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npx versuz@latest install hiyenwong-ai-collection-collection-skills-hamiltonian-autonomous-emergent-dgmgit clone https://github.com/hiyenwong/ai_collection.gitcp ai_collection/SKILL.MD ~/.claude/skills/hiyenwong-ai-collection-collection-skills-hamiltonian-autonomous-emergent-dgm/SKILL.md--- name: hamiltonian-autonomous-emergent-dgm description: "Autonomous emergence of Hamiltonian parameters in deep generative models via Riemannian diffusion score fields. Extracting implicit physical laws from trained neural networks using algebraic framework. Activation: Hamiltonian, deep generative model, Riemannian diffusion, score field, spin glass, equivariant attention, physical law discovery, force estimator, emergent physics." --- # Autonomous Emergence of Hamiltonian in Deep Generative Models > Deep generative models autonomously discover and internalize underlying physical laws — achieving 99.7% cosine similarity in recovering microscopic Hamiltonian parameters without any energetic priors. ## Metadata - **Source**: arXiv:2604.20821 - **Authors**: Wenjie Xi, Wei-Qiang Chen - **Published**: 2026-04-22 - **Categories**: cond-mat.dis-nn, cond-mat.stat-mech ## Core Methodology ### Key Innovation Establishes a rigorous algebraic framework to extract implicit physical interactions learned by generative models. The zero-noise limit of a Riemannian diffusion score field is proven exactly equivalent to the thermodynamic restoring force, enabling the trained neural network to serve as a direct force estimator. ### Technical Framework 1. **Riemannian Diffusion Score Field**: Map the score field of a diffusion model to thermodynamic forces 2. **Force Estimator**: Use trained neural network directly as a physical force estimator 3. **Linear Inversion**: Apply overdetermined linear inversion to recover Hamiltonian parameters 4. **O(3)-Equivariant Architecture**: Train attention model on thermal equilibrium snapshots of 1D frustrated spin glass ### Key Results - 99.7% cosine similarity with ground-truth interaction parameters - 87% variance explained in continuous force field - No energetic priors needed — physical rules emerge autonomously ## Implementation Guide ### Prerequisites - Diffusion model training framework (PyTorch) - O(3)-equivariant attention architecture - Spin glass simulation environment ### Step-by-Step 1. Train O(3)-equivariant attention model on equilibrium configurations 2. Extract score field from trained diffusion model 3. Map zero-noise score field to restoring forces 4. Apply linear inversion to recover Hamiltonian parameters 5. Validate against ground-truth parameters ## Applications - Discovering physical laws from simulation data - Validating that generative models learn rules, not just patterns - Extracting interaction parameters in many-body systems - Protein structure analysis (AlphaFold-like applications) ## Pitfalls - Requires high-quality equilibrium training data - Linear inversion assumes correct functional form - Computational cost of training equivariant architectures ## Related Skills - energy-based-neurocomputation - lattice-field-theory-neurons - neural-emulator-theory