Neural Canonical Transformation with Symplectic Flows

Abstract

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. arXiv:1910.00024v2 [cond-mat.stat-mech]

Equivariant Flow-Based Sampling for Lattice Gauge Theory

“…This is true, in particular, for the lattice formulation of quantum chromodynamics (QCD) [21][22][23], which enables calculations of nonperturbative phenomena arising from the standard model of particle physics. Recently, there has been progress in the development of flow-based generative models which can be trained to directly produce samples from a given probability distribution; early success has been demonstrated in theories of bosonic matter, spin systems, molecular systems, and for Brownian motion [24][25][26][27][28][29][30][31][32][33][34]. This progress builds on the great success of flow-based approaches for image, text, and structured object generation [35][36][37][38][39][40][41][42], as well as non-flow-based machine learning techniques applied to sampling for physics [43][44][45][46][47][48].…”

Ab-initio study of interacting fermions at finite temperature with neural canonical transformation

“…Parametrizing and optimizing a rich family of equivariant many-body unitary transformations U turn out to be a fairly nontrivial task. In this paper, we present an elegant solution to this problem by constructing U as unitary representation of the canonical transformation of phase space variables in classical mechanics, extending the previous work on neural canonical transformations [14] from classical to quantum domain. The resulting approach naturally generalizes the ground-state variational Monte Carlo (VMC) method [15,16] to finite temperatures and is not hindered by the fermion sign problem.…”

“…However, they still demand advances in quantum technologies to be practically useful. Third, variational free energy studies of statistical mechanics and field theory problems [14,[35][36][37][38][39][40] can be regarded as the classical counterparts of the present approach. Last but not least, the so-called quantum flow approach [41] performs similar unitary transformation to a single-particle basis.…”

Learning Non-linear Wavelet Transformation via Normalizing Flow

“…Normalizing flows, on the other hand, are a family of non-linear invertible deep learning models (Dinh et al, 2015;Kingma & Dhariwal, 2018;Chen et al, 2018;Li et al, 2020;Li & Wang, 2018). Their applications cover many topics, including image generation (Kingma & Dhariwal, 2018;Dinh et al, 2016), independent component analysis (ICA) (Dinh et al, 2015;Sorrenson et al, 2020), variational inference (Kingma et al, 2016b;Rezende & Mohamed, 2015), Monte Carlo sampling (Song et al, 2017), and scientific applications (Li et al, 2020;Li & Wang, 2018;Hu et al, 2020a;Noé et al, 2019). Normalizing flows are a promising candidate for learnable wavelet transformation.…”