Kohn-Sham Equations as Regularizer: Building Prior Knowledge into Machine-Learned Physics

Li, Li; Hoyer, Stephan; Pederson, Ryan; Sun, Ruoxi; Cubuk, Ekin D.; Riley, Patrick; Burke, Kieron

doi:10.1103/physrevlett.126.036401

Cited by 147 publications

(152 citation statements)

References 49 publications

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“…We train the model inside a standard numerical method for solving the underlying PDEs as a differentiable program, with the neural networks and the numerical method written in a framework [JAX ( 38 )] supporting reverse-mode automatic differentiation. This allows for end-to-end gradient-based optimization of the entire algorithm, similar to prior work on density functional theory ( 39 ), molecular dynamics ( 40 ), and fluids ( 33 , 34 ). The methods we derive are equation-specific and require high-resolution ground-truth simulations for training data.…”

mentioning

confidence: 97%

Machine learning–accelerated computational fluid dynamics

Kochkov

Smith

Alieva

et al. 2021

Proc. Natl. Acad. Sci. U.S.A.

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596

254

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Numerical simulation of fluids plays an essential role in modeling many physical phenomena, such as weather, climate, aerodynamics, and plasma physics. Fluids are well described by the Navier–Stokes equations, but solving these equations at scale remains daunting, limited by the computational cost of resolving the smallest spatiotemporal features. This leads to unfavorable trade-offs between accuracy and tractability. Here we use end-to-end deep learning to improve approximations inside computational fluid dynamics for modeling two-dimensional turbulent flows. For both direct numerical simulation of turbulence and large-eddy simulation, our results are as accurate as baseline solvers with 8 to 10× finer resolution in each spatial dimension, resulting in 40- to 80-fold computational speedups. Our method remains stable during long simulations and generalizes to forcing functions and Reynolds numbers outside of the flows where it is trained, in contrast to black-box machine-learning approaches. Our approach exemplifies how scientific computing can leverage machine learning and hardware accelerators to improve simulations without sacrificing accuracy or generalization.

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mentioning

confidence: 97%

Machine learning–accelerated computational fluid dynamics

Kochkov

Smith

Alieva

et al. 2021

Proc. Natl. Acad. Sci. U.S.A.

Self Cite

596

254

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“…It adds to existing coarse-grained and statistical potentials used for simulation [ 47 – 50 ] and model scoring approaches [ 51 – 53 ]. More broadly, it points to DMS as a useful technology falling under the banner of differentiable programming [ 54 ], an expansion of the principles of deep learning to the concept of taking gradients through arbitrary algorithms and utilising the known structure of the system under study [ 55 , 56 ].…”

Section: Introductionmentioning

confidence: 99%

Differentiable molecular simulation can learn all the parameters in a coarse-grained force field for proteins

Greener

Jones

2021

PLoS ONE

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Finding optimal parameters for force fields used in molecular simulation is a challenging and time-consuming task, partly due to the difficulty of tuning multiple parameters at once. Automatic differentiation presents a general solution: run a simulation, obtain gradients of a loss function with respect to all the parameters, and use these to improve the force field. This approach takes advantage of the deep learning revolution whilst retaining the interpretability and efficiency of existing force fields. We demonstrate that this is possible by parameterising a simple coarse-grained force field for proteins, based on training simulations of up to 2,000 steps learning to keep the native structure stable. The learned potential matches chemical knowledge and PDB data, can fold and reproduce the dynamics of small proteins, and shows ability in protein design and model scoring applications. Problems in applying differentiable molecular simulation to all-atom models of proteins are discussed along with possible solutions and the variety of available loss functions. The learned potential, simulation scripts and training code are made available at https://github.com/psipred/cgdms.

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“…Indeed, machine-learning offers a new generation of accurate, highly non-local, xc functionals [7]. While those functionals are designed to perform tasks of different degree of complexity, all share the aim of learning one of the maps of DFT, namely, the Hohenberg-Kohn map between the external potential v(r) and the density ρ(r) [8][9][10][11][12][13], or the Kohn-Sham map between the density ρ(r) and the xc functional E xc [ρ(r)] and its functional derivative v xc [ρ(r)] = δE xc [ρ(r)]/δρ(r) [14][15][16].…”

Section: Introductionmentioning

confidence: 99%

Machine learning the derivative discontinuity of density-functional theory

Gedeon

Schmidt

Hodgson

et al. 2021

Mach. Learn.: Sci. Technol.

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Machine learning is a powerful tool to design accurate, highly non-local, exchange-correlation functionals for density functional theory. So far, most of those machine learned functionals are trained for systems with an integer number of particles. As such, they are unable to reproduce some crucial and fundamental aspects, such as the explicit dependency of the functionals on the particle number or the infamous derivative discontinuity at integer particle numbers. Here we propose a solution to these problems by training a neural network as the universal functional of density-functional theory that (a) depends explicitly on the number of particles with a piece-wise linearity between the integer numbers and (b) reproduces the derivative discontinuity of the exchange-correlation energy. This is achieved by using an ensemble formalism, a training set containing fractional densities, and an explicitly discontinuous formulation.

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Kohn-Sham Equations as Regularizer: Building Prior Knowledge into Machine-Learned Physics

Cited by 147 publications

References 49 publications

Machine learning–accelerated computational fluid dynamics

Machine learning–accelerated computational fluid dynamics

Differentiable molecular simulation can learn all the parameters in a coarse-grained force field for proteins

Machine learning the derivative discontinuity of density-functional theory

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