🚀 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.

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:

Begin with coarse-grained or conservative assumptions.
Refine step-by-step if necessary.
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.

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.

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

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

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

6️⃣ Visual Architecture

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]

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

🧩 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.