size. The 1.5B model achieves reasonable results on simple datasets
but struggles with complex temporal patterns. In contrast, the 3B model demonstrates significantly
better accuracy and generalization, while the 7B model achieves the best overall performance,
particularly in capturing long-term dependencies and handling out-of-distribution scenarios. These
results indicate that larger models can substantially enhance temporal reasoning capabilities.
5.6
Visualization of the Reasoning Process
Long-CoT SFT
Long-CoT SFT + RL
Pretend Thinking, Pseudo-Reasoning
MSE:2.516
Slow Thinking, Deep Understanding
MSE:1.372
Question:
Here is the High Useful Load data of the ETTh1 dataset.
I will now give you the past 96 recorded dates, and please help me forecast the next 96 recorded dates. The data is as follows: <Time Series Data>
<think>
the content should look like a thought process.
Step 1: Acknowledge the meta-request.
Step 2: Formulate a generic plan.
Step 3: Conclude the thinking process.
</think>
<answer>
<Time Series Data>
</answer>
<think>
To forecast the next 96 dates, I would use time series forecasting models
such as ARIMA or Prophet. These models are suitable for time series
data and can handle trends and seasonality effectively.
</think>
<answer>
<Explanation>
<Time Series Data>
</answer>
Only RL
Poor Readability
MSE:1.753
<think>
To the next 96 dates, There is of a trend: daily values
from 7.4 on July 1 to 12.7 on July Peaks occur at
each day, but overall, the highest are trending over
the four …
</think>
<answer>
<Time Series Data>
</answer>
Figure 5: A reasoning case study of long-CoT SFT, RL, and Hybrid Methods on ETTh1 dataset.
As shown in Figure 5, our case study highlights key differences among training paradigms. SFT
enables imitation of reasoning patterns but often results in superficial replication, leading to flawed
9
logic and suboptimal performance. Pure RL achieves reasonable accuracy but generates think with
poor readability. In contrast, the SFT+RL paradigm not only teaches extended reasoning effectively
but, through its reinforcement phase, also improves prediction accuracy while helping the model
identify which reasoning components most contribute to performance gains.
Time-R1
GPT4TS
Autoformer
TimeXer
TimeLLM
Time-R1 Only RL
iTransformer
PatchTST
Time-R1 Only SFT
Figure 6: Illustration of forecasting showcases comparing Time-R1 and baseline models. The look-
back window and the predicted window are set to 96 for the ETTh2 dataset.
5.7
Visualization of Prediction Results
In this visualization (Figure 6), Time-R1 is compared against six baseline methods, including
LLM-based approaches (GPT4TS and TimeLLM) and traditional models (PatchTST, iTransformer,
Autoformer, and TimeXer). Time-R1 consistently demonstrates more accurate and smoother predic-
tions, closely aligning with the ground truth. In contrast, the version of Time-R1 trained with only
SFT performs poorly, yielding subpar forecasting results. On the other hand, Time-R1 with only RL
achieves improved performance, outperforming the baselines to some extent.
6
Conclusion
In this work, we proposed Time-R1, a generative time series forecasting framework that enables
LLMs to perform deliberate reasoning for improved prediction. We introduced time series reasoning
by training LLMs to adopt a slow-thinking paradigm, generating explainable intermediate reasoning
steps before producing final forecasts, which achieves state-of-the-art TSF performance. Time-
R1 consists of a two-stage RFT framework, which first employs CoT-based SFT on synthetic
reasoning data to instill multi-step temporal analysis capabilities in the model. This is followed
by a novel RL phase with fine-grained, multi-objective reward signals specifically design for TSF.
Notably, GRIP (group-based relative importance for policy optimization), a policy optimization
method was proposed to leverage non-uniform sampling to refine reasoning paths and improve
forecasting accuracy. Additionally, by open-sourcing the training code, we provide the broader
research community and society with practical access to a time series reasoning solution, enabling all
to benefit from these advancements.
