Organizations: Fudan University, China · Shanghai Artificial Intelligence Laboratory, China · University of Cambridge, UK · Nankai University, China · UNC-Chapel Hill, US · University of Oxford, UK
Recovering analytical solutions of physical fields from visual observations is a fundamental yet underexplored capability for AI-assisted scientific reasoning. We study visual-to-symbolic analytical solution inference (ViSA) for two-dimensional linear steady-state fields: given field visualizations (and first-order derivatives) plus minimal auxiliary metadata, the model must output a single executable SymPy expression with fully instantiated numeric constants. We introduce ViSA-R2 and align it with a self-verifying, solution-centric chain-of-thought pipeline that follows a physicist-like pathway: structural pattern recognition solution-family (ansatz) hypothesis parameter derivation consistency verification. We also release ViSA-Bench, a VLM-ready synthetic benchmark covering 30 linear steady-state scenarios with verifiable analytical/symbolic annotations, and evaluate predictions by numerical accuracy, expression-structure similarity, and character-level accuracy. Using an 8B open-weight Qwen3-VL backbone, ViSA-R2 outperforms strong open-source baselines and the evaluated closed-source frontier VLMs under a standardized protocol.
Figure 2: Overall pipeline for ViSA dataset construction and evaluation.
Figure 3: Overall pipeline for feature extraction, evidence matching, parameter inference, and synthesize high-quality reasoning chain.
VLM
LLM-only
Model
Overall
Char
Struct
Num
Overall
Char
Struct
Num
GPT-5.2 ( OpenAI, 2025 )
0.474
0.290
0.768
0.370
0.200
0.273
0.323
0.295
Grok-4-1-Fast ( xAI, 2025a )
0.242
0.169
0.613
0.048
0.116
0.217
0.195
0.176
Claude-Haiku-4-5 ( Anthropic, 2025a )
0.316
0.204
0.638
0.167
0.159
0.221
0.261
0.221
ViSA-R2 (Ours)
0.512
0.482
0.860
0.385
–
–
–
–
Table 1: Overall comparison in VLM vs. LLM settings. We report the OverallScore and its three components for both VLM inputs and LLM-only inputs under the standardized protocol in § 4.1 . We only include models that successfully return valid outputs for more than 90% of the test instances.
Model
Input
Overall
Success
ViSA-R2 framework
Vision
0.512
100.0%
Claude-Haiku-4-5 ( Anthropic, 2025a )
Vision
0.316
93.3%
Claude Sonnet 4.5 ( Anthropic, 2025c )
Text
0.352
63.3%
Gemini-3-Pro ( Google DeepMind, 2025 )
Vision
0.572
60.0%
Claude Opus 4 ( Anthropic, 2025b )
Text
0.135
26.7%
DeepSeek-R1 ( DeepSeek, 2025 )
Text
0.915
10.0%
Table 2: Comparison of representative models by overall score and success rate.
Variant
Char
Struct
Num
Overall
Qwen-3-8B-baseline
0.191
0.616
0.210
0.328
w/o CoT alignment
0.016
0.053
0.017
0.024
w/ CoT alignment (ViSA-R2)
0.482
0.860
0.385
0.512
ViSA-R2 framework
0.480
0.842
0.780
0.669
Table 3: Ablation B. CoT alignment and coefficient refinement.
Category
Scenario Name
Operator
Sing.
#Samples
Electrostatics
Point Charge Potential
Laplacian
✓
500
Electric Dipole Potential
Laplacian
✓
500
Coulomb Potential Field
Laplacian
×
500
Heat Transfer
Gaussian Heat Source
Laplacian
×
500
Multi-Gaussian Heat Sources
Laplacian
×
500
Thermal Boundary Layer
Laplacian
×
500
Table 4: Distribution of scenarios in the training dataset by operator type and physical domain.
Department of Computer Science, University of Oxford · Department of Computer Science and Engineering, Hong Kong University of Science and Technology · Institute of Logic and Computation, TU Wien