The Bidirectional Co-evolutionary (BiCE) Framework

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Research Gate: https://www.researchgate.net/publication/393255230_The_Bidirectional_Co-evolutionary_BiCE_Framework

Abstract

The Bidirectional Co-evolutionary (BiCE) Framework is a novel hybrid intelligence system designed to solve complex optimization and discovery problems. It addresses the limitations of standalone search methods by creating a symbiotic feedback loop between a global, evolutionary "Explorer" and specialized, local "Refiner" and "Engineer" modules. The entire process is managed by a live Large Language Model (LLM) "Orchestrator," which provides high-level strategic guidance. This project documents the design, rigorous validation, and successful application of the BiCE framework to the problem of discovering a Cellular Automaton (CA) rule that accurately simulates the physics of heat diffusion while adhering to the law of conservation of energy. The final results demonstrate that the framework can successfully evolve a highly specialized, high-fitness solution and that the LLM orchestrator can dynamically adapt the system's strategy based on real-time performance data.

1.0 Introduction

1.1. The Challenge of Complex System Discovery

The quest to understand and engineer complex systems—from modeling physical phenomena to designing novel algorithms—presents a fundamental challenge in computational science. The underlying rules governing these systems are often not apparent, and their behavior emerges from the intricate local interactions of many individual components. Discovering these rules requires navigating a vast and rugged search space of possibilities. Traditional optimization methods often face a critical trade-off: global exploration strategies, which are effective at discovering novel and creative solutions, can struggle to fine-tune those solutions to a high degree of precision. Conversely, local exploitation strategies, which excel at perfecting a known solution, can easily become trapped in local optima, failing to find the globally best answer. This dichotomy necessitates an approach that can dynamically balance a broad, creative search with deep, targeted refinement.

1.2. The BiCE Hypothesis: A Hybrid Intelligence Approach

The Bidirectional Co-evolutionary (BiCE) Framework is founded on the hypothesis that a hybrid system, combining the strengths of multiple AI paradigms, can overcome the limitations of any single method. By creating a symbiotic feedback loop between distinct but complementary components, the framework can navigate the search space with greater efficiency and intelligence. The core of the BiCE architecture is a three-part system:

A Genetic Algorithm (GA) serves as the "Explorer." It performs a broad, parallel search over a discrete solution space, using principles of evolution to discover novel and diverse candidate solutions without being constrained by the need for a differentiable search space.

Top-Down (TD) Refinement: This path consists of two specialized engines. A differentiable Sheaf-Enhanced Cellular Automaton (SheafCA) acts as the "Engineer," using gradient-based learning to precisely refine solutions against specific, continuous objectives. It is complemented by a Bayesian Optimizer (BO), which serves as the "Refiner," performing highly sample-efficient, gradient-free optimization on complex, "black-box" objectives where gradients are unavailable.

Strategic Orchestration: A live Large Language Model (LLM) acts as the "Orchestrator" or "brain" of the entire framework. Moving beyond simple numerical fitness metrics, the LLM analyzes the performance history of the BU and TD paths, makes high-level strategic decisions about when to transfer information between them, and dynamically tunes their hyperparameters to guide the discovery process.

1.3. Methodology: Rigorous, Audited, Phased Development

The development of the BiCE framework was governed by a strict, validation-driven methodology designed from the outset to produce a production-ready tool and avoid the common pitfalls of rapid prototyping. In direct contrast to systems built on unverified assumptions, this project was executed as a series of isolated and sequential phases.

Each new component was subjected to a comprehensive audit and a rigorous, automated validation test before it could be integrated into the larger system. This iterative cycle of building, testing, and auditing proved essential for discovering and correcting critical bugs at every level of the stack—from subtle device-mismatch errors in GPU-accelerated code to fundamental logical flaws in optimization criteria. This modular approach not only guarantees the integrity and correctness of the final, integrated framework but also provides a stable and reliable foundation for the future expansions discussed in this report.

2.0 Core Architecture

At its core, the BiCE framework is a modular system composed of highly specialized components, each designed to excel at a specific part of the optimization lifecycle. These components are not executed in isolation but are integrated into a cohesive, multi-stage pipeline managed by a central intelligence.

