# 🚀 Contrastive Deep Learning in Generative Simulation (GS)

## 🧠 Abstract

This document proposes a methodological framework for integrating **contrastive deep learning** within the **GenerativeSimulation (GS)** paradigm. GS-agents are language-guided agents equipped with simulation kernels (e.g., SFPPy, Radigen, Pizza3) that respond to user prompts by executing simulations. We demonstrate how contrastive models—trained on **ratios** rather than absolute values—can efficiently recover scaling laws from sparse simulation outputs. This approach enhances traceability, interpretability, and generalization and enables GS to bridge symbolic physics-based models with machine-learned surrogates.

> This document proposes a methodological framework for integrating **contrastive deep learning** within the **GenerativeSimulation (GS)** paradigm. GS-agents are language-guided agents equipped with simulation kernels (e.g., SFPPy, Radigen, Pizza3) that respond to user prompts by executing simulations. We demonstrate how contrastive models—trained on **ratios** rather than absolute values—can efficiently recover scaling laws from sparse simulation outputs. This approach enhances traceability, interpretability, and generalization and enables GS to bridge symbolic physics-based models with machine-learned surrogates.
>
> **Contrastive GS** is developed to support fast-reasoning when the simulation dataset is incomplete or when kernels cannot directly provide an answer to complex engineering or scientific questions. **Contrastive GS** can be operated by GS-agents based on data available on a **GS-hub**.
>
> We additionally explore connections to dimensionality reduction (e.g., PCA in log-space, Vaschy-Buckingham theorem) and sparse additive modeling. Contrastive GS is positioned as a hybrid modeling framework between symbolic reasoning and empirical prediction.

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## 1️⃣ GS-Agents and Simulation Reasoning

**GenerativeSimulation (GS)** is a hybrid computing paradigm in which **language-first agents** (GS-agents) handle simulation-based reasoning. Agents are connected to domain-specific kernels that operate simulations in:

- 🐍 Native Python environments (e.g., `SFPPy`, `Radigen`), or
- 🔧 Cascading environments that manipulate input templates (e.g., `DSCRIPT` in `Pizza3`) before calling external codes (e.g., LAMMPS).

GS-agents operate within a **sandboxed** context: they do not submit hardware jobs but interpret and diagnose the results. Their conclusions are marked by **pertinence**, including:

- ✅ Relevance/failure status
- 📊 Degree of acceptability
- 🔍 Explanation of physical significance

To limit computational cost, agents follow a **tiered strategy**:

1. Begin with coarse-grained or conservative assumptions.
2. Refine step-by-step if necessary.
3. Terminate early if an answer is confidently derived.

All decisions are **traceable** and logged. Past simulations can be:

- 🔁 Reused or recombined,
- 🤝 Shared across agents,
- 👩‍⚖️ Reviewed by human supervisors via **GS-hubs**.

> **GS-hubs** serve as peer-review platforms that enhance and curate simulation logic, training datasets, and modeling protocols.

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## 2️⃣ Motivation: Scaling Laws in Sparse Data

In many simulation settings:

- 📐 The input space is high-dimensional: $\mathbf{x} = (x_1, \dots, x_n)$
- 🧪 Outputs $y_k = f(\mathbf{x}_k)$ are scalar and observed at limited $\mathbf{x}_k$
- 📏 Some input variables span several orders of magnitude

Many problems exhibit **self-similarity** or **scaling laws**:



$$
\frac{f(\mathbf{x}_u)}{f(\mathbf{x}_v)} \propto \prod_i \left(\frac{x_{i,u}}{x_{i,v}}\right)^{a_i^{(u,v)}}
$$

Here, the **exponents** $a_i^{(u,v)}$ are not constant globally, but tend to be:

- 🧭 Stable within **local domains**
- ⚖️ Governed by structure, symmetries, or conservation laws

The strategy of **contrastive learning** builds predictive models using the log-ratio:

$$
\log\left(\frac{f_u}{f_v}\right) \approx \sum_i a_i^{(u,v)} \cdot \log\left(\frac{x_{i,u}}{x_{i,v}}\right)
$$

With $m$ simulations, one may derive up to $m(m-1)/2$ independent ratios, vastly improving learning capacity.