10
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14
Appendix
A
Related Work
A.1
TimeSeries Forcasting
Time series forecasting has evolved from classical models like ARIMA, effective under ideal condi-
tions [45, 3], to modern deep learning approaches. While ARIMA offers theoretical guarantees, it
struggles with real-world data complexities. Machine learning methods [6, 7] remain highly robust
due to their interpretability and ability to model nonlinear relationships. The advent of deep learn-
ing introduced sequence-to-sequence models such as Recurrent Neural Networks (RNNs), which
initially captured temporal dynamics well [10, 11]. However, RNNs face limitations like restricted
receptive fields and error accumulation [11]. Advanced architectures incorporating self-attention
and convolutional networks have since been developed to capture long-range dependencies [33, 46].
Concurrently, integrating traditional techniques like trend-seasonal decomposition into neural net-
works has improved performance [47, 48]. Notably, even simple linear networks enhanced with
decomposition strategies can achieve competitive results [37]. Additionally, slice-based methods
have shown promise in long-term forecasting by segmenting time series for better accuracy [36, 49].
These advancements blend classical principles with deep learning to tackle the challenges of TSF.
A.2
LLM-based TimeSeries Forcasting
In recent years, large language models (LLMs) have attracted attention for their ability to understand
and generate human-like text, now extending into time series analysis [50–52]. The application
of LLMs in this field primarily follows two approaches: fine-tuning and prompt-based zero-shot
learning. Fine-tuning involves further training pre-trained LLMs on specific time series data to
improve performance [18–21], though it requires significant labeled data and computational resources.
Conversely, prompt-based zero-shot methods utilize the model’s existing knowledge through carefully
designed prompts, avoiding task-specific training [22, 53]. While more flexible and resource-efficient,
these methods may not match the performance of fine-tuned models, especially in specialized tasks
[54, 9]. Both paradigms illustrate the growing interest in using LLMs for time series, despite
challenges in optimizing their effectiveness for such tasks.
A.3
Large Language Models and Reinforcement Learning
Reinforcement Learning (RL) [55] allows an agent to learn decision-making through interactions with
its environment, aiming to maximize cumulative rewards. RLHF introduced RL to LLMs via human
feedback [56, 57], initially training a reward model on human preferences and then using it for tuning
the policy LLM, often with Proximal Policy Optimization (PPO). However, PPO’s complexity due
to multiple optimization rounds poses challenges. To simplify this, methods like Direct Preference
Optimization (DPO) [58] and SimPO [59] have been proposed, offering computational efficiency
but suffering from off-policy issues (Pang et al., 2024). Another approach, Group Relative Policy
Optimization (GRPO) [60], avoids a critic model by estimating baselines from group scores, while
RLOO [61] uses a simplified REINFORCE-style framework. Despite these advances, applying RL to
enhance LLM-driven reasoning for time series forecasting tasks is still underexplored.
B
Preliminaries and Analysis of RL in LLMs
B.1
Reinforcement Learning in LLMs
To leverage Reinforcement Learning (RL) for optimizing Large Language Models (LLMs) in Natural
Language Processing (NLP) tasks, the initial and crucial step is to formulate the LLM’s generation
process as a Markov Decision Process (MDP) [62, 56]. This involves clearly defining the fundamental
components of an RL framework: the agent, the environment, states, actions, and rewards. In this
context, the LLM itself can be viewed as the agent, interacting with an environment that encompasses
the task it is trying to solve (e.g., text generation, question answering). The agent’s goal is to learn a
policy that maximizes a cumulative reward signal, which reflects how well it performs the given task.
As shown in Figure 7, within this MDP framework, the operationalization of RL concepts is specifi-
cally tailored to the sequential token-by-token generation characteristic of LLMs. At any discrete
15
Agent
Env
𝐴𝑡
𝑅𝑡
𝑆𝑡
𝑆𝑡+1
𝑅𝑡+1
Language Model
Prompt
𝐴𝑡
𝐴𝑡+1
𝑅𝑡
𝑅𝑡+1
𝑉𝑡
𝑉𝑡+1
…
𝑆𝑡
𝑆𝑡+1
𝑆𝑡+2
RL
RL in LLM
A token in response
Use preview to predict next
𝐴𝑡
Action
𝑅𝑡
Reward
𝑉𝑡
Value
𝑆𝑡
State
Figure 7: Modeling large language models with reinforcement learning.
time step t, the system’s current condition is captured by the state st. This state typically comprises
the initial prompt concatenated with the sequence of tokens generated up to that point; thus, the
initial state s0 consists solely of the prompt. From a state st, the LLM executes an action at, which
manifests as the selection and generation of the subsequent token from its predefined vocabulary V .
