Shuo-Hui Li

Shuo-Hui Li, Chen Chen, Yao-Wen Zhang et al.

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.

Shuo-Hui Li, Yao-Wen Zhang, Ding Pan

We propose a variational modelling method with differentiable temperature for canonical ensembles. Using a deep generative model, the free energy is estimated and minimized simultaneously in a continuous temperature range. At optimal, this generative model is a Boltzmann distribution with temperature dependence. The training process requires no dataset, and works with arbitrary explicit density generative models. We applied our method to study the phase transitions (PT) in the Ising and XY models, and showed that the direct-sampling simulation of our model is as accurate as the Markov Chain Monte Carlo (MCMC) simulation, but more efficient. Moreover, our method can give thermodynamic quantities as differentiable functions of temperature akin to an analytical solution. The free energy aligns closely with the exact one to the second-order derivative, so this inclusion of temperature dependence enables the otherwise biased variational model to capture the subtle thermal effects at the PTs. These findings shed light on the direct simulation of physical systems using deep generative models

Shuo-Hui Li

Wavelet transformation stands as a cornerstone in modern data analysis and signal processing. Its mathematical essence is an invertible transformation that discerns slow patterns from fast ones in the frequency domain. Such an invertible transformation can be learned by a designed normalizing flow model. With a generalized lifting scheme as coupling layers, a factor-out layer resembling the downsampling, and parameter sharing at different levels of the model, one can train the normalizing flow to filter high-frequency elements at different levels, thus extending traditional linear wavelet transformations to learnable non-linear deep learning models. In this paper, a way of building such flow is proposed, along with a numerical analysis of the learned transformation. Then, we demonstrate the model's ability in image lossless compression, show it can achieve SOTA compression scores while achieving a small model size, substantial generalization ability, and the ability to handle high-dimensional data.

Shuo-Hui Li, Chen-Xiao Dong, Linfeng Zhang et al.

Canonical transformation plays a fundamental role in simplifying and solving classical Hamiltonian systems. We construct flexible and powerful canonical transformations as generative models using symplectic neural networks. The model transforms physical variables towards a latent representation with an independent harmonic oscillator Hamiltonian. Correspondingly, the phase space density of the physical system flows towards a factorized Gaussian distribution in the latent space. Since the canonical transformation preserves the Hamiltonian evolution, the model captures nonlinear collective modes in the learned latent representation. We present an efficient implementation of symplectic neural coordinate transformations and two ways to train the model. The variational free energy calculation is based on the analytical form of physical Hamiltonian. While the phase space density estimation only requires samples in the coordinate space for separable Hamiltonians. We demonstrate appealing features of neural canonical transformation using toy problems including two-dimensional ring potential and harmonic chain. Finally, we apply the approach to real-world problems such as identifying slow collective modes in alanine dipeptide and conceptual compression of the MNIST dataset.

Shuo-Hui Li, Lei Wang

We present a variational renormalization group (RG) approach using a deep generative model based on normalizing flows. The model performs hierarchical change-of-variables transformations from the physical space to a latent space with reduced mutual information. Conversely, the neural net directly maps independent Gaussian noises to physical configurations following the inverse RG flow. The model has an exact and tractable likelihood, which allows unbiased training and direct access to the renormalized energy function of the latent variables. To train the model, we employ probability density distillation for the bare energy function of the physical problem, in which the training loss provides a variational upper bound of the physical free energy. We demonstrate practical usage of the approach by identifying mutually independent collective variables of the Ising model and performing accelerated hybrid Monte Carlo sampling in the latent space. Lastly, we comment on the connection of the present approach to the wavelet formulation of RG and the modern pursuit of information preserving RG.