Differentiable free energy surface: a variational approach to directly observing rare events using generative deep-learning models
Authors: Shuo-Hui Li, Chen Chen, Yao-Wen Zhang, Ding Pan
Organizations: Department of Physics, The Hong Kong University of Science and Technology, Hong Kong, China · Department of Chemistry, The Hong Kong University of Science and Technology, Hong Kong, China · IAS Center for AI for Scientific Discoveries, Hong Kong University of Science and Technology, Hong Kong, China
Rare events are central to the evolution of complex many-body systems, characterized as key transitional configurations on the free energy surface (FES). Conventional methods require adequate sampling of rare event transitions to obtain the FES, which is computationally very demanding. Here we introduce the variational free energy surface (VaFES), a dataset-free framework that directly models FESs using tractable-density generative models. Rare events can then be immediately identified from the FES with their configurations generated directly via one-shot sampling of generative models. By extending a coarse-grained collective variable (CV) into its reversible equivalent, VaFES constructs a latent space of intermediate representation in which the CVs explicitly occupy a subset of dimensions. This latent-space construction preserves the physical interpretability and transparent controllability of the CVs by design, while accommodating arbitrary CV formulations. The reversibility makes the system energy exactly accessible, enabling variational optimization of the FES without pre-generated simulation data. A single optimization yields a continuous, differentiable FES together with one-shot generation of rare-event configurations. Our method can reproduce the exact analytical solution for the bistable dimer potential and identify a chignolin native folded state in close alignment with the experimental NMR structure. Our approach thus establishes a scalable, systematic framework for advancing the study of complex statistical systems.
Figures & tables
Figure 1: Schematic illustration of the VaFES framework. The coarse-grained collective variable (CV) transformation is expanded into a bijective transformation T , simultaneously establishing two reversible mappings. It maps configuration variables of physical space, x , to the intermediate representation (s,u) , where the CVs s explicitly occupy a subset of the dimensions and the remaining degrees of freedom are collected in the auxiliary variables u . The same T transformation also maps the physical energy function into the latent-space energy of intermediate representations, thereby defining the conditional distribution π(u∣s) for the auxiliary variables u . A tractable-density generative model, illustrated here by a multi-coupling normalizing flow, parameterizes pθ(u∣s) and enables one-shot direct sampling of auxiliary variables conditioned on the CVs. Minimizing the divergence between pθ(u∣s) and π(u∣s) yields a variational estimate of the free energy surface and, through the inverse map T−1 , the corresponding physical configurations.
Figure 2: Free energy surface of the bistable dimer. (a) The results obtained by VaFES are shown by the purple solid line and the exact results by the red dashed line. The temperature is fixed at T=1.0 . The inset shows the potential energy landscape of the dimer. The CV is the bond length between the two particles. (b) The free energy landscape as a continuous function of temperature and bond length.
Figure 3: Free energies of diazene ( NX2HX2 ) at 120 K. (a) Estimated free energy surface of diazene. The CVs are a machine-learning coordinate indicating the trans state at 0 and the cis state at 1 on the x -axis, and the z -displacement of H2 on the y -axis. Two reaction paths identified by NEB are shown: the inversion path (red line) and the torsion path (green line). (b) Free energy profile along the inversion pathway projected onto the x -axis, together with representative diazene configurations. The local-frame coordinate definitions are illustrated using the first configuration. (c) Free energy profile along the torsion pathway projected onto the x -axis, together with representative diazene configurations.
Figure 4: Structural change and free energy landscape of alanine dipeptide at 300 K. (a) The structure of the dipeptide molecule defines the local-frame coordinate and the two backbone dihedral angles ψ and ϕ . (b) Molecular configurations along the reaction path identified by the NEB method. (c) Estimated free energy surface of alanine dipeptide obtained by VaFES. The CV coordinates are the two dominant components identified by TICA. A reaction path (red line) was identified using NEB, with endpoints fixed at the two local minima separated by the greatest distance. (d) Free energy profile along the reaction path in (c). The backbone dihedral angles, ψ and ϕ , are shown by the dashed blue and dash-dot pink lines, respectively.
Figure 5: Free energy profile and key configurations of chignolin (PDB 1UAO) at 350 K. (a) The estimated free energy surface. The CVs are the two inter- \text{C}{\vphantom{\text{X}}}_{\smash[t]{\text{\alpha}}} distances ∣CXαX1−CXαX10∣ and ∣CXαX3−CXαX8∣ . Six regions in CV space are boxed and labeled as key states. (b) Two folding pathways implied by the estimated FES. In the upper pathway, folding is initiated by the hydrophobic collapse from unfolded state iv to intermediate state v then vi, followed by turn formation from intermediate state vi to native folded state i. In the lower pathway, folding is initiated by turn formation: the two β -strands approach each other with modest hydrophobic closure and nascent turn formation from unfolded state iv to intermediate state iii, followed by tight turn formation from intermediate state iii to intermediate state ii and final hydrophobic collapse from intermediate state ii to native folded state i.
Appendix figures & tables5 assets
Supplementary material from the paper’s appendix.
Appendix
Parameter
Value
Spline bins K
50
Coupling layers
16
MLP hidden dimensions
20,50,100,150
Activation
ELU
Prior distribution
Uniform
Optimizer
Adamax
Appendix
Table S1 : VaFES hyperparameters for the bistable dimer potential application.
Figure S1 : The analytical results of the FES of the bistable dimer potential system as a continuous function of the temperature and the bond length CV.
Parameter
Value
Spline bins K
50
Coupling layers
16
MLP hidden dimensions
(20,50,100,150)
Activation
ELU
Prior distribution
Uniform
Optimizer
Adamax
Appendix
Table S2 : Hyperparameters for the diazene application.
Parameter
Value
Temperature
2.49\mathrm{kJ}\text{,}{\mathrm{mol}}^{-1}(300K)
Spline bins K
50
Coupling layers
12
MLP hidden dimensions
(512,512,512,1024,1024)
Activation
ELU
Prior distribution
Uniform
Appendix
Table S3 : Hyperparameters for the alanine dipeptide application.
Parameter
Value
Temperature β
0.35\mathrm{mol}\text{,}{\mathrm{kJ}}^{-1}(≈350K)
Spline bins K
32
Coupling layers (stage 1 / 2 / 3)
3 / 14+1 / 8
MLP hidden dims
(512,1024,1024)
ResNet1d channels / hidden channels
256 / 128
ResNet1d kernel size
25
Appendix
Table S4 : VaFES hyperparameters for the chignolin application.
Center for Data Science, Peking University · LMAM and School of Mathematical Sciences, Peking University · Center for Machine Learning Research, Peking University
Chemical Physics Program and Institute for Physical Science and Technology, University of Maryland, College Park 20742, USA · Biophysics Program and Institute for Physical Science and Technology, University of Maryland, College Park 20742, USA · Department of Chemistry and Biochemistry and Institute for Physical Science and Technology, University of Maryland, College Park 20742, USA +1