Autonomous Emergence of Hamiltonian in Deep Generative Models

Abstract

The unprecedented predictive success of deep generative models in complex many-body systems, such as AlphaFold 3, raises an epistemological question: do these networks merely memorize data distributions via high-dimensional interpolation, or do they autonomously deduce the underlying physical laws? To address this, we introduce a framework to extract the implicit physical interactions learned by generative models. Using the exact equivalence between the zero-noise limit of a diffusion score field and the thermodynamic restoring force, we directly compare the internal interaction structure of a trained neural network with the physical Hamiltonian. Applying this framework to a sequence-dependent, frustrated 1D $O(3)$ spin glass, we probe the latent representations of an $O(3)$-equivariant attention architecture trained solely on thermal equilibrium snapshots. Without imposing a locality cutoff or configuration independence, the dense scalar matrix in our architecture develops locality and nearly configuration-independent coefficients that recover the microscopic interactions of the spin glass with Pearson correlation $0.993$ on $2{,}000$ unseen sequences. This provides quantitative, falsifiable evidence for the \emph{emergence of the Hamiltonian} in a deep generative model. Similarly, in an explicit-water phenol experiment, the network is trained only on coordinate data. With sufficient expressive capacity, a molecular-graph-guided Pairformer recovers the dominant bond, angle, and proper-torsion potential-of-mean-force (PMF) sectors. A sector-preserving projection of configuration-dependent coefficients onto constants yields $R_{\rm const}^2=0.9669$ on unseen molecular dynamics (MD), providing molecular-scale evidence for partial \emph{emergence of the Hamiltonian}.