Equivariant Flows: sampling configurations for multi-body systems with symmetric energies

By Jonas Köhler, Leon Klein, Frank No'e. 2019

108 citations recorded by the source corpus.

Flows are exact-likelihood generative neural networks that transform samples from a simple prior distribution to the samples of the probability distribution of interest. Boltzmann Generators (BG) combine flows and statistical mechanics to sample equilibrium states of strongly interacting many-body systems such as proteins with 1000 atoms. In order to scale and generalize these results, it is essential that the natural symmetries of the probability density - in physics defined by the invariances of the energy function - are built into the flow. Here we develop theoretical tools for constructing such equivariant flows and demonstrate that a BG that is equivariant with respect to rotations and particle permutations can generalize to sampling nontrivially new configurations where a nonequivariant BG cannot.

Subfields: Materials Chemistry, Molecular Biology, Statistical and Nonlinear Physics, AI for Physical Sciences

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Citations in the corpus

Stochastic Normalizing Flows (2020, methodology): “∆ S t can be interpreted as the entropy in the variational free energy of the ﬂow transformation (No´e et al., 2019; Li & Wang, 2018).”
Equivariant Flows: exact likelihood generative learning for symmetric densities (2020, methodology): “A preliminary account of equivariant normalizing flows has been given in two recent workshop submissions (Rezende et al., 2019; Köhler et al., 2019).”
Equivariant Hamiltonian Flows (2019, neutral): “U(n) and SU(n)), multi-particle systems [20], etc.”
Normalizing Flows for Probabilistic Modeling and Inference (2019, methodology): “The concept of equivariant flows was proposed concurrently by Köhler et al. (2019) and Rezende et al. (2019), who described methods for building equivariant transformations in the context of continuous-time flows.”
Continuous-in-Depth Neural Networks (2020, background): “Other theoretical research fronts include interfacing Neural ODEs with normalizing flows [22, 79, 40, 56], graph NNs [61, 67], stochastic differential equations [52, 37, 24, 47] and RNNs [10, 65, 27, 18, 49].”
Wavelet Networks: Scale Equivariant Learning From Raw Waveforms (2020, background): “Similar intuitions have recently propelled the development of several deep learning approaches to model physical systems [54, 30, 73, 20].”
Normalizing Flows: An Introduction and Review of Current Methods (2020, background): “Applications include image generation [41], [57], noise modelling [1], video generation [60], audio generation [27], [53], [77], graph generation [65], reinforcement learning [67], [70], [93], computer graphics [69], and physics [51], [58], [71], [104], [105].”
Geometric and Physical Quantities improve E(3) Equivariant Message Passing (2021, unknown): “We further input the norm of the velocity as scalar, which altogether results in an input (Thomas et al., 2018) .0155 .0182 NMP (Gilmer et al., 2017) .0107 .0017 Radial Field (K¨ohler et al., 2019) .0104 .0019 EGNN (Satorras et al., 2021) .0070 ± .00022 .0029 SE linear ( l f = 2 ,l a = 2 ) .0116 ± .”
How to train your neural ODE (2020, background): “Neural ODEs have shown great promise in the physical sciences (Köhler et al., 2019), in modeling irregular time series (Rubanova et al., 2019), mean field games (Ruthotto et al., 2019), and for generative modeling through normalizing flows with free-form Jacobians (Grathwohl et al., 2019).”