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## 3️⃣ Contrastive GS: Principles and Interpretations

### 🔄 3.1 Learning from Log-Ratios

Instead of modeling $f(\mathbf{x})$ directly, Contrastive GS models **scaling transformations**:

- 🧮 Inputs: $\Delta_i = \log(x_{i,u} / x_{i,v})$, or $(1/T_u - 1/T_v)$ for temperatures
- 🎯 Target: $\log(f_u / f_v)$

This focuses on **relative change**, not absolute behavior.

### 📐 3.2 Relation to Generalized Derivatives

This contrastive formulation mimics **directional derivatives**:

If $\mathbf{x}_u$ and $\mathbf{x}_v$ lie on a generalized trajectory,
 then $\log(f_u / f_v)$ quantifies directional acceleration along that path.

This resonates with:

- 🔗 Lie algebraic structures
- 🌊 Flow-like interpretation of simulations
- 🧲 Conservative physical systems

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## 4️⃣ Dimensionality Reduction and Scaling Structure



### ✨ 4.1 Vaschy-Buckingham $\pi$-Theorem

- 🔣 Dimensional analysis constructs dimensionless quantities $\pi_i = \prod_j x_j^{\alpha_{ij}}$
- 🔄 If $f = g(\pi_1, ..., \pi_r)$, then contrastive inputs are aligned with log-ratios of $\pi$ terms
- 🧠 Suggests built-in alignment with physical constraints
- 

### ✨ 4.2 PCA and PCoA in Log-Transformed Spaces

- 📉 Applying **PCA** or **PCoA** to $\log(x_{i,u}/x_{i,v})$ uncovers **principal axes of variation**

- 🌀 **PCoA** (Principal Coordinates Analysis) may be more robust with non-Euclidean or semimetric distance measures

- 🛠️ These transformations help compress features before feeding them to contrastive models

  

### ✨ 4.3 Sparse Additive Models for Scaling

- 🧩 Sparse scaling models assume:

$$
\log\left(\frac{f_u}{f_v}\right) \approx \sum_{i \in S} g_i\left(\log\left(\frac{x_{i,u}}{x_{i,v}}\right)\right)
$$

- 🧵 Only a subset $S$ of variables is relevant

- 🕵️ These models offer interpretability and facilitate feature selection

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## 5️⃣ Methodological Synthesis

Contrastive GS combines **symbolic insight** with **data-driven generalization**:

| ⚙️ Physics-based Kernels | 🤖 Empirical ML Models        | 🧬 Contrastive GS                         |
| ----------------------- | ---------------------------- | ---------------------------------------- |
| Symbolic equations      | Black-box models             | Scaling structure via log-ratios         |
| Hard-coded dependencies | Flexible pattern recognition | Physically aligned, data-efficient       |
| Idealized assumptions   | Overfitting risks            | Interpretable exponents, domain-adaptive |

> "**Contrastive GS bridges symbolic kernels and black-box inference by recovering scaling logic embedded in numerical experiments.**"

This enables:

- 🔁 Reuse of sparse simulations for broader extrapolation

- 🌐 Learning local scaling regimes and hybrid surrogates

- 🔍 Grounding black-box predictions in physical reasoning

  

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## 6️⃣ Visual Architecture

```mermaid
graph TD
    A[🧑‍💻 User Prompt] -->|Query| B[🤖 GS-Agent]
    B --> C{⚙️ Kernel Type}
    C -->|Python-native| D[🧪 Radigen / SFPPy]
    C -->|Cascading| E[📦 Pizza3 / LAMMPS]
    D --> F[📊 Simulation Results]
    E --> F
    F --> G[🔁 Contrastive Learning Layer]
    G --> H[📈 Scaling Laws]
    H --> I[🧾 Answer with Explanation]
    G --> J[📂 Training Dataset Augmentation]
```

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## 7️⃣ Future Directions

- 🧭 Partition input space into domains of local exponents
- 🔢 Apply symbolic regression to learned scaling structures
- 🎲 Couple contrastive learning with uncertainty estimation
- 🧮 Extend to multi-output and vector-valued simulations

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## 🧩 Conclusion

Contrastive deep learning offers a powerful method to reveal **scaling laws** from simulation outputs, even under data sparsity. In the context of **GenerativeSimulation**, this approach unifies symbolic modeling and black-box prediction. It provides a structured, interpretable, and efficient path for enhancing simulation-based reasoning, setting a foundation for next-generation scientific agents.