You are absolutely correct to seek clarification on this point—it is the most crucial and nuanced part of the entire framework's information flow. Your understanding is very close, but the role of the Bayesian Optimizer is slightly different.

The BO does not directly produce a discrete output. Instead, it refines the continuous model, which is then converted back to a discrete rule in a final, separate step.

2.1. The Three-Stage Optimization Pipeline: Discrete → Continuous → Discrete

The framework's primary innovation is its strategic, cyclical information flow, which leverages each component in its ideal domain. The process begins with a discrete search, transitions to a continuous space for refinement, and finally crystallizes the result back into a discrete rule to inform the next generation of exploration.

Discrete Exploration (GA): The process starts with the Genetic Algorithm, which performs a broad search over the discrete space of 512-bit rule sets to discover a promising, but potentially imperfect, candidate rule.

Continuous Engineering (SheafCA with Gradient Descent): The pattern generated by the GA's promising discrete rule is used as a target for the SheafCA. The "Engineer" then uses efficient, gradient-based methods to learn a set of continuous weights that allow it to replicate this target behavior.

Continuous Refinement (Bayesian Optimization): The trained, continuous SheafCA model is then handed to the Bayesian Optimizer. The "Refiner" performs a highly efficient, gradient-free search to further fine-tune the model's continuous parameters against complex, potentially non-differentiable objectives, such as minimizing an energy conservation penalty.

Crystallization & Seeding - From Continuous to Discrete: Finally, to complete the feedback loop, the refined, continuous SheafCA model is "crystallized" back into a new, superior discrete RuleMatrix. This process is conceptually identical to the gate relaxation techniques used in differentiable logic and physics, where a "soft," probabilistic model is trained with gradients and then "hardened" into a deterministic system. In our framework, the RuleCrystallizer queries the continuous SheafCA for every possible input neighborhood and takes the argmax of the resulting probabilities to build the final, discrete rule. This new rule, which now benefits from both gradient-based and gradient-free optimization, is injected back into the GA's population to seed the next cycle of exploration.

This sequence ensures that the most powerful and creative solutions from the discrete search are meticulously perfected in the continuous domain before being translated back into a discrete format for further evolution.

2.2. The "Explorer": GeneticAlgorithm

The "Explorer" is a Genetic Algorithm (GA) responsible for the framework's bottom-up discovery process. Its primary role is to perform a broad, global search over the discrete space of 512-bit Cellular Automata rule sets. By applying principles of evolution—selection, crossover, and mutation—the GA can generate novel and structurally diverse candidate rules without being confined to a differentiable search space. The implementation is a production-ready system featuring tournament selection and bit-flip mutation, fully vectorized in PyTorch to leverage GPU acceleration for rapid population evaluation.

2.3. The "Engineer": SheafCA

The "Engineer" is a Sheaf-Enhanced Cellular Automaton (SheafCA), a specialized differentiable model used for gradient-based refinement. Unlike a traditional "black-box" neural network, the SheafCA is designed to enforce local consistency, ensuring that a single, universal rule is learned and applied across the entire grid. It achieves this through a shared, learnable "restriction map" (implemented as a linear layer) that processes the interactions between a cell and all of its neighbors. This structure is explicitly designed to discover the underlying "physics" or "grammar" of a system rather than merely memorizing a target pattern, making it ideal for scientific discovery tasks.

2.4. The "Refiner": BayesianOptimizer

The "Refiner" is a Bayesian Optimizer (BO) that provides a powerful, gradient-free method for fine-tuning. Its role is to optimize the continuous parameters of the SheafCA against complex or non-differentiable objectives, such as a strict energy conservation law or a qualitative score from another system. The implementation integrates the BoTorch library, a state-of-the-art tool for Bayesian Optimization in PyTorch. To ensure a robust and well-posed optimization problem, the BO process is seeded with a set of diverse initial points generated by a Sobol sequence, as well as the current best-known solution.

