VIPER-R1: Multimodal Scientific Discovery

• The paper introduces VIPER-R1, which automates discovery of fundamental physical laws by fusing visual plots and kinematic data with symbolic reasoning.
• It employs a two-stage curriculum combining causal chain-of-thought and reward-guided reinforcement learning to refine symbolic hypothesis generation.
• The model demonstrates significant performance gains on the PhysSymbol corpus, achieving high structural fidelity and improved interpretability over previous methods.

VIPER-R1 is a multimodal model for automated discovery of fundamental physical laws from empirical data, designed to emulate the inductive reasoning processes of a physicist. In contrast to prior uni-modal or solely regression-based symbolic discovery methods, VIPER-R1 tightly integrates visual perception of trajectory plots, structured numerical data, and explicit symbolic reasoning. The framework is built as a two-stage curriculum—first encouraging explainable causal inference and hypothesis generation from multimodal evidence, then refining symbolic formulae using a reward-guided reinforcement learning objective. A final inference step incorporates external symbolic regression for residual correction. Performance and interpretability are benchmarked on the newly introduced PhysSymbol corpus, showing significant advances over state-of-the-art vision-LLMs (Liu et al., 24 Aug 2025).

1. Model Architecture and Multimodal Integration

VIPER-R1 processes empirical evidence as a multimodal set $E = \{\mathcal{V}, \mathcal{D}\}$, where $\mathcal{V}$ represents visualizations (e.g., phase portraits and time-series trajectory plots) and $\mathcal{D}$ denotes kinematic data (positions, velocities, accelerations). The core architecture comprises:

• Multimodal Encoder: Jointly embeds $\mathcal{V}$ and $\mathcal{D}$ for downstream symbolic reasoning.
• Motion Structure Induction (MSI) Module: Supervises the agent to simultaneously generate a causal chain-of-thought (C-CoT) connecting visual patterns to dynamic hypotheses, along with a preliminary symbolic formula $S$.
• Reward-Guided Symbolic Calibration (RGSC): Refines the symbolic formula by maximizing a composite reward over format consistency, parameter-agnostic structural similarity (measured via Jaccard similarity of skeletonized term sets), and exact algebraic match.
• Agentic Inference with Symbolic Residual Realignment (SR\textsuperscript{2}): At prediction time, the model generates a high-confidence ansatz, computes the residual $r(t) = a_\text{GT}(t) - a_\text{VLM}(x, v, t)$, and invokes an external symbolic regression engine to precisely model any discrepancy.

This composite pipeline enables VIPER-R1 to move beyond superficial pattern matching, instead synthesizing visual, quantitative, and explainable symbolic clues in a computationally tractable manner.

2. Curriculum Learning: Motion Structure Induction and Symbolic Calibration

Training proceeds over two carefully staged curricula:

A. Motion Structure Induction (MSI)

• Joint Causal Reasoning and Symbolic Hypothesis: From $E$, the model predicts $Y = (C, S)$, with the loss

$$\mathcal{L}_{\text{MSI-1}} = -\mathbb{E}_{(E,Y)} \sum_t \log \pi_\theta(y_t \mid E, y_{<t}).$$

This aligns the generative process to produce not just symbolic outputs but also the explanatory reasoning steps akin to a physicist's mentorship.

• C-CoT–Guided Symbolic Formulation: The model is further optimized for symbol generation conditioned on the causal explanation:

$$\mathcal{L}_{\text{MSI-2}} = -\mathbb{E}_{(E, C, S)} \sum_t \log\pi_\theta(s_t \mid E, C, s_{<t}).$$

B. Reward-Guided Symbolic Calibration (RGSC)

• Following supervised pretraining, the model samples $G$ candidate formulas $\{S_1, S_2, \dots, S_G\}$, evaluates each with a reward

$$R(S_i) = w_f R_\text{format}(S_i) + w_s R_\text{structural}(S_i, S_{\text{GT}}) + w_a R_\text{accuracy}(S_i, S_{\text{GT}}),$$

and computes normalized relative advantages:

$$A_i = \frac{r_i - \operatorname{mean}(r_1,\ldots, r_G)}{\operatorname{std}(r_1,\ldots, r_G) + \epsilon}.$$

Policy updates are performed using Group Relative Policy Optimization (GRPO), with an additional KL penalty against the MSI reference to avoid policy drift.