The set of all possible tokens constitutes the action space, whose cardinality is denoted as |V |. The
selection of a particular action at is governed by the agent’s policy π(at|st), representing a probability
distribution over the vocabulary conditional on the current state st. Following the execution of action
at, the system undergoes a state transition to st+1. In the context of LLM generation, this transition
is deterministic, defined by the concatenation st+1 = [st, at]. Crucially for the learning process, a
reward signal rt is provided, and a corresponding value function Vt can be estimated. These elements
are contingent upon the specific NLP objective and serve to quantify the desirability of the generated
outputs or intermediate actions, thereby guiding the optimization of the LLM’s policy.
B.2
Reinforcement Learning with Verified Reward
Group Relative Policy Optimization.
The GRPO method [60] diverges from typical approaches
by not requiring a critic model, which often has a comparable size to the policy model. Instead,
GRPO calculates the baseline using scores obtained from a group of generated outputs.
Specifically, for every question q sampled from the dataset distribution P(Q), GRPO first employs
the existing policy model πθold to produce G completions, denoted as {o1, o2, · · · , oG}. Subsequently,
the policy model πθ is optimized by maximizing a defined objective:
JGRP O(θ) = Eq∼P (Q),{oi}G
i=1∼πθold(o|q)
(
1
G
G
X
i=1
1
|oi|
|oi|
X
t=1
n
min
h πθ(oi,t|q, oi,<t)
πθold(oi,t|q, oi,<t)Ai,
clip
 πθ(oi,t|q, oi,<t)
πθold(oi,t|q, oi,<t), 1 −ϵ, 1 + ϵ

Ai
i
−βDKL [πθ||πref]
o)
,
(9)
DKL [πθ||πref] = πref(oi,t|q, oi,<t)
πθ(oi,t|q, oi,<t) −log πref(oi,t|q, oi,<t)
πθ(oi,t|q, oi,<t) −1,
(10)
In this formulation, ϵ and β function as hyperparameters. The reference model is denoted by πθref,
typically representing the model’s initial state before reinforcement learning is applied. Furthermore,
Ai signifies the advantage, calculated from a set of rewards {r1, r2, . . . , rG} that correspond to the
various completions within each group:
Ai = ri −mean({r1, r2, . . . , rG})
std({r1, r2, . . . , rG})
.
(11)
Rule-based Reward Function.
Rather than relying on an auxiliary trained reward model, GRPO
employs a rule-based system for reward computation. This system calculates the total reward ri for a
given output oi by aggregating two distinct components, as formalized in Equation 12:
2ri = Rformat(oi) + Raccuracy(oi),
(12)
16
Here, the first component, the format reward Rformat(oi), serves to ensure that the output adheres to
the required structural specifications. The second component, the accuracy reward Raccuracy(oi), is
designed to assign substantially higher values to responses that are correct and precise.
B.3
Analysis of GRPO
Analysing Training Cost and Performance Trade-off.
Equation 9 reveals a linear relationship
between GRPO’s training overhead and the number of sampled completions. This is primarily due to
the necessity of computing probability distributions over all completions for the policy, reference, and
old policy models. Taking the DeepSeek-Math experiment as an example, sampling 16 completions
per question requires 48 forward passes (16 × 3), leading to a sharp increase in computational cost.
Experimental results (Figure 8) show that increasing the number of completions can improve model
MSE and MAE on the ETTh1 dataset, with diminishing returns in performance gains. More critically,
reducing the number of completions to lower computational load significantly degrades the reasoning
capability of the Qwen2.5-7B-Instruct model, highlighting the limitations of conventional approaches.
Analyzing Optimization Opportunities from Contribution Heterogeneity.
Recent work [63]
reveals substantial heterogeneity in the contribution of individual completions to training effectiveness.