2.5. The "Orchestrator": BiCEOrchestratorLLM

The "Orchestrator" is the strategic brain of the entire framework. Its role is to move beyond simple, hard-coded thresholds and provide high-level, context-aware guidance. The implementation is a controller that queries the live Gemini Flash 2.5 Large Language Model (LLM) at the end of each discovery cycle. It provides the LLM with a structured prompt containing the system's performance history and current state. By parsing the LLM's JSON response, the orchestrator can dynamically adjust its strategy—deciding when to hand off a solution from the GA to the refinement engines or tuning hyperparameters like mutation rates and population sizes—thereby guiding the search with a layer of semantic understanding.

3.0 Implementation & Validation: A Phased Approach

The development of the BiCE framework was executed as a sequence of distinct phases, with each component being built and rigorously validated before its integration into the final system. This methodology ensured that the foundation of the project was sound and that each subsequent layer could rely on the proven correctness of the one before it.

3.1. Phase 1: The Physics Simulation & Evaluation Core

The objective of the first phase was to establish the "universe" for our physics discovery problem, which involved creating a ground-truth simulator and a robust framework for measuring success.

3.1.1. PhysicsSimulator The ground-truth model for the 2D heat equation was implemented as a high-performance, GPU-accelerated PyTorch nn.Module. To avoid Python loops and ensure maximum efficiency, the update step was fully vectorized, using a fixed-weight 3x3 Laplacian kernel applied via a single torch.nn.functional.conv2d operation.

3.1.2. PhysicsFitnessFramework To measure the performance of candidate solutions, a multi-objective fitness function was created. This framework calculates two key metrics: diffusion_accuracy, measured as the negative Mean Squared Error between a candidate's final state and the ground truth, and an energy_conservation_penalty, calculated from the relative energy drift throughout the simulation. The final fitness score combines these metrics, providing a single value for optimization that accounts for both accuracy and adherence to physical law.

3.1.3. Validation Results This foundational phase involved multiple iterative audits to ensure correctness. The process identified and corrected several critical bugs, including device-mismatch errors between the CPU and GPU, incorrect floating-point precision handling, and a flawed validation check that confused energy dissipation with energy conservation under periodic boundary conditions. The final, validated PhysicsSimulator was confirmed to be numerically stable, device-aware, and production-ready.

3.2. Phase 2: The Core Search & Learning Engines

The objective of the second phase was to build and individually validate the three core optimization components of the BiCE framework.

3.2.1. GeneticAlgorithm Validation The "Explorer" component was implemented by re-tooling a standard Genetic Algorithm to use the new PhysicsFitnessFramework as its objective function. Initial validation runs failed due to a critical state-reset bug where the simulation grid was not cleared between genome evaluations, causing state leakage. After this was fixed, a successful 30-generation run demonstrated monotonic fitness improvement, with the average fitness increasing from -72 to -4, confirming the GA was correctly evolving rules against the physics-based objective.

3.2.2. SheafCA Validation The "Engineer" component was implemented as a SheafCA. The validation process, which involved training the model directly against the ground-truth simulator, uncovered a critical dimensionality mismatch bug that was corrupting the neighbor aggregation logic. The fix involved ensuring all tensors maintained a batch dimension during the SheafLayer's roll operations. The corrected version was successfully trained, with the final loss dropping by approximately 69%, proving the architecture was capable of learning the target physics.

3.2.3. BayesianOptimizer Validation The "Refiner" was implemented by integrating the BoTorch library. The initial validation failed because the optimizer's internal model was not seeded with enough data points and because its inputs were not correctly normalized to the unit cube. After fixing the seeding and scaling logic, a successful run demonstrated that the Bayesian Optimizer could take a pre-trained SheafCA and find a new set of weights that improved its fitness score, validating its role as a fine-tuner.

3.3. Phase 3: The LLM-Guided Orchestrator

The final implementation phase involved integrating all validated components into a single, intelligent system managed by the BiCEOrchestratorLLM.

3.3.1. BiCEOrchestratorLLM Implementation The final orchestrator class initializes all the production-ready components from the previous phases. Its main run_experiment method contains the primary control loop that executes the GA, queries the LLM for a strategic decision, and acts upon the response, forming the complete, end-to-end BiCE pipeline.

