This work presents Neural Equivariant Interatomic Potentials (NequIP), an E(3)-equivariant neural network approach for learning interatomic potentials from ab-initio calculations for molecular dynamics simulations. While most contemporary symmetry-aware models use invariant convolutions and only act on scalars, NequIP employs E(3)-equivariant convolutions for interactions of geometric tensors, resulting in a more information-rich and faithful representation of atomic environments. The method achieves state-of-the-art accuracy on a challenging and diverse set of molecules and materials while exhibiting remarkable data efficiency. NequIP outperforms existing models with up to three orders of magnitude fewer training data, challenging the widely held belief that deep neural networks require massive training sets. The high data efficiency of the method allows for the construction of accurate potentials using high-order quantum chemical level of theory as reference and enables high-fidelity molecular dynamics simulations over long time scales.

Large-scale imaging surveys will increase the number of galaxy-scale strong lensing candidates by maybe three orders of magnitudes beyond the number known today. Finding these rare objects will require picking them out of at least tens of millions of images, and deriving scientific results from them will require quantifying the efficiency and bias of any search method. To achieve these objectives automated methods must be developed. Because gravitational lenses are rare objects, reducing false positives will be particularly important. We present a description and results of an open gravitational lens finding challenge. Participants were asked to classify 100,000 candidate objects as to whether they were gravitational lenses or not with the goal of developing better automated methods for finding lenses in large data sets. A variety of methods were used including visual inspection, arc and ring finders, support vector machines (SVM) and convolutional neural networks (CNN). We find that many of the methods will be easily fast enough to analyse the anticipated data flow. In test data, several methods are able to identify upwards of half the lenses after applying some thresholds on the lens characteristics such as lensed image brightness, size or contrast with the lens galaxy without making a single false-positive identification. This is significantly better than direct inspection by humans was able to do. Having multi-band, ground based data is found to be better for this purpose than single-band space based data with lower noise and higher resolution, suggesting that multi colour data is crucial. Multi-band space based data will be superior to ground based data. The most difficult challenge for a lens finder is differentiating between rare, irregular and ring-like face-on galaxies and true gravitational lenses. The degree to which the efficiency and biases of lens finders can be quantified largely depends on the realism of the simulated data on which the finders are trained.Article number, page 1 of 26

Deep learning has been immensely successful at a variety of tasks, ranging from classification to artificial intelligence. Learning corresponds to fitting training data, which is implemented by descending a very high-dimensional loss function. Understanding under which conditions neural networks do not get stuck in poor minima of the loss, and how the landscape of that loss evolves as depth is increased remains a challenge. Here we predict, and test empirically, an analogy between this landscape and the energy landscape of repulsive ellipses. We argue that in fully-connected deep networks a phase transition delimits the over-and under-parametrized regimes where fitting can or cannot be achieved. In the vicinity of this transition, properties of the curvature of the minima of the loss (the spectrum of the hessian) are critical. This transition shares direct similarities with the jamming transition by which particles form a disordered solid as the density is increased, which also occurs in certain classes of computational optimization and learning problems such as the perceptron. Our analysis gives a simple explanation as to why poor minima of the loss cannot be encountered in the overparametrized regime. Interestingly, we observe that the ability of fully-connected networks to fit random data is independent of their depth, an independence that appears to also hold for real data. We also study a quantity ∆ which characterizes how well (∆ < 0) or badly (∆ > 0) a datum is learned. At the critical point it is power-law distributed on several decades, P+(∆) ∼ ∆ θ for ∆ > 0 and P−(∆) ∼ (−∆) −γ for ∆ < 0, with exponents that depend on the choice of activation function. This observation suggests that near the transition the loss landscape has a hierarchical structure and that the learning dynamics is prone to avalanche-like dynamics, with abrupt changes in the set of patterns that are learned.

Supervised deep learning involves the training of neural networks with a large number N of parameters. For large enough N , in the so-called over-parametrized regime, one can essentially fit the training data points. Sparsity-based arguments would suggest that the generalization error increases as N grows past a certain threshold N * . Instead, empirical studies have shown that in the over-parametrized regime, generalization error keeps decreasing with N . We resolve this paradox through a new framework. We rely on the so-called Neural Tangent Kernel, which connects large neural nets to kernel methods, to show that the initialization causes finite-size random fluctuations f N −f N ∼ N −1/4 of the neural net output function f N around its expectationf N . These affect the generalization error N for classification: under natural assumptions, it decays to a plateau value ∞ in a power-law fashion ∼ N −1/2 . This description breaks down at a so-called jamming transition N = N * . At this threshold, we argue that f N diverges. This result leads to a plausible explanation for the cusp in test error known to occur at N * . Our results are confirmed by extensive empirical observations on the MNIST and CIFAR image datasets. Our analysis finally suggests that, given a computational envelope, the smallest generalization error is obtained using several networks of intermediate sizes, just beyond N * , and averaging their outputs. arXiv:1901.01608v5 [cond-mat.dis-nn]

