index.qmd

---
title: "LLM-Guided Morphogenesis"
subtitle: "Closed-loop discovery in coupled chemical-particle systems"
author: "Cédric Allier"
date: "2026-01-31"
---

## Introduction

Understanding morphogenesis requires more than generating patterns; it requires understanding how interacting processes converge toward stable, functional forms. This work presents a **closed-loop experimental framework** in which experiments, reasoning, and long-term memory are tightly coupled, with a large language model (LLM) operating as an **active scientific agent**. Note, the LLM does not only explore parameter space; it is allowed, at controlled points, to **modify and replace the governing partial differential equations (PDEs)** that define the system dynamics. In this framework, PDEs are treated as *hypotheses*, not fixed truths.

The system combines reaction–diffusion chemistry, diffusiophoretic particle transport, and particle–field feedback to explore a rich landscape of emergent morphologies. Rather than treating particles or fields as primary objects, the framework treats **interactions as fundamental**, with structure arising from their mutual constraint. The LLM evaluates simulation outcomes, formulates mechanistic hypotheses, and directs subsequent interventions through structured exploration and persistent memory, enabling cumulative understanding across regimes rather than isolated optimization.

A key methodological insight motivating this work is that **LLM-guided discovery is fundamentally limited by the laws it is allowed to explore**. Early stages of the project revealed strong attractors imposed by the Brusselator reaction–diffusion model, which restricted accessible morphologies despite extensive parameter search. This limitation led to an explicit design choice: elevate *model-space exploration*—including PDE modification and variant creation. After 80 iterations exhausting all particle-level features, the LLM modified the Brusselator PDE itself by introducing **nonlinear diffusion** (concentration-dependent $D_1$), unlocking qualitatively new labyrinthine and vermiform pattern types inaccessible to any parameter tuning of the standard model.

The current results focus on diffusiophoresis coupled to classical reaction–diffusion systems. The framework now extends to include **mechanochemical coupling** via the Material Point Method (MPM), enabling study of how mechanical admissibility constrains morphogenetic outcomes. See the [Mechanochemical Morphogenesis](mechanochemical.qmd) section for the unified framework combining chemistry and continuum mechanics.


---

## The Exploration Loop

```{mermaid}
%%| fig-width: 8
flowchart LR
    A[Simulation] --> B[Pattern Analysis]
    B --> C[LLM Evaluation]
    C --> D[Hypothesis Update]
    D --> E[Parameter Mutation]
    E --> A

    C --> F[(Memory)]
    F --> C

    style A fill:#e1f5fe
    style C fill:#fff3e0
    style F fill:#f3e5f5
```

The framework implements a **closed-loop exploration engine** composed of five interacting components:

1. **Simulation Engine**  
   A physics-based simulator executes coupled particle–field dynamics defined by a selected PDE model (e.g., reaction–diffusion chemistry coupled to diffusiophoresis). For a given model, parameters, and initialization, the simulator produces reproducible field and particle trajectories together with quantitative diagnostics.

2. **Pattern and Metric Analysis**  
   High-dimensional simulation outputs are compressed into montages, scalar metrics, and regime descriptors. These representations enable comparison across iterations and across different governing models.

3. **LLM Evaluation**  
   The LLM interprets results in the context of accumulated memory. Its role is not to optimize a single score, but to perform scientific operations: identify regimes, detect saturation or failure, generate mechanistic hypotheses, and assess whether observed limitations arise from parameter choices or from the structure of the governing laws.

4. **Hypothesis and Memory Update**  
   Observations, failed attempts, and validated principles are written into explicit long-term memory. This memory persists across experimental blocks and constrains future reasoning, enabling cumulative understanding rather than episodic trial-and-error.

5. **Intervention Selection**  
   Guided by Upper Confidence Bound (UCB) tree search, the LLM selects the next intervention. Interventions are stratified by scope:
   - *Within blocks*: parameter mutations within a fixed model.
   - *At block boundaries*: higher-impact actions, including modification of interaction terms, creation of new PDE variants, or replacement of the reaction–diffusion model itself.

