TIRE: Top-Down Formula Instantiation with Reliable Execution for Physics Problem Synthesis
Authors: Xiaoyu Wu, Zhaoxin Yu, Xinglin Xiao, Nan Xu
Organizations: ScienceOne AI · State Key Laboratory of Multimodal Artificial Intelligence Systems, Institute of Automation, Chinese Academy of Sciences · School of Artificial Intelligence, University of Chinese Academy of Sciences
Advancing physics reasoning in large language models requires scalable training data that connect physical laws, concrete problem scenarios, and verifiable numerical answers. Expert-designed problems offer top-down control over physical laws, conditions, and scenarios, but require costly human effort. LLM-based synthesis from texts scales efficiently, but model-generated answers and assessments leave supervision vulnerable to hallucinations. We introduce TIRE, a framework for Top-Down Formula Instantiation with Reliable Execution that brings knowledge-driven problem construction to large-scale data generation. Top-down instantiation uses reusable physical formulas and their applicability conditions to control the physical relations and numerical configurations of generated problems. Reliable execution establishes reference answers before problem generation, providing a common numerical foundation for problem construction and teacher distillation. Together, these mechanisms combine control over problem content with execution-grounded supervision for both supervised fine-tuning and reinforcement learning. Using TIRE, we construct TIRE-19k, spanning nine subfields of physics. Across 3 backbone LLMs and 4 benchmarks, TIRE-19k outperforms 5 representative datasets with equal amounts of SFT data, exceeding the strongest baseline by 2.86 percentage points on average. Further GRPO raises Qwen3-8B's average accuracy by 1.52 percentage points over its TIRE-19k SFT checkpoint. These results demonstrate the value of generating physics training data from explicit physical knowledge with reliable execution.
Figures & tables
Figure 1: Overview of TIRE. Physical knowledge conditions formula instantiation, which produces questions for teacher distillation. Blue blocks denote LLM-based modules; green blocks denote program-based operations. Solid arrows indicate data transfer, and dotted arrows indicate knowledge conditioning. Execution-derived answers support both question construction and teacher-answer verification; retained examples support student training.
Formula source
Instantiation
Formulas
Examples
PHYSICS
Direct
2,561
15,133
SciBench
Perturbed
492
4,011
Total
3,051
19,144
Table 1: Composition of the TIRE-19k training set. Formula counts are deduplicated within each source. Two records occur in both sources, so the total is their globally deduplicated union.
Figure 2: Subject distributions of formula records associated with TIRE-19k and training examples. Formula records can belong to multiple subjects, giving 3,069 subject memberships for 3,051 globally unique records; formula percentages use subject memberships as the denominator.
Figure 3: Question and answer length distributions in TIRE-19k. Dashed lines mark medians.
Training data
SciBench
PhysReason
OlympiadBench
TheoremQA
Average
MegaScience
56.71
66.17
26.69
32.46
45.51
PhysLogic
66.97
68.58
26.69
64.04
56.57
Spark
74.56
74.31
43.64
65.79
64.58
OpenSci
78.84
82.26
47.88
64.04
68.26
SCP-116k
79.22
82.62
43.64
69.30
68.70
TIRE-19k
83.76
82.99
51.27
71.05
72.27
Table 2: Comparison of training datasets on Qwen3.5-9B. All scores are percentages.
Configuration
SciB.
PhysR.
Olympiad.
TheoremQA.
Average
TIRE
81.29
84.10
49.58
71.05
71.51
w/o unit tests
79.69 (-1.60)
82.99 (-1.11)
47.46 (-2.12)
70.18 (-0.87)
70.08 (-1.43)
w/o perturbation
79.72 (-1.57)
83.18 (-0.92)
48.31 (-1.27)
69.30 (-1.75)
70.13 (-1.38)
w/o all
79.25 (-2.04)
81.33 (-2.77)
40.25 (-9.33)
70.18 (-0.87)
67.75 (-3.76)
Table 5: Ablation study of TIRE on Qwen3.5-9B, all scores are percentages.
Teacher
SciBench
PhysReason
OlympiadBench
TheoremQA
Average
Qwen3-235B-A22B
83.76
82.99
51.27
71.05
72.27
Qwen3.7-plus
81.76
84.66
47.46
71.05
71.23
Qwen3.8-flash
84.52
85.58
48.31
68.42
71.71
DeepSeek-v4.1-flash
82.74
85.95
46.61
64.04
69.84
Table 6: Sensitivity to the distillation teacher with Qwen3.5-9B as the student.