In this paper we first recall the recent result that in deep networks a phase transition, analogous to the jamming transition of granular media, delimits the over-and under-parametrized regimes where fitting can or cannot be achieved. The analysis leading to this result support that for proper initialization and architectures, in the whole over-parametrized regime poor minima of the loss are not encountered during training, because the number of constraints that hinders the dynamics is insufficient to allow for the emergence of stable minima. Next, we study systematically how this transition affects generalization properties of the network (i.e. its predictive power). As we increase the number of parameters of a given model, starting from an under-parametrized network, we observe for gradient descent that the generalization error displays three phases: (i) initial decay, (ii) increase until the transition point -where it displays a cusp -and (iii) slow decay toward an asymptote as the network width diverges. However if early stopping is used, the cusp signaling the jamming transition disappears. Thereby we identify the region where the classical phenomenon of over-fitting takes place as the vicinity of the jamming transition, and the region where the model keeps improving with increasing the number of parameters, thus organizing previous empirical observations made in modern neural networks. ‡ The often used cross-entropy loss function also displays a transition where all data are well-fitted. However, in the over-parametrized regime the dynamic never stops, as the total loss vanishes only if the output and therefore the weights diverge. Imposing a time cut-off is done in practice, but it blurs the criticality near jamming, as exemplified below with the early stopping procedure.

We analyze numerically the training dynamics of deep neural networks (DNN) by using methods developed in statistical physics of glassy systems. The two main issues we address are (1) the complexity of the loss landscape and of the dynamics within it, and (2) to what extent DNNs share similarities with glassy systems. Our findings, obtained for different architectures and datasets, suggest that during the training process the dynamics slows down because of an increasingly large number of flat directions. At large times, when the loss is approaching zero, the system diffuses at the bottom of the landscape. Despite some similarities with the dynamics of mean-field glassy systems, in particular, the absence of barrier crossing, we find distinctive dynamical behaviors in the two cases, showing that the statistical properties of the corresponding loss and energy landscapes are different. In contrast, when the network is under-parametrized we observe a typical glassy behavior, thus suggesting the existence of different phases depending on whether the network is under-parametrized or over-parametrized.

How many training data are needed to learn a supervised task? It is often observed that the generalization error decreases as n −β where n is the number of training examples and β is an exponent that depends on both data and algorithm. In this work we measure β when applying kernel methods to real datasets. For MNIST we find β ≈ 0.4 and for CIFAR10 β ≈ 0.1, for both regression and classification tasks, and for Gaussian or Laplace kernels. To rationalize the existence of non-trivial exponents that can be independent of the specific kernel used, we study the teacher–student framework for kernels. In this scheme, a teacher generates data according to a Gaussian random field, and a student learns them via kernel regression. With a simplifying assumption—namely that the data are sampled from a regular lattice—we derive analytically β for translation invariant kernels, using previous results from the kriging literature. Provided that the student is not too sensitive to high frequencies, β depends only on the smoothness and dimension of the training data. We confirm numerically that these predictions hold when the training points are sampled at random on a hypersphere. Overall, the test error is found to be controlled by the magnitude of the projection of the true function on the kernel eigenvectors whose rank is larger than n. Using this idea we predict the exponent β from real data by performing kernel PCA, leading to β ≈ 0.36 for MNIST and β ≈ 0.07 for CIFAR10, in good agreement with observations. We argue that these rather large exponents are possible due to the small effective dimension of the data.

Context. Future large-scale surveys with high-resolution imaging will provide us with approximately 10 5 new strong galaxy-scale lenses. These strong-lensing systems will be contained in large data amounts, however, which are beyond the capacity of human experts to visually classify in an unbiased way. Aims. We present a new strong gravitational lens finder based on convolutional neural networks (CNNs). The method was applied to the strong-lensing challenge organized by the Bologna Lens Factory. It achieved first and third place, respectively, on the space-based data set and the ground-based data set. The goal was to find a fully automated lens finder for ground-based and space-based surveys that minimizes human inspection. Methods. We compared the results of our CNN architecture and three new variations ("invariant" "views" and "residual") on the simulated data of the challenge. Each method was trained separately five times on 17 000 simulated images, cross-validated using 3 000 images, and then applied to a test set with 100 000 images. We used two different metrics for evaluation, the area under the receiver operating characteristic curve (AUC) score, and the recall with no false positive (Recall 0FP ). Results. For ground-based data, our best method achieved an AUC score of 0.977 and a Recall 0FP of 0.50. For space-based data, our best method achieved an AUC score of 0.940 and a Recall 0FP of 0.32. Adding dihedral invariance to the CNN architecture diminished the overall score on space-based data, but achieved a higher no-contamination recall. We found that using committees of five CNNs produced the best recall at zero contamination and consistently scored better AUC than a single CNN. Conclusions. We found that for every variation of our CNN lensfinder, we achieved AUC scores close to 1 within 6%. A deeper network did not outperform simpler CNN models either. This indicates that more complex networks are not needed to model the simulated lenses. To verify this, more realistic lens simulations with more lens-like structures (spiral galaxies or ring galaxies) are needed to compare the performance of deeper and shallower networks.