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A recipe for cracking the quantum scaling limit with machine learned electron densities

Mario Geiger

A long-standing goal of science is to accurately simulate large molecular systems using quantum mechanics. The poor scaling of current quantum chemistry algorithms on classical computers, however, imposes an effective limit of about a few dozen atoms on traditional electronic structure calculations. We present a machine learning (ML) method to break through this scaling limit for electron densities. We show that Euclidean neural networks can be trained to predict molecular electron densities from limited data. By learning the electron density, the model can be trained on small systems and make accurate predictions on large ones. In the context of water clusters, we show that an ML model trained on clusters of just 12 molecules contains all the information needed to make accurate electron density predictions on cluster sizes of 50 or more, beyond the scaling limit of current quantum chemistry methods.

IOP Publishing Ltd2023

Finding symmetry breaking order parameters with Euclidean neural networks

Mario Geiger

Curie's principle states that "when effects show certain asymmetry, this asymmetry must be found in the causes that gave rise to them." We demonstrate that symmetry equivariant neural networks uphold Curie's principle and can be used to articulate many symmetry-relevant scientific questions as simple optimization problems. We prove these properties mathematically and demonstrate them numerically by training a Euclidean symmetry equivariant neural network to learn symmetry breaking input to deform a square into a rectangle and to generate octahedra tilting patterns in perovskites.

AMER PHYSICAL SOC2021

Landscape and training regimes in deep learning

Mario Geiger, Leonardo Petrini, Matthieu Wyart

Deep learning algorithms are responsible for a technological revolution in a variety oftasks including image recognition or Go playing. Yet, why they work is not understood.Ultimately, they manage to classify data lying in high dimension – a feat genericallyimpossible due to the geometry of high dimensional space and the associatedcurse ofdimensionality. Understanding what kind of structure, symmetry or invariance makesdata such as images learnable is a fundamental challenge. Other puzzles include that(i) learning corresponds to minimizing a loss in high dimension, which is in generalnot convex and could well get stuck bad minima. (ii) Deep learning predicting powerincreases with the number of fitting parameters, even in a regime where data areperfectly fitted. In this manuscript, we review recent results elucidating (i, ii) andthe perspective they offer on the (still unexplained) curse of dimensionality paradox.We base our theoretical discussion on the (h,α) plane wherehcontrols the numberof parameters andαthe scale of the output of the network at initialization, andprovide new systematic measures of performance in that plane for two common imageclassification datasets. We argue that different learning regimes can be organized intoa phase diagram. A line of critical points sharply delimits an under-parametrized phasefrom an over-parametrized one. In over-parametrized nets, learning can operate intwo regimes separated by a smooth cross-over. At large initialization, it correspondsto a kernel method, whereas for small initializations features can be learnt, togetherwith invariants in the data. We review the properties of these different phases, ofthe transition separating them and some open questions. Our treatment emphasizesanalogies with physical systems, scaling arguments and the development of numericalobservables to quantitatively test these results empirically. Practical implications arealso discussed, including the benefit of averaging nets with distinct initial weights, orthe choice of parameters (h,α) optimizing performance.

2021