Training data
Setting
SciBench
PhysReason
Olympiad.
TheoremQA
Average
MegaScience
SFT
44.41
62.29
19.49
44.74
42.73
SFT → GRPO
46.09
65.25
16.95
42.11
42.60 (-0.13)
PhysLogic
SFT
61.01
62.85
23.31
52.63
49.95
SFT → GRPO
62.38
62.85
19.92
50.88
49.01 (-0.94)
SFT
78.08
78.19
40.68
69.30
66.56
TIRE-19k
SFT → GRPO
78.64
77.63
43.22
72.81
68.08 (+1.52)
Table 7: Qwen3-8B accuracy (%) after SFT and after further GRPO, with Average subscripts indicating absolute accuracy changes in percentage points from the corresponding SFT checkpoint.
Appendix figures & tables5 assets
Supplementary material from the paper’s appendix.
Appendix
Construction source
SciBench
PhysReason
Olympiad.
TheoremQA
Average
Qwen3.5-9B
PHYSICS + SciBench
83.76
82.99
51.27
71.05
72.27
PHYSICS only
82.01
83.73
48.73
72.81
71.82
Qwen3-8B
PHYSICS + SciBench
78.08
78.19
40.68
69.30
66.56
PHYSICS only
74.98
80.59
40.68
71.05
66.82
Appendix
Table 8: Evaluation after excluding SciBench-derived training data. Both variants contain 19,144 examples before the validation split. All scores are percentages; Average is the unweighted mean of the four benchmark accuracies.
Stage
Model and main settings
Formula extraction
DeepSeek-R1-250528; at most 16,000 output tokens
Formula filtering
DeepSeek-R1-250528; True/False judgment; temperature 0.6
Program and unit-test generation
DeepSeek-V3.1, thinking mode; temperature 0.6
Parameter generation
DeepSeek-V3.1, thinking mode; temperature 0.6
Parameter perturbation
Rule-based, ±5%
Question verbalization
DeepSeek-V3.1, thinking mode; temperature 0.6; at most 20,000 output tokens
Appendix
Table 9: Construction configuration of TIRE-19k.
Configuration
Setting
SFT optimization
Full-parameter ms-swift, 5 epochs; AdamW ( β1=0.9 , β2=0.95 ); learning rate 10−6 , cosine decay, 10% warmup; weight decay 0.1; gradient clipping 1.0; global batch size 2; bf16, ZeRO-2, no packing; the epoch-5 checkpoint is evaluated
SFT data use
Main dataset comparisons: 2% of each 19,144-example set is held out for validation (seed 42), leaving 18,762 training examples; the ablation datasets also contain 19,144 examples and use the shorter configuration in Appendix B.2
Maximum length and thinking mode
40,960 with thinking (TIRE-19k, SCP-116k, and the Qwen3.7-plus teacher in Table 6 ); 20,480 with thinking (OpenSci; Qwen3.8-flash and DeepSeek-v4.1-flash teachers in Table 6 ); 16,384 without thinking (Spark); 8,192 without thinking (MegaScience) or with thinking (PhysLogic); rationale given below the table
GRPO (Table 7 )
Qwen3-8B; each run starts from the epoch-5 SFT checkpoint of its dataset and trains for 3 epochs on the prompts whose four-sample solve rate under that checkpoint lies in (0,1) ; ms-swift GRPO with vLLM rollouts; binary reward (Equation 4 ); 8 rollouts per prompt, 128 completions per update; learning rate 10−6 (cosine, 10% warmup); 16,384 tokens, at most 14,336 generated; temperature 0.6, top-p 0.95, top-k 80; KL β=0.04 , clipping ϵ=0.2
Training hardware
SFT: 4 NVIDIA H200 GPUs (141 GB), some baselines on 2 GPUs at the same global batch size; ablations (Table 5 ): 8 H200 GPUs; GRPO: 8 H200 GPUs
Appendix
Table 10: Training configurations.
Benchmark
Subset
Questions
Scoring
SciBench
Physics: textbooks fund , class , and thermo of the official release
193
Numeric match with 5% relative tolerance; macro-average over the three textbooks
PhysReason
Text-only questions with a single boxed reference answer
Official MathJudger with the per-question tolerance of the dataset
TheoremQA
Physics questions without images
114
Numeric match: 4% relative tolerance for reals, exact for integers, lists compared after sorting
Appendix
Table 11: Benchmark subsets, question counts, and scoring.