A small subset of high-value samples can provide optimization signals up to tens of times stronger than
ordinary ones. This non-uniform distribution opens new avenues for improving training efficiency:
by dynamically identifying and prioritizing high-contribution samples, it becomes possible to reduce
computational overhead to as low as one-third or even one-fifth of the original level while maintaining
model performance (see Section C for detailed optimization strategies). These findings not only
explain the efficiency bottlenecks in existing methods but also lay a theoretical foundation for the
design of adaptive sampling strategies in future work.
Analysising Completion Contribution.
To evaluate the contribution of each completion to the
training of the policy model, we analyze the derivative structure of the objective function in Eq. (9)
with respect to the model parameters θ as:
∇θJGRP O(θ) = E[q∼P (Q),{oi}G
i=1∼πθold(O|q)]
(
1
G
G
X
i=1
1
|oi|
|oi|
X
t=1
"
∇θ
 πθ (oi,t|q, oi,<t)
πθold (oi,t|q, oi,<t)Ai

−β

∇θ
πref (oi,t|q, oi,<t)
πθ (oi,t|q, oi,<t) −∇θ log πref (oi,t|q, oi,<t)
πθ (oi,t|q, oi,<t)
 #)
=E[q∼P (Q),{oi}G
i=1∼πθold(O|q)]
(
1
G
G
X
i=1
1
|oi|
|oi|
X
t=1
"
∇θπθ (oi,t|q, oi,<t)
πθold(oi,t|q,oi,<t)
Ai
+ β
πref (oi,t|q, oi,<t) ∇θπθ (oi,t|q, oi,<t)
π2
θ (oi,t|q, oi,<t)
−∇θπθ (oi,t|q, oi,<t)
πθ (oi,t|q, oi,<t)
 #)
=E[q∼P (Q),{oi}G
i=1∼πθold(O|q)]
(
1
G
G
X
i=1
1
|oi|
|oi|
X
t=1
"
πθ (oi,t|q, oi,<t)
πθold (oi,t|q, oi,<t)Ai
+ β
πref (oi,t|q, oi,<t)
πθ (oi,t|q, oi,<t) −1
 #
∇θπθ (oi,t|q, oi,<t)
πθ (oi,t|q, oi,<t)
)
=E[q∼P (Q),{oi}G
i=1∼πθold(O|q)]
(
1
G
G
X
i=1
1
|oi|
|oi|
X
t=1
"
πθ (oi,t|q, oi,<t)
πθold (oi,t|q, oi,<t)
|
{z
}
Probability ratio
Ai
|{z}
Advantage
+ β
πref (oi,t|q, oi,<t)
πθ (oi,t|q, oi,<t) −1

|
{z
}
KL divergence constraint
#
∇θ log πθ (oi,t|q, oi,<t)
|
{z
}
Policy model gradient
)
.
(13)
17
Figure 8: Completion number vs. (left) MSE ↓and (right) MAE ↓. Experiments are conducted on
ETTh1 using Qwen2.5B-7B-Instruct.
This analysis reveals four key factors influencing policy updates:
1. Advantage, which assesses the value of a completion in improving expected returns through the
advantage function. A higher advantage indicates stronger reward alignment, making the completion
more influential in guiding policy updates.
2. Probability ratio, wich compares the likelihood of an action under the current policy πθ to that
under the old policy πθold. It amplifies actions favored by the new policy and suppresses those
preferred by the old one, guiding the policy toward higher rewards. A higher ratio signifies greater
confidence in the action, influencing the optimization process more significantly. This term is crucial
for identifying high-value completions when combined with the advantage function.
3. KL divergence, which measures the deviation of the current policy from the reference model. It
enforces stability during training by penalizing excessive changes, but does not directly contribute to
reasoning pattern formation.