This separation ensures stability and reproducibility while enabling **controlled exploration of model space**. From a machine learning perspective, the system operates over a constrained hypothesis class of lawful transformations, combining black-box experimentation, symbolic model search, and continual learning with external memory. The objective is not performance optimization, but **epistemic progress**: discovering which laws admit morphogenesis, which do not, and why.

The exploration uses **Upper Confidence Bound (UCB)** tree search to balance exploitation of promising configurations with exploration of novel parameter regimes. Iterations are grouped into blocks of 8, with memory consolidation at block boundaries.

---

## Four Coupled Interactions

| Interaction | Physics | Description |
|-------------|---------|-------------|
| [**Field–Field**](field-field.qmd) | Reaction-diffusion PDE | Turing patterns via activator-inhibitor dynamics |
| [**Field–Particle**](field-particle.qmd) | Diffusiophoresis | Field gradients drive particle motion |
| [**Particle–Field**](particle-field.qmd) | Consumption/production | Particles locally modify concentrations |
| [**Particle–Particle**](particle-particle.qmd) | Attraction-repulsion | Short-range forces between particles |

---

## Mechanochemical Extension

::: {.callout-tip}
## New: Coupling Chemistry with Mechanics
The framework extends beyond pattern formation to include **continuum mechanics** via the Material Point Method (MPM). This enables studying morphogenesis under mechanical constraints.

[**→ Mechanochemical Morphogenesis**](mechanochemical.qmd)
:::

```{mermaid}
%%| fig-width: 8
flowchart LR
    subgraph Current["Current: Chemistry"]
        RD["Reaction-Diffusion"]
        DP["Diffusiophoresis"]
    end

    subgraph Extension["Extension: Mechanics"]
        DEF["Deformation F"]
        STR["Stress σ"]
    end

    subgraph Unified["Unified Framework"]
        U["Mechanochemical<br>Morphogenesis"]
    end

    Current --> Unified
    Extension --> Unified

    style Current fill:#e3f2fd
    style Extension fill:#fff3e0
    style Unified fill:#e8f5e9
```

Key additions:

- **Deformation gradient $\mathbf{F}$** and **stress $\boldsymbol{\sigma}$** per particle
- **Active stress** driven by morphogen concentration
- **Strain-dependent diffusion** for mechanochemical feedback
- **Edge embeddings** for learnable cell-cell interactions (synaptic class)

---

## Reaction-Diffusion Models

Four PDE models have been explored for the chemical patterning component, revealing a clear hierarchy of accessible morphologies:

### [Brusselator](brusselator.qmd)

[![](log/Claude_exploration/instruction_diffusiophoresis_parallel/montage/montage_iter_012.png){.lightbox}](brusselator.qmd)

### [Gray-Scott](grayscott.qmd)

[![](log/Claude_exploration/instruction_diffusiophoresis_parallel/montage/montage_iter_014.png){.lightbox}](grayscott.qmd)

### [FitzHugh-Nagumo](fhn.qmd)

[![](log/Claude_exploration/instruction_diffusiophoresis_parallel/montage/montage_iter_018.png){.lightbox}](fhn.qmd)

### [Schnakenberg](schnakenberg.qmd)

[![](log/Claude_exploration/instruction_diffusiophoresis_parallel/montage/montage_iter_041.png){.lightbox}](schnakenberg.qmd)

---

## References

1. Prigogine, I., & Lefever, R. (1968). Symmetry breaking instabilities in dissipative systems. *J. Chem. Phys.*, 48(4), 1695-1700.

2. Pearson, J. E. (1993). Complex patterns in a simple system. *Science*, 261(5118), 189-192.

3. Anderson, J. L. (1989). Colloid transport by interfacial forces. *Annu. Rev. Fluid Mech.*, 21(1), 61-99.

4. Turing, A. M. (1952). The chemical basis of morphogenesis. *Phil. Trans. R. Soc. Lond. B*, 237(641), 37-72.