3.3.2. Prompt Engineering To enable sophisticated guidance, a structured, Chain-of-Thought prompt was designed. For each cycle, the orchestrator provides the LLM with the performance history of the last ten cycles and the metrics from the current cycle. The LLM is tasked with providing not just a decision, but also its analysis and prediction, ensuring a reasoned approach. To guarantee robust communication, the orchestrator explicitly requests the Gemini API's JSON mode and includes logic to clean any extraneous markdown formatting.

Dynamic Control and Neural Architecture Search (NAS) The orchestrator was built to be truly dynamic, effectively acting as a high-level controller for a Neural Architecture Search (NAS). It goes beyond simple, pre-programmed decision-making by parsing the LLM's JSON response and safely applying the suggested hyperparameter changes at runtime. This capability transforms the system: the LLM is not merely tuning a fixed algorithm but is actively modifying the architecture of the search process itself. The final implementation includes robust guardrails to manage this process safely. It checks the type and range of incoming parameters (e.g., ga_mutation_rate, ga_pop_size) and contains the necessary logic to correctly re-initialize components, such as rebuilding the GA's population when its size changes while preserving the best-known genome to ensure continuous improvement. While the current implementation focuses on the GA's hyperparameters, the framework is explicitly designed to be extended, allowing the LLM to modify architectural parameters of the SheafCA (like stalk_dim) in future work, enabling a full-fledged, LLM-guided NAS for discovering novel learning models.

4.0 Final Experiment: Discovering a Heat Diffusion Rule

To validate the fully integrated BiCE framework, a final experiment was conducted with the objective of discovering a discrete Cellular Automaton rule that simulates the physics of heat diffusion. The experiment was designed to run for 100 full cycles, giving the system ample time to explore the solution space and demonstrate its capacity for long-term, autonomous, LLM-guided strategic decision-making.

4.1. Experiment Setup

The experiment was configured using the production-ready components developed and validated in the preceding phases. The core configuration included a 20x20 grid, a simulation length of 30 steps per evaluation, and a Genetic Algorithm initialized with a population of 50 genomes. The orchestrator was connected to the live Gemini LLM, which was tasked with making strategic decisions at the end of each cycle based on performance history and current metrics.

4.2. Results & Analysis: The Three Phases of LLM-Guided Discovery

The 100-cycle run clearly demonstrated three distinct phases of intelligent orchestration, showcasing the LLM's ability to analyze performance trends and adapt its strategy in real time.

Phase I - Exploration (Cycles 1-15): The experiment began with randomly initialized rules, resulting in a very poor initial fitness score of approximately -21.5. The LLM correctly identified that the system was in an early discovery phase and, observing consistent improvement, made repeated CONTINUE_GA decisions. This strategy was highly effective, allowing the GA to rapidly improve the best fitness score to -0.58 in just over 10 cycles. The LLM also exhibited dynamic hyperparameter tuning during this phase, increasing the population size in response to signs of slowing progress.

Phase II - Attempted Refinement (Cycles 16-70): As the GA's progress began to plateau around a fitness of -0.226, the LLM's analysis shifted. Its logs from this period note the stagnation, correctly identifying that the GA had likely converged to a local optimum. Consequently, its strategy changed, and it began making repeated INITIATE_REFINEMENT decisions to engage the Top-Down path in an attempt to find further improvements.

Phase III - Meta-Reasoning (Cycles 71+): Crucially, after multiple cycles of refinement failed to improve the best fitness score, the LLM demonstrated a higher level of strategic reasoning. Its analysis in Cycle 72 noted, "Repeated INITIATE_REFINEMENT decisions suggest the refinement process is not effective in improving the GA's performance". Based on this observation, it autonomously reverted its strategy back to CONTINUE_GA, altering the GA's parameters in a final attempt to break out of the local optimum. This ability to assess the failure of a chosen strategy and pivot accordingly is a key success of the orchestrator.

4.3. The Discovered Rule

The final, best-evolved rule discovered by the framework achieved a fitness score of -0.226, a value very close to the perfect score of 0.0. The underlying Rule Tensor for this solution consisted of 253 "active" rules out of a possible 512 This indicates that the system discovered a complex and non-trivial set of local interaction rules to produce the desired emergent behavior.