4. Policy model gradient, which indicates the direction of parameter updates.
Previous research [64] has shown that removing the KL divergence constraint does not significantly
affect the model’s reasoning performance, as the core learning signal primarily comes from the
advantage term aligned with rewards. Furthermore, we decompose the core expression for policy
updates into two components: the probability ratio and the advantage term. For a completion to
make a significant contribution to training, both components must have substantial values. If either of
them is close to zero, the overall contribution will also be negligible. By removing the KL divergence
term and decoupling its regularization effect, we derive the simplified form of GRPO’s objective
derivative as:
∇θJGRP O(θ) ≈E[q∼P (Q),{oi}G
i=1∼πθold(O|q)]
(
1
G
G
X
i=1
1
|oi|
|oi|
X
t=1
h πθ (oi,t|q, oi,<t)
πθold (oi,t|q, oi,<t)
|
{z
}
Probability ratio
·
Ai
|{z}
Advantage
i
∇θ log πθ (oi,t|q, oi,<t)
|
{z
}
Policy model gradient
)
,
(14)
This simplified formulation focuses on the reward-driven learning signal while preserving the essential
gradient dynamics required for effective policy optimization. With the help of this simplified
expression, we can evaluate the advantage value of each completion before the full model computation
is carried out. Specifically, if a completion exhibits a very small absolute advantage value, its
contribution to the policy update is negligible. We can filter out such low-value completions at an
early stage, thereby avoiding redundant forward and backward computations.
To further improve training efficiency and learning effectiveness, we could introduce a sampling strat-
egy based on reward values or advantage estimates. Unlike uniform sampling across all completions,
we prioritize those with higher advantage values for inclusion in the training process. These samples
typically contain stronger learning signals and are more valuable for policy updates. By increasing
the sampling probability of these high-impact completions during batch construction—or through
18
upsampling techniques—we not only reduce computational overhead but also significantly accelerate
convergence and enhance final model performance.
This approach ensures efficient training iterations while maintaining the quality of policy updates,
achieving a favorable balance between computational cost and learning effectiveness. As a result, it
enables a dual improvement in both training efficiency and model capability.
C
Group-based Relative Importance for Policy Optimization
We introduce GRIP (Group-based Relative Importance for Policy Optimization), a general RL op-
timization method designed for optimizing entire trajectories generated by LLMs as time series
forecasting reasoners. Unlike GRPO, which linearly increases inference cost with the number of com-
pletions due to uniform sampling and equal weighting, GRIP adopts a non-uniform sampling strategy
that selects a small subset of high-reward trajectories from a larger candidate pool, significantly
reducing forward passes. Furthermore, GRIP employs adaptive weighting to amplify gradient signals
from informative samples, enabling more efficient learning from high-value completions. This design
not only reduces compute burden but also mitigates the diminishing returns of increased sample
count, thereby enhancing both training efficiency and forecasting accuracy. The GRIP objective
function, formally defined in Equation 15, operates within a policy gradient framework and integrates
a non-uniform sampling strategy with adaptive trajectory weighting.
JGRIP(θ) =E
q∼P (Q),
{oj}k·G
j=1∼πθold(o|q),
{oi}G
i=1∼Sample({oj}k·G
j=1;R(oj))
(
G
X
i=1
wU
i
1
|oi|
 |oi|
X
t=1
min
 πθ(oi,t|q, oi,<t)
πθold(oi,t|q, oi,<t)Ai,
clip
 πθ(oi,t|q, oi,<t)
πθold(oi,t|q, oi,<t), 1 −ϵ, 1 + ϵ

Ai

−βDKL[πθ||πref]
)
.
(15)
where ϵ and β are hyperparameters. πref is the reference model, typically initialized as the pre-trained
model before reinforcement learning begins. The output {oi} is selected through a sampling process
from policy πθold. The hyperparameter k controls the size of the rollout space, while G referred to as
the group size. DKL represents the KL divergence, which is incorporated into the loss function as
a regularization term during training. And Ai is the advantage computed using a group of rewards
{r1, r2, . . . , rG} corresponding to the completion trajectories within each group. The weight wU
i
denotes the adaptive weighting assigned to each trajectory. This objective balances exploration and
exploitation while mitigating gradient dilution. The sampling strategy and adaptive weight will be
discussed in the following section.
GRIP Pipeline.
To elucidate its operational mechanics, the GRIP algorithm is implemented through
a structured pipeline. The distinct stages of this process are outlined as follows:
(1) The old policy model samples k groups of G completions for each question, a total of k · G.
