arXiv subjects

Christoph Ortner

Publications and source records attributed to Christoph Ortner.

At least 55 records · Page 3Linked to original sources

arXiv2022

The Design Space of E(3)-Equivariant Atom-Centered Interatomic Potentials

The rapid progress of machine learning interatomic potentials over the past couple of years produced a number of new architectures. Particularly notable among these are the Atomic Cluster Expansion (ACE), which unified many of the earlier ideas around atom density-based descriptors, and Neural Equivariant Interatomic Potentials (NequIP), a message passing neural network with equivariant features that showed state of the art accuracy. In this work, we construct a mathematical framework that unifies these models: ACE is generalised so that it can be recast as one layer of a multi-layer architecture. From another point of view, the linearised version of NequIP is understood as a particular sparsification of a much larger polynomial model. Our framework also provides a practical tool for systematically probing different choices in the unified design space. We demonstrate this by an ablation study of NequIP via a set of experiments looking at in- and out-of-domain accuracy and smooth extrapolation very far from the training data, and shed some light on which design choices are critical for achieving high accuracy. Finally, we present BOTNet (Body-Ordered-Tensor-Network), a much-simplified version of NequIP, which has an interpretable architecture and maintains accuracy on benchmark datasets.

stat.ML↗

arXiv2022

Hyperactive Learning (HAL) for Data-Driven Interatomic Potentials

Data-driven interatomic potentials have emerged as a powerful class of surrogate models for {\it ab initio} potential energy surfaces that are able to reliably predict macroscopic properties with experimental accuracy. In generating accurate and transferable potentials the most time-consuming and arguably most important task is generating the training set, which still requires significant expert user input. To accelerate this process, this work presents \text{\it hyperactive learning} (HAL), a framework for formulating an accelerated sampling algorithm specifically for the task of training database generation. The key idea is to start from a physically motivated sampler (e.g., molecular dynamics) and add a biasing term that drives the system towards high uncertainty and thus to unseen training configurations. Building on this framework, general protocols for building training databases for alloys and polymers leveraging the HAL framework will be presented. For alloys, ACE potentials for AlSi10 are created by fitting to a minimal HAL-generated database containing 88 configurations (32 atoms each) with fast evaluation times of <100 microsecond/atom/cpu-core. These potentials are demonstrated to predict the melting temperature with excellent accuracy. For polymers, a HAL database is built using ACE, able to determine the density of a long polyethylene glycol (PEG) polymer formed of 200 monomer units with experimental accuracy by only fitting to small isolated PEG polymers with sizes ranging from 2 to 32.

physics.comp-ph↗

arXiv2022

A framework for a generalisation analysis of machine-learned interatomic potentials

Machine-learned interatomic potentials (MLIPs) and force fields (i.e. interaction laws for atoms and molecules) are typically trained on limited data-sets that cover only a very small section of the full space of possible input structures. MLIPs are nevertheless capable of making accurate predictions of forces and energies in simulations involving (seemingly) much more complex structures. In this article we propose a framework within which this kind of generalisation can be rigorously understood. As a prototypical example, we apply the framework to the case of simulating point defects in a crystalline solid. Here, we demonstrate how the accuracy of the simulation depends explicitly on the size of the training structures, on the kind of observations (e.g., energies, forces, force constants, virials) to which the model has been fitted, and on the fit accuracy. The new theoretical insights we gain partially justify current best practices in the MLIP literature and in addition suggest a new approach to the collection of training data and the design of loss functions.

math.NA↗

arXiv2022

BIP: Boost Invariant Polynomials for Efficient Jet Tagging

Deep Learning approaches are becoming the go-to methods for data analysis in High Energy Physics (HEP). Nonetheless, most physics-inspired modern architectures are computationally inefficient and lack interpretability. This is especially the case with jet tagging algorithms, where computational efficiency is crucial considering the large amounts of data produced by modern particle detectors. In this work, we present a novel, versatile and transparent framework for jet representation; invariant to Lorentz group boosts, which achieves high accuracy on jet tagging benchmarks while being orders of magnitudes faster to train and evaluate than other modern approaches for both supervised and unsupervised schemes.

physics.comp-ph↗

arXiv2022

Equivariant analytical mapping of first principles Hamiltonians to accurate and transferable materials models

We propose a scheme to construct predictive models for Hamiltonian matrices in atomic orbital representation from ab initio data as a function of atomic and bond environments. The scheme goes beyond conventional tight binding descriptions as it represents the ab initio model to full order, rather than in two-centre or three-centre approximations. We achieve this by introducing an extension to the Atomic Cluster Expansion (ACE) descriptor that represents Hamiltonian matrix blocks that transform equivariantly with respect to the full rotation group. The approach produces analytical linear models for the Hamiltonian and overlap matrices. Through an application to aluminium, we demonstrate that it is possible to train models from a handful of structures computed with density functional theory, and apply them to produce accurate predictions for the electronic structure. The model generalises well and is able to predict defects accurately from only bulk training data.

cond-mat.mtrl-sci↗

arXiv2022

Stability of the Minimum Energy Path

The minimum energy path (MEP) is the most probable transition path that connects two equilibrium states of a potential energy landscape. It has been widely used to study transition mechanisms as well as transition rates in the fields of chemistry, physics, and materials science. In this paper, we derive a novel result establishing the stability of MEPs under perturbations of the energy landscape. The result also represents a crucial step towards studying the convergence of various numerical approximations of MEPs, such as the nudged elastic band and string methods.

math.NA↗

arXiv2022

Atomic cluster expansion and wave function representations

The atomic cluster expansion (ACE) has been highly successful for the parameterisation of symmetric (invariant or equivariant) properties of many-particle systems. Here, we generalize its derivation to anti-symmetric functions. We show how numerous well-known linear representations of wave functions naturally arise within this framework and we explore how recent successful nonlinear parameterisations can be further enhanced by employing ACE methodology. From this analysis we propose a wide design space of promising wave function representations.

cond-mat.mtrl-sci↗

arXiv2022

Theoretical Study of Elastic Far-Field Decay from Dislocations in Multilattices

We precisely and rigorously characterise the decay of elastic fields generated by dislocations in crystalline materials, focusing specifically on the role of multilattices. Concretely, we establish that the elastic field generated by a dislocation in a multilattice can be decomposed into a continuum field predicted by a linearised Cauchy-Born elasticity theory, and a discrete and nonlinear core corrector representing the defect core. We demonstrate both analytically and numerically the consequences of this result for cell size effects in numerical simulations.

math.AP↗

arXiv2022

Multilayer atomic cluster expansion for semi-local interactions

Traditionally, interatomic potentials assume local bond formation supplemented by long-range electrostatic interactions when necessary. This ignores intermediate range multi-atom interactions that arise from the relaxation of the electronic structure. Here, we present the multilayer atomic cluster expansion (ml-ACE) that includes collective, semi-local multi-atom interactions naturally within its remit. We demonstrate that ml-ACE significantly improves fit accuracy compared to a local expansion on selected examples and provide physical intuition to understand this improvement.

cond-mat.mtrl-sci↗

arXiv2022

Comment on "Manifolds of quasi-constant SOAP and ACSF fingerprints and the resulting failure to machine learn four body interactions"

The "quasi-constant" SOAP and ACSF fingerprint manifolds recently discovered by Parsaeifard and Goedecker are a direct consequence of the presence of degenerate pairs of configurations, a known shortcoming of all low-body-order atom-density correlation representations of molecular structures. Contrary to the configurations that are rigorously singular -- that we demonstrate can only occur in finite, discrete sets -- the continuous "quasi-constant" manifolds exhibit low, but non-zero, sensitivity to atomic displacements. Thus, it is possible to build interpolative machine-learning models of high-order interactions along the manifold, even though the numerical instabilities associated with proximity to the exact singularities affect the accuracy and transferability of such models, to an extent that depends on numerical details of the implementation.

physics.chem-ph↗

arXiv2022

Optimal Evaluation of Symmetry-Adapted $n$-Correlations Via Recursive Contraction of Sparse Symmetric Tensors

We present a comprehensive analysis of an algorithm for evaluating high-dimensional polynomials that are invariant under permutations and rotations. The key bottleneck is the contraction of a high-dimensional symmetric and sparse tensor with a specific sparsity pattern that is directly related to the symmetries imposed on the polynomial. We propose an explicit construction of a recursive evaluation strategy and show that it is optimal in the limit of infinite polynomial degree.

math.NA↗

arXiv2021

Local invertibility and sensitivity of atomic structure-feature mappings

The increasingly common applications of machine-learning schemes to atomic-scale simulations have triggered efforts to better understand the mathematical properties of the mapping between the Cartesian coordinates of the atoms and the variety of representations that can be used to convert them into a finite set of symmetric descriptors or features. Here, we analyze the sensitivity of the mapping to atomic displacements, showing that the combination of symmetry and smoothness leads to mappings that have singular points at which the Jacobian has one or more null singular values (besides those corresponding to infinitesimal translations and rotations). This is in fact desirable, because it enforces physical symmetry constraints on the values predicted by regression models constructed using such representations. However, besides these symmetry-induced singularities, there are also spurious singular points, that we find to be linked to the incompleteness of the mapping, i.e. the fact that, for certain classes of representations, structurally distinct configurations are not guaranteed to be mapped onto different feature vectors. Additional singularities can be introduced by a too aggressive truncation of the infinite basis set that is used to discretize the representations.

physics.chem-ph↗

arXiv2021

Asymptotic Expansion of the Elastic Far-Field of a Crystalline Defect

Lattice defects in crystalline materials create long-range elastic fields which can be modelled on the atomistic scale using an infinite system of discrete nonlinear force balance equations. Starting with these equations, this work rigorously derives a novel far-field expansion of these fields: The expansion is computable and is expressed as a sum of continuum correctors and discrete multipole terms which decay with increasing algebraic rate as the order of the expansion increases. Truncating the expansion leaves a remainder describing the defect core structure, which is localised in the sense that it decays with an algebraic rate corresponding to the order at which the truncation occurred.

math.AP↗

arXiv2021

QM/MM Methods for Crystalline Defects. Part 3: Machine-Learned Interatomic Potentials

We develop and analyze a framework for consistent QM/MM (quantum/classic) hybrid models of crystalline defects, which admits general atomistic interactions including traditional off-the-shell interatomic potentials as well as state of art "machine-learned interatomic potentials". We (i) establish an a priori error estimate for the QM/MM approximations in terms of matching conditions between the MM and QM models, and (ii) demonstrate how to use these matching conditions to construct practical machine learned MM potentials specifically for QM/MM simulations.

math.NA↗

arXiv2021

Physics-inspired structural representations for molecules and materials

The first step in the construction of a regression model or a data-driven analysis, aiming to predict or elucidate the relationship between the atomic scale structure of matter and its properties, involves transforming the Cartesian coordinates of the atoms into a suitable representation. The development of atomic-scale representations has played, and continues to play, a central role in the success of machine-learning methods for chemistry and materials science. This review summarizes the current understanding of the nature and characteristics of the most commonly used structural and chemical descriptions of atomistic structures, highlighting the deep underlying connections between different frameworks, and the ideas that lead to computationally efficient and universally applicable models. It emphasizes the link between properties, structures, their physical chemistry and their mathematical description, provides examples of recent applications to a diverse set of chemical and materials science problems, and outlines the open questions and the most promising research directions in the field.

physics.chem-ph↗

arXiv2021

Rigorous body-order approximations of an electronic structure potential energy landscape

We show that the local density of states (LDOS) of a wide class of tight-binding models has a weak body-order expansion. Specifically, we prove that the resulting body-order expansion for analytic observables such as the electron density or the energy has an exponential rate of convergence both at finite Fermi-temperature as well as for insulators at zero Fermi-temperature. We discuss potential consequences of this observation for modelling the potential energy landscape, as well as for solving the electronic structure problem.

math-ph↗

arXiv2021

Crouzeix-Raviart finite element method for non-autonomous variational problems with Lavrentiev gap

We investigate the convergence of the Crouzeix-Raviart finite element method for variational problems with non-autonomous integrands that exhibit non-standard growth conditions. While conforming schemes fail due to the Lavrentiev gap phenomenon, we prove that the solution of the Crouzeix-Raviart scheme converges to a global minimiser. Numerical experiments illustrate the performance of the scheme and give additional analytical insights.

math.NA↗

arXiv2021

Atomic Cluster Expansion: Completeness, Efficiency and Stability

The Atomic Cluster Expansion (Drautz, Phys. Rev. B 99, 2019) provides a framework to systematically derive polynomial basis functions for approximating isometry and permutation invariant functions, particularly with an eye to modelling properties of atomistic systems. Our presentation extends the derivation by proposing a precomputation algorithm that yields immediate guarantees that a complete basis is obtained. We provide a fast recursive algorithm for efficient evaluation and illustrate its performance in numerical tests. Finally, we discuss generalisations and open challenges, particularly from a numerical stability perspective, around basis optimisation and parameter estimation, paving the way towards a comprehensive analysis of the convergence to a high-fidelity reference model.

math.NA↗