Setting
SciBench
PhysReason
OlympiadBench
TheoremQA
Prompt
User prompt with unit sentence; zero-shot
System and user prompt; zero-shot
Official English open-ended prompt; zero-shot
System and user prompt; zero-shot
Reasoning mode
Thinking enabled for the backbones and for models fine-tuned on reasoning traces, including all TIRE models; disabled for models fine-tuned on MegaScience or Spark.
0.6 / 0.95; presence penalty 1.0; judge temperature 0
0.6 / 0.95; presence penalty 1.0
0.6 / 0.95; presence penalty 1.0
Output-token limit
Raised stepwise up to 60,000 tokens (server context 65,536; YaRN factor 2 for Qwen3-8B), regenerating only unfinished responses; responses still truncated are regenerated up to twice at 60,000 tokens. The judge uses at most 32,768 tokens.
Responses per question
One response per question; accuracy is the percentage of correct responses.
Answer extraction
Last \boxed{} after the reasoning block; scientific-notation normalization
Final response after the reasoning block, passed to the judge
Last \boxed{} after the reasoning block
Last \boxed{} after the reasoning block; numbers and lists parsed
Appendix
Table 12: Prompts, decoding, and answer processing.
Training large language models for complex reasoning is bottlenecked by the scarcity of verifiable, high-quality data. In domains like physics, standard text augmentation often introduces hallucinations, while static benchmarks lack the reasoning traces required for fine-tuning. We introduce the Infinite Problem Generator (IPG), an agentic framework that synthesizes physics problems with guaranteed solvability through a Formula-as-Code paradigm. Unlike probabilistic text generation, IPG constructs solutions as executable Python programs, enforcing strict mathematical consistency. As a proof-of-concept, we release ClassicalMechanicsV1, a high-fidelity corpus of 1,335 classical mechanics problems expanded from 165 expert seeds. The corpus demonstrates high structural diversity, spanning 102 unique physical formulas with an average complexity of 3.05 formulas per problem. Furthermore, we identify a Complexity Blueprint, demonstrating a strong linear correlation (R2≈0.95) between formula count and verification code length. This relationship establishes code complexity as a precise, proxy-free metric for problem difficulty, enabling controllable curriculum generation. We release the full IPG pipeline, the ClassicalMechanicsV1 dataset, and our evaluation report to support reproducible research in reasoning-intensive domains.
We have witnessed remarkable advances in LLM reasoning capabilities with the advent of DeepSeek-R1. However, much of this progress has been fueled by the abundance of internet question-answer (QA) pairs, a major bottleneck going forward, since such data is limited in scale and concentrated mainly in domains like mathematics. In contrast, other sciences such as physics lack large-scale QA datasets to effectively train reasoning-capable models. In this work, we show that physics simulators can serve as a powerful alternative source of supervision for training LLMs for physical reasoning. We generate random scenes in physics engines, create synthetic question-answer pairs from simulated interactions, and train LLMs using reinforcement learning on this synthetic data. Our models exhibit zero-shot sim-to-real transfer to real-world physics benchmarks: for example, training solely on synthetic simulated data improves performance on IPhO (International Physics Olympiad) problems by 5-10 percentage points across model sizes. These results demonstrate that physics simulators can act as scalable data generators, enabling LLMs to acquire deep physical reasoning skills beyond the limitations of internet-scale QA data. Code available at: https://sim2reason.github.io/.
Physics reasoning requires constructing a consistent model of the underlying physical system rather than relying solely on symbolic or formula-based manipulation. Although large language models have shown strong ability in solving math and coding problems, they still struggle with physics problems, as these problems entangle the physical modeling process with mathematical calculations. Humans approach physics by first building a representation of the system before performing calculations. Inspired by this, we introduce a unified framework that distills intermediate representations that explicitly encode the physical modeling process and adopt a two-stage post-training strategy, where supervised fine-tuning establishes structured modeling, and reinforcement learning with rubric-based feedback improves the quality of the modeling process. Experiments on multiple multimodal physics benchmarks show that our approach generally improves physical reasoning performance across different models and datasets. Across PhysReason, PhyX, and SeePhys, physical modeling outperforms GRPO by ~3% on average. showing that explicit physical modeling is an effective strategy for improving physics reasoning in small VLMs.
Ye Zhang, Xuehang Guo, Rui Pan +4
University of Pennsylvania · William & Mary · University of Illinois Urbana-Champaign +3