(2) The reward function computes the reward for each completion (Sec. 3.4.1).
(3) GRIP non-uniformly samples G completions based on rewards to balance exploration and
exploitation (Sec. 3.4.2).
(4) The advantage of each completion is calculated, and adaptive weighting is employed to assign
greater significance to high-quality reasoning paths among these completions (Sec. 3.4.2).
(5) Subsequently, the policy model is updated, with its gradient signals effectively formed as a
weighted average of the selected completions, reflecting this assigned path-dependent significance.
GRIP differs significantly from GRPO in both its initial sampling strategy and the mechanism for
weighting completion contributions during policy updates. GRPO typically generates G completions
directly and employs an arithmetic mean when processing their outcomes. In contrast, GRIP first
explores a broader set by sampling k · G completions, from which G are subsequently selected for
the update phase. Critically, while GRPO’s use of an arithmetic mean implies equal consideration
for its G samples, GRIP’s policy update leverages an adaptive weighted average. This ensures that
high-quality reasoning paths within the selected G completions exert a more substantial influence on
the gradient signals, thereby fostering more robust and effective learning.
19
D
CoT-based Supervised Fine-tuning Data collection
Algorithm 1 Three-stage CoT data construction for OURS
1: Input: T := Set of training time series samples with historical data and ground truth labels
2: Output: DSFT := Structured Chain-of-Thought dataset for SFT
3: DSFT ←∅
Initialize the output dataset
4: for t ∈T do
5:
xhist ←ExtractHistorical(t)
6:
Extract historical time series input
7:
y(1)
pred, . . . , y(k)
pred ←DeepSeek-R1(xhist; strict formatting)
8:
Generate k candidate predictions
9:
MAPEi ←ComputeMAPE(y(i)
pred, ytrue) for each candidate
10:
Evaluate using MAPE
11:
y∗←SelectMinMAPE({y(i)
pred})
12:
Select best prediction
13:
CoT(1), . . . , CoT(k) ←DeepSeek-R1(prompt = xhist, ytrue, CoT of y∗)
14:
Prompt to generate reasoning paths based on ground truth label
15:
CoT∗←SelectHighQuality({CoT(i)}, ytrue)
16:
Select CoT aligned with ground truth label
17:
sample ←Concatenate(<think>, CoT∗, <\think>, <answer>, ytrue, <\answer>)
18:
Combine reasoning and true answer
19:
DSFT ←DSFT ∪{sample}
20:
Add to final dataset
21: end for
E
Data Statistics
We presents the statistical characteristics of our datasets, shown in table 4.
Table 4: Statistics information of experimental datasets.
Dataset
Domain
Timestamps
Features
Frequency
ETTh1&ETTh2
Electricity
17,420
7
1 hour
ETTm1&ETTm2
Electricity
69,680
7
15 mins
AQWan&AQShunyi
Environment
35,064
11
1 hour
Exchange
Economy
7,588
8
1 day
Wind
Energy
48,673
7
15 mins
NASDAQ
Stock
1,244
5
1 day
F
Additional Result and Analysis
F.1
Main Results
Due to page limitations, we present the main results using the MAE evaluation metric here in Table 5.
Compared to both traditional methods and LLM-based approaches, Time-R1 achieves competitive
improvements.
F.2
Analysis about Reward Design
Distance Metric in Accuracy Reward.
To investigate the impact of different distance metrics on
the accuracy reward during the reinforcement learning phase, we conducted experiments on multiple
real-world time series forecasting tasks using the Exchange, AQShunyi, and NASDAQ datasets,
shown in Figure 9. Specifically, we evaluated Mean Squared Error (MSE), Mean Absolute Error
(MAE), Dynamic Time Warping (DTW), and Mean Absolute Percentage Error (MAPE) as accuracy
20
Table 5: Performance comparison of Time-R1 and baseline models with best values in bold and
second-best underlined. MAE ↓is used as the evaluation metric. Deepseek-R1 denotes zero-shot
using our reasoning template. Overall, Time-R1 achieves superior performance across dataset.
Methods
ETTh1
ETTh2
ETTm1
ETTm2
Exchange
AQWan
AQShunyi
Wind
NASDAQ
Traditional
PatchTST
1.6376
2.0076
1.9446
1.3973
0.0200
39.5089
42.5427
20.4654
0.0213
DLinear
1.5116
2.0855
1.7815
1.9136
0.0256
51.7896
50.0134
18.6036
0.0217
FEDFormer
2.2584
2.3487
2.5178
2.7729
0.0247
56.3129
50.3959
24.2476
0.0218
iTransformer
1.5126
1.9295
1.6609
1.4598
0.0204
40.3070
42.5116
18.1864
0.0237
AutoFormer
1.5945
2.3246
2.0989
1.7696
0.0230
48.4358
51.4833
19.6086
0.0229
TimeXer
1.5596
2.0895
1.7785
1.4419
0.0195
41.8280
41.4929
18.4876
0.0233
LLM-based
CrossTimeNet
1.5492
2.1731
1.9863
1.5281
0.0217
43.6267
46.0941
20.6205
0.0252
GPT4TS
1.4299
1.9134
1.9635
1.4943
0.0202
40.2815
42.3969
18.4919
0.0256
TimeLLM
1.4286
1.9143
1.9626
1.4926
0.0201
40.2746
42.3926
18.4890
0.0255
DeepSeek-R1
1.4316
2.0165
1.8876
1.6289
0.0209
64.9627
57.6412
30.6469
0.0231
Ours
Time-R1
1.2325
1.7192
1.6510
1.3771
0.0161
39.1846
46.5032
15.1095
0.0217
reward signals within our GRIP optimization framework. The experimental results demonstrate
that MSE consistently outperformed other metrics, followed by MAE, while DTW and MAPE
showed relatively limited improvements. MSE penalizes the squared prediction errors, making it
more sensitive to outliers and thereby guiding the model to focus on overall consistency between
predictions and ground truth values. This leads to enhanced stability in multi-step forecasts. In
contrast, although MAE is more robust due to its linear response to errors, it suffers from less
concentrated gradient updates, affecting convergence efficiency. DTW, despite its ability to handle
temporal misalignment, introduces computational complexity and asymmetry issues, making it
challenging to integrate effectively into end-to-end training. Additionally, MAPE can suffer from
numerical instability when target values are close to zero, limiting its practical utility in training.
In conclusion, we recommend prioritizing MSE as the primary accuracy reward metric during the
reinforcement learning optimization phase to achieve superior predictive accuracy and reasoning
stability. Auxiliary reward terms may be incorporated based on specific task requirements to further
enhance model robustness.
Exchange Dataset
AQShunyi Dataset
NASDAQ Dataset
Figure 9: Experimental Results Display Base on MSE, MAE, DTW, MAPE Distance Metrics.
F.3
Impact of Predicted Window Length
The results in Table 6 demonstrate that Time-R1 consistently outperforms TimeLLM across multiple
datasets and prediction horizons, particularly in long-sequence rolling forecasting tasks. Our approach
employs a rolling prediction strategy: first using the historical 96 time steps to predict the subsequent
96 steps, and then recursively feeding the predicted values back as input to forecast further into the
future. In the ETTh1 dataset, Time-R1 achieves an MSE of 6.4146 and an MAE of 1.2693 under
the 192-step forecasting window, substantially lower than TimeLLM’s MSE of 7.7990 and MAE
of 1.4989, highlighting its superior capability in multi-step rolling prediction. Under the longer
336-step forecasting window, Time-R1 obtains an MSE of 6.6886, still significantly outperforming
TimeLLM’s MSE of 9.3575, which indicates that the model maintains high accuracy and stability
even after multiple recursive prediction steps.
21
Table 6: Time series forecasting results of Time-R1 comparing with SOTA baseline TimeLLM across
four datasets. All results are averaged from 3 different predicted window of {96, 192, 336}.
