Klaus-Robert Müller, Frank Noé
Brooke Husic (FU)
01.01.2019 – 31.12.2021
We want to pioneer the new field of Quantum Kinetics, i.e., the study of quantum mechanical (QM) processes at long timescales. Simulating QM at long timescales has previously been unfeasible but is now coming into reach due to the convergence of mathematics and machine learning methods that we will develop here.
The equilibrium behavior of molecular systems is described by the stationary multi-body Schrödinger equation in the Born-Oppenheimer approximation. Let H be the Hamilton total energy operator and U(x) the energy of the molecule in configuration (atomic coordinates) x. For given x, the stationary Schrödinger equation must be solved over the space of antisymmetric functions in electronic coordinates r and spins s. Computing a high-accuracy numerical approximations for the Schrödinger equation, e.g., using the Coupled-Cluster method, may take from several hours to several days for a tiny 10-atom molecule and a single configuration x.
To address this problem, the PI Klaus-Robert Müller has developed Quantum Machine Learning (QML) together with A. Tkatchenko and others. In QML, one uses a dataset where U(x) has been solved by approximating the Schrödinger equation ab initio for a library of thousands of molecules, and trains a machine-learning structure to predict approximations of U(x) for new molecules and configurations x by using transferable features. Recently, the prediction error of QML has become so small that its predictions are on par with those of high-level QM methods such as Coupled-Cluster. However, QML methods can make predictions of U(x) in milliseconds, resulting in many orders of magnitude speedup.
Most molecular systems can change their configuration significantly, and therefore molecular dynamics (MD) simulations are conducted using an approximation to the energy, U(x), e.g., using the Langevin model. Unfortunately, the simulation time steps are on the order of one femtosecond, while configuration changes govern- ing molecular function are often rare events that may take milliseconds to hours. Even when the energy function is given by a fast, classical approximation or a QML model, the direct simulation of a single protein folding and unfolding event would take years to centuries on a supercomputer.
KML: Key to solving this sampling problem is conformation dynamics theory, developed by Christof Schütte and colleagues. For MD simulated under equilibrium conditions, which implies the existence of a unique stationary distribution and detailed balance, the kinetics, i.e. the statistical behavior of long-time MD is governed by the Markov propagator P(τ) with a real-valued dominant spectrum. The classical model for this propagator is a Markov State Model (MSM), where P is approximated by a transition matrix that switches between discrete clusters of configurations. The PI Frank Noé has been one of the pioneers in developing MSM methods and has made large- scale applications, such as the first all-atom model of protein-protein association and dissociation. A recent development is a family of variational approaches to optimally approximate the dominant spectral components of the Markov propagator. This variational approach turns the approximation of P into an ML problem. Recently, we have developed VAMPnets – neural networks that are trained on simulation data and approximate molecular kinetics with unprecendented quality.
As a first step, we will directly combine both state-of-the-art estimators by performing QML on a given QM data set, generating simulation trajectories from it, and performing KML learning on that data. However, in order to find an optimal solution, we will aim at uniting the two learning structures and solve the combined learning problem in an end-to-end fashion. We will address unsolved problems in the encoding of symmetries and invariances. We will also develop a reversible neural network version of VAMPnet.
The accessibility of Quantum Kinetics has far-reaching consequences in physics, chemistry and biology. Application questions include: what is the statistics of protonation changes in a protein and how do they couple with folding and binding? How do quantum effects influence the permeation of an ion through a membrane channel? What is the molecular mechanism of plastic deformation in a metal, when quantum effects are considered?
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