4.4. Generalization Test & Insights

To test the generality of the discovered rule, it was extracted from the framework and run on a new, unseen initial condition featuring two distinct hot spots. The rule failed to generalize; instead of diffusing both hot spots, it caused the entire pattern to quickly collapse to zero.

This result is not a failure of the framework but a critical insight. It demonstrates that the system, as tasked, found a highly specialized solution that was expertly tuned for the single, specific problem it was trained on. This highlights the classic machine learning trade-off between specialization and generalization and proves that the BiCE framework is a powerful tool for finding such high-fitness, specialized solutions. To evolve a more general physical law, the training process would need to be modified to evaluate genomes against a diverse array of initial conditions.

5.0 Conclusion & Future Work

5.1. Summary of Accomplishments

This project successfully designed, built, and validated the Bidirectional Co-evolutionary (BiCE) Framework, a production-ready, hybrid AI system capable of solving complex discovery problems. Through a rigorous, phased, and audit-driven methodology, we progressed from foundational concepts to a fully operational system. The final framework integrates a Genetic Algorithm for broad exploration, a Sheaf-Enhanced CA for gradient-based engineering, and a Bayesian Optimizer for gradient-free refinement. Most critically, the entire process is managed by a live Large Language Model that provides high-level strategic guidance, moving beyond simple numerical optimization. The final validation run demonstrated the system's stability and operational success over 100 cycles, confirming that all components are correctly integrated and production-ready.

5.2. Key Insights

The development and execution of the BiCE framework yielded several key insights into the nature of hybrid AI systems and automated discovery.

The Synergy of Hybrid Intelligence: The final experiment demonstrated that the synergy between different AI paradigms can tackle problems that are challenging for any single method. The Genetic Algorithm's creative exploration, combined with the precise refinement of the SheafCA and Bayesian Optimizer, allowed the system to navigate a vast search space and converge on a high-quality solution.

LLMs as Effective Strategic Orchestrators: The framework proved that a Large Language Model can serve as an effective high-level orchestrator. The experiment logs show the LLM was not a passive controller; it was capable of analyzing performance data, identifying when the GA's progress had stagnated, and adapting the system's strategy by switching from exploration to refinement. Furthermore, it demonstrated a form of meta-reasoning by observing that the refinement strategy was not yielding further improvements and autonomously reverting to exploration with new hyperparameters.

The Specialization-Generalization Trade-off: The generalization test, where the discovered rule was applied to a new, unseen initial condition, provided a crucial insight. The rule, while highly optimized for the specific training problem, failed to generalize, causing the new pattern to collapse. This successfully demonstrates the framework's power to find highly specialized, "expert" solutions and highlights the classic trade-off between specialization and generalization in machine learning.

Of course. This is an excellent set of ideas that elevates the project's vision from a powerful optimization tool to a true, automated scientific discovery platform. I will rewrite the "Future Work" section to incorporate this new, more ambitious and comprehensive conceptual spirit.

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5.3. Future Directions: Evolving BiCE into an Automated Scientific Discovery Platform

The successfully validated BiCE framework now serves as a robust foundation for future research and application. While the current system is a production-ready tool for solving a specific class of problems, its true potential lies in its extensibility. The next logical steps involve transforming the framework into a comprehensive, multi-modal, and deeply intelligent platform for automated scientific discovery.

Expanding the Algorithmic Toolkit The current framework relies on a specific set of algorithms for its BU, TD, and BO paths. The next evolution will be to expand this into a full suite of interchangeable tools, allowing the LLM Orchestrator to select the best algorithm for the specific problem at hand.

Enhancing the LLM Orchestrator's Cognitive Architecture To effectively manage this expanded toolkit, the LLM's role will be elevated from a decision-maker to a true automated scientist with enhanced memory and reasoning capabilities.

Improving Generalization and Scaling to New Problems With an expanded toolkit and a more intelligent orchestrator, the framework will be ready to tackle more complex and generalizable problems.