A vast proportion of total global computer time is spent on structure search and property prediction on the atomic scale in the fields of materials, physics and chemistry. Typical tasks include solving the Schrodinger equation in various approximations, molecular dynamics using Newtonian equations of motion, but also classification and design based on predicted interactions of molecules and periodic structures. Efficient representations of material and molecular structure, coupled with regularisation (in kernel methods and neural networks) are beginning to show dramatic speedups and reductions in the scaling of computational complexity at the same time as making step-changes in simulation accuracy. The first science applications that take advantage of the million-factor speedups include accurate calculations of the structure of amorphous materials and molecular liquids. Outstanding problems include, but with good prospects, finding better low dimensional representations (e.g. body-ordered expansions), clarifying the relationships between low dimensionality and interaction range, and providing global error bounds. One of the most pressing problems is "extrapolation", because typical simulations generate samples using stochastic processes based on the fitted models and thus underestimating the potential energy can lead to exponential amplification of errors.
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