MOD-SR: A New Symbolic Regression Method Integrating Multimodal Learning and Gradient-Guided Diffusion Models

This post introduces MOD-SR, an ICML 2026 accepted paper that innovatively integrates multimodal learning, direct optimization, and gradient-guided diffusion models to solve symbolic regression problems. The method is proposed by KROX777 and available on GitHub (https://github.com/KROX777/MOD-SR, updated 2026-05-25). Its core contributions address key challenges in traditional symbolic regression, opening new avenues for AI-driven scientific discovery and explainable AI.

Symbolic Regression (SR) aims to find interpretable mathematical expressions from data, which is crucial for scientific insights. Traditional SR methods face three main issues:

1. Search space explosion: Exponential growth of possible expressions makes optimal solution finding NP-hard.
2. Discrete-continuous gap: Mathematical expressions are discrete structures, but modern deep learning optimizes continuous spaces.
3. Underutilization of multimodal data: Real-world scientific data (numerical, image, text) are not effectively integrated in traditional methods.

MOD-SR (Multimodal Optimization with Diffusion for Symbolic Regression) has three key innovations:

1. Multimodal unified framework: Processes numerical, image, and text data via specialized encoders and cross-modal attention, mapping to a unified semantic space.
2. Direct optimization + diffusion model fusion: Combines direct optimization (fast gradient descent in continuous space) with diffusion models (high-quality discrete expression generation) in an end-to-end framework.
3. Gradient-guided mechanism: Uses gradient of the target function to guide diffusion model's sampling, improving search efficiency.

MOD-SR's architecture includes four modules:

1. Multimodal encoder: Encodes numerical (MLP), image (ViT/CNN), text (pre-trained LM) data, fused via cross-modal attention.
2. Expression diffusion model: Represents expressions as trees, uses tree-structured diffusion process (similar to DDPM) to generate candidate expressions.
3. Direct optimizer: Fine-tunes numerical parameters (coefficients, exponents) of generated expressions via gradient descent.
4. Gradient calculator: Computes gradient of expression structure against fitting error to guide diffusion sampling.

MOD-SR was evaluated on multiple benchmarks:

• Classic SR benchmarks: Outperformed traditional methods on Nguyen and Koza benchmarks in expression recovery accuracy and efficiency.
• Multimodal tests: Validated effectiveness of multimodal fusion using image/text auxiliary data.
• Ablation experiments: Confirmed the value of each core component (multimodal learning, direct optimization, gradient guidance). The paper is accepted by ICML 2026.

MOD-SR has wide applications:

1. Scientific discovery: Automatically find physical laws, chemical equations from experimental data.
2. Engineering optimization: Extract approximate formulas from simulation data for fast prediction/optimization (aerospace, auto design).
3. Education assistance: Generate analytical solutions for math problems to aid learning.
4. Explainable AI: Provide symbolic explanations for black-box models to enhance transparency.

MOD-SR has room for improvement:

1. Computation cost: Diffusion model training/inference requires large resources; need to improve efficiency.
2. Complex expressions: Handling highly nested complex expressions needs better performance.
3. Domain adaptability: Integrate domain-specific prior knowledge for better results in different scientific fields.
4. Neural network integration: Explore hybrid modeling of symbolic expressions and neural networks to leverage complementary strengths.