3. Causal Chain-of-Thought and Symbolic Reasoning

A defining component of VIPER-R1 is its explicit Causal Chain-of-Thought (C-CoT). The model is trained to articulate how and why specific visual or numerical patterns—such as spiral phase portraits (implying damping) or harmonic trajectories—logically lead to terms in the hypothesized dynamic equation. This output not only increases interpretability but also regularizes symbolic extraction by providing a rational "scientific" context, improving both structural and semantic fidelity of the resulting laws.

4. Agentic Inference and Symbolic Residual Realignment

At inference, VIPER-R1 operates in a multi-step manner that closely parallels a physicist’s iterative discovery workflow:

1. Primary Hypothesis Generation: The trained model predicts an initial symbolic form $a_\text{VLM}(x, v, t)$.
2. Empirical Residual Computation: The difference $r(t) = a_\text{GT}(t) - a_\text{VLM}(x, v, t)$ isolates factors missed by the primary ansatz.
3. External Symbolic Regression (SR\textsuperscript{2}): An external regression tool fits $a_\text{residual}(x, v, t)$ to the residual signal, yielding the final combined law:

$$a_\text{final}(x, v, t) = a_\text{VLM}(x, v, t) + a_\text{residual}(x, v, t).$$

This aligns the framework with perturbative refinement practices in physical sciences, ensuring both interpretability and empirical adequacy.

5. Performance Evaluation and PhysSymbol Corpus

Evaluation utilizes the PhysSymbol benchmark, which contains 5,000 instances, each comprising dual trajectory visualizations, high-resolution kinematic data, a canonical governing equation, and expert C-CoT annotations. Key metrics are:

• Structural Score ($S_\text{struct}$): Jaccard similarity of skeletonized equation terms.
• Accuracy Score ($S_\text{acc}$): Exact algebraic match with ground truth.
• Mean Squared Error (MSE): Empirical error after SR\textsuperscript{2} realignment.

On this corpus, VIPER-R1-7B achieved $S_\text{struct} = 0.812$, $S_\text{acc} = 0.487$, and post-SR\textsuperscript{2} MSE $=0.032$, outperforming prior leading VLMs such as Claude-4-Sonnet.

Corpus Structure: PhysSymbol

Attribute	Description
Instances	5,000
Modalities	Phase plots, time series plots, kinematic series
Annotation	Governing ODEs and expert stepwise reasoning
Generation Method	Combinatorial synthesis from a physics term library

6. Scientific Significance and Impact

VIPER-R1 advances automated scientific discovery by coupling visual and quantitative observation with interpretable, structured symbolic inference. Its core contributions include:

• Emulating the iterative, rational process of scientific hypothesis formation, rather than end-to-end regression.
• Tightly integrating visual feature extraction, human-interpretable causal reasoning, and algebraically precise symbolic generation.
• Offering not only high accuracy and structural fidelity but also transparent, step-wise explanations for each proposed law.

This methodology closes key gaps observed in previous "sensory deprived" approaches to symbolic regression, establishing a new paradigm for vision-grounded law induction with applications in physics, engineering, and scientific machine learning.

7. Future Directions

Potential refinements and open research problems arising from VIPER-R1 include:

• Extension beyond kinematics to multi-body, field-theoretic, or non-canonical systems.
• Exploration of more advanced visual encoders or symbolic reasoning modules, possibly leveraging future advances in VLM and LLM capabilities.
• Systematic evaluation of C-CoT strategies, RL reward design, and the influence of external regression tool choice on robustness and generalization.
• Scaling the curriculum to larger, more diverse empirical corpora and assessing transfer to real-world (non-simulated) phenomena.

These directions will be critical for the generalization of VIPER-R1–style frameworks to new domains and for pushing toward automated, interpretable scientific reasoning of broader scope.