Method
Predicted window
Time-R1
TimeLLM
MSE
MAE
MSE
MAE
ETTh1
96
5.8752
1.2325
6.8780
1.4286
192
6.4146
1.2693
7.7990
1.4989
336
6.6886
1.3288
9.3575
1.5617
ETTh2
96
8.7093
1.7192
9.9814
1.9143
192
9.4605
1.8868
10.4066
2.0755
336
14.3501
2.2914
15.7851
2.6351
ETTm1
96
13.1034
1.6510
15.8845
1.9626
192
15.0766
2.2640
16.5843
2.2904
336
18.0009
2.3299
19.8010
2.5629
ETTm2
96
5.6673
1.3771
5.6695
1.4926
192
6.8172
1.5331
7.1581
1.5264
336
8.5726
1.7476
9.0012
1.7224
F.4
Case Study: an Input and Output Example of Time-R1
We provide a complete input-output example of Time-R1 for reference, with some time steps omitted,
shown in Table 7.
G
Limitations
Despite significant advancements in enhancing the model’s ability to capture high-frequency temporal
patterns and improve time series reasoning, challenges remain due to computational limitations that
prevent experimentation with larger open-source models. Additionally, extending the length of the
prediction window is not feasible. Consequently, our method may encounter inaccurate predictions
when dealing with complex temporal structures or long-term forecasting requirements.
H
Broader Impact
This paper presents work whose goal is to advance the field of Machine Learning. There are many
potential societal consequences of our work, none which we feel must be specifically highlighted.
22
Table 7: Case study of complete input-output example of Time-R1.
Input:
Here is the High UseFul Load data of the transformer.I will now give you data for the past 96
recorded dates, and please help me forecast the data for next 96 recorded dates. The data is as
follows:
“‘
date
HUFL
2016-07-01 00:00:00
5.827
2016-07-01 01:00:00
5.693
2016-07-01 02:00:00
5.157
2016-07-01 03:00:00
5.090
2016-07-01 04:00:00
5.358
2016-07-01 05:00:00
5.626
2016-07-01 06:00:00
7.167
2016-07-01 07:00:00
7.435
2016-07-01 08:00:00
5.559
2016-07-01 09:00:00
4.555
. . .”’
Please give me the complete data for the next 96 recorded dates, remember to give me the
complete data. You must first conduct reasoning inside <think> ...</think>. When you have the
final answer, you can output the answer inside <answer>...</answer>.
Output:
<think>Summary of HUFL 96-Hour Forecast Reasoning.
1. Basic Patterns and Trends in the Data: The HUFL dataset spans 96 hours from July 1 to July
4, 2016, with values ranging from 4.2 to 12.7. Daily maximums increase from 7.4 to 12.7 over
four days, showing an upward trend despite varying peak times.
2. Observed Seasonality or Daily Fluctuations: Daily patterns show lower values at night
(00:00–5:00) and higher values during mornings and midday. Morning peaks occur between
6–8 AM, midday peaks around 11 AM–noon, both increasing daily. Evening rises (6–10 PM)
also show growth but with some irregularity in peak timing and magnitude.
3. Forecasting Methods and Assumptions Made: Forecasting assumes daily seasonality and
a slowing upward trend based on observed data. Using July 4th as a base, each subsequent
day’s hourly values are incremented by small amounts (0.1–0.3 or 2–5%), more for peak hours.
Trend may plateau after recent highs; alternative patterns like reversals were considered but not
confirmed due to limited data.
4. Potential Limitations or Uncertainties: Only four days of data limit identification of long-
term trends or weekly cycles. Trend continuation is speculative, external factors are unknown,
and inherent variability reduces forecast accuracy.
Conclusion: The forecast extends observed daily seasonality and recent upward trends, incre-
mentally adjusting July 4th’s values. Despite maintaining daily fluctuation structures, the short
data window and lack of validation mean the forecast is an informed estimate. </think>
<answer>
“‘
date
attr
2016-07-05 00:00:00
11.989
2016-07-05 01:00:00
12.525
2016-07-05 02:00:00
12.324
2016-07-05 03:00:00
10.717
2016-07-05 04:00:00
11.320
2016-07-05 05:00:00
10.851
2016-07-05 06:00:00
13.329
2016-07-05 07:00:00
11.454
2016-07-05 08:00:00
11.052
2016-07-05 09:00:00
10.985
... ”’
</answer>
23