arXiv subjects

Christoph Ortner

Publications and source records attributed to Christoph Ortner.

At least 19 recordsLinked to original sources

arXiv2026

Stable Recovery of Matrix Gauge Classes from Pointwise Invariants

A parameterized matrix family $x\mapsto H(x)$ on a configuration domain is determined by its physical content only up to a constant orthogonal change of basis. This gauge ambiguity is intrinsic to data-driven Hamiltonian models, such as tight-binding parameterizations, reduced-order electronic structure methods, or excited-state models. It raises a basic inverse problem: what observations of $H(x)$ suffice to identify the family up to this gauge? The pointwise spectrum is incomplete already for linear families on $\mathbb{R}$. Here, we prove that, under natural non-degeneracy and connectivity assumptions, augmenting the spectrum with loop products of the coupling matrices in the instantaneous eigenframe yields a complete invariant and that inversion is stable. We support the theory with numerical experiments.

math.NA↗

arXiv2026

A fast summation method for the DFT-D3 dispersion correction

The DFT-D3 dispersion correction is routinely added to machine learning force fields (MLFFs) trained on dispersion-deficient functionals such as PBE. Its environment-dependent pair coefficients, however, break the atom-centered separability that fast summation methods require, forcing practitioners either to truncate D3 or to accept a substantial slowdown. We introduce FourierD3, a method that uses a functional low-rank decomposition to restore this separability and enable particle-mesh evaluation in $O(N\log N)$ time without a real-space cutoff on the dispersion sum.

physics.comp-ph↗

arXiv2026

Equivariant Many-body Message Passing Interatomic Potentials for Magnetic Materials

Magnetism governs key properties of materials used in energy, data storage, and spintronic technologies, yet its complex coupling to lattice and electronic degrees of freedom challenges conventional first-principles approaches. We introduce an equivariant message-passing graph neural network that embeds atomic magnetic moments as explicit degrees of freedom, enabling the learning of magnetic interactions beyond collinear approximations. The model learns physically consistent and transferable representations of magnetic behaviour and can incorporate spin-orbit coupling, achieving near density-functional-theory accuracy with strong data efficiency across diverse magnetic systems by fine-tuning from a pre-trained model. Applications to structural transformations, finite-temperature magnetic phenomena, and materials screening for strongly spin-orbit coupled materials demonstrate transferable magnetic behaviour, establishing a practical foundation for data-driven, high-throughput discovery of complex magnetic materials.

cond-mat.mtrl-sci↗

arXiv2026

The High Cost of Data Augmentation for Learning Equivariant Models

According to Noether's theorem the presence of a continuous symmetry in a Hamiltonian systems is equivalent to the existence of a conserved quantity, yet these symmetries are not always explicitly enforced in data-driven models. There remains a debate whether or not encoding of symmetry into a model architecture is the optimal approach. A competing approach is to target approximate symmetry through data augmentation. In this work, we study two approaches aimed at improving the symmetry properties of such an approximation scheme: one based on a quadrature rule for the Haar measure on the compact Lie group encoding the continuous symmetry of interest and one based on a random sampling of that Haar measure. We demonstrate both theoretically and empirically that the quadrature augmentation leads to exact symmetry preservation in polynomial models, while the random augmentation has only square-root convergence of the symmetrization error.

math.NA↗

arXiv2026

Regularity Priors for the Linear Atomic Cluster Expansion

Machine-learned interatomic potentials enable large systems to be simulated for long time scales at near ab-initio accuracy. This accuracy is achieved by fitting extremely flexible model architectures to high quality reference data. In practice, this flexibility can cause unwanted behavior such as jagged predicted potential energy surfaces and generally poor out-of-distribution behavior. We investigate a general strategy for incorporating prior beliefs on the regularity of the target energy into linear Atomic Cluster Expansion (ACE) models and explore to what extent this approach improves the quality of the fitted models. Our main focus is an over-regularisation that replicates the Gaussian broadening used in Smooth Overlap of Atomic Positions (SOAP) descriptors within the ACE framework. Numerical tests indicate that the exact form of the prior is non-critical but that including such a prior leads to significant improvement in test errors, consistent repulsion at close-approach, eliminates spurious false minima in the potential energy and enhances stability during molecular dynamics simulations.

physics.chem-ph↗

arXiv2025

Stochastic Reconfiguration with Warm-Started SVD

The combination of the variational Monte Carlo (VMC) method with deep learning wave function architectures has led to several successes in ground-state calculations of quantum many-body systems in recent years. However, commonly used stochastic gradient-based methods often perform poorly on these parameter training problems and typically lack convergence guarantees. The stochastic reconfiguration (SR) method provides a robust preconditioner of the stochastic gradient, whose computational cost becomes prohibitive for large parameter spaces owing to the repeated inversion of large covariance matrices. To overcome this bottleneck, we propose a warm-started stochastic reconfiguration (WSSR) method, which integrates warm-start techniques from singular value decomposition (SVD) to refine low-rank approximations of the preconditioning matrix iteratively. Numerical experiments on typical atomic and molecular systems highlight the effectiveness of the WSSR method within VMC calculations.

math-ph↗

arXiv2025

Machine learning intermolecular transfer integrals with compact atomic cluster representations

Calculating intermolecular charge transfer integrals in organic semiconductors requires substantial computer resource for each individual calculation. We might alternatively construct a machine learning model for transfer integrals, which model the full six-degrees of freedom for the relative position of dimer pairs, trained on representative calculations for the molecules of interest. Recent developments have produced effective machine learning force fields, which model the total energy of atomic assemblies. We extend the Atomic Cluster Expansion (ACE) with the correct symmetries for transfer (kinetic-energy) integrals. Combined with a spherical harmonic basis makes, this forms a strong inductive bias and makes for a data efficient model. We introduce coarse-grained and heavy-atom representations, and assess the methodology on representative conjugated semiconductors: ethylene, thiophene, and naphthalene.

cond-mat.dis-nn↗

arXiv2025

Flexible Uncertainty Calibration for Machine-Learned Interatomic Potentials

Reliable uncertainty quantification (UQ) is essential for developing machine-learned interatomic potentials (MLIPs) in predictive atomistic simulations. Conformal prediction (CP) is a statistical framework that constructs prediction intervals with guaranteed coverage under minimal assumptions, making it an attractive tool for UQ. However, existing CP techniques, while offering formal coverage guarantees, often lack accuracy, scalability, and adaptability to the complexity of atomic environments. In this work, we present a flexible uncertainty calibration framework for MLIPs, inspired by CP but reformulated as a parameterized optimization problem. This formulation enables the direct learning of environment-dependent quantile functions, producing sharper and more adaptive predictive intervals at negligible computational cost. Using the foundation model MACE-MP-0 as a representative case, we demonstrate the framework across diverse benchmarks, including ionic crystals, catalytic surfaces, and molecular systems. Our results show order-of-magnitude improvements in uncertainty-error correlation, enhanced data efficiency in active learning, and strong generalization performance, together with reliable transfer of calibrated uncertainties across distinct exchange-correlation functionals. This work establishes a principled and data-efficient approach to uncertainty calibration in MLIPs, providing a practical route toward more trustworthy and transferable atomistic simulations.

physics.chem-ph↗

arXiv2025

An Atomic Cluster Expansion Potential for Twisted Multilayer Graphene

Twisted multilayer graphene, characterized by its moir\'e patterns arising from inter-layer rotational misalignment, serves as a rich platform for exploring quantum phenomena. Machine learning interatomic potentials (MLIPs) are a promising approach to model such systems. Our work develops a method to generate training and test datasets for fitting MLIPs that capture all possible misalignments but remain small-scale to facilitate efficient data generation and parameter estimation. To achieve this, we generate configurations with periodic boundary conditions suitable for DFT calculations, and then introduce an internal twist and shift within those supercell structures. Using this technique, supplemented with an active learning workflow, we fit an Atomic Cluster Expansion potential for simulating twisted multilayer graphene and test it for accuracy and robustness on a range of simulation tasks.

physics.comp-ph↗

arXiv2025

Scalable Data-Driven Basis Selection for Linear Machine Learning Interatomic Potentials

Machine learning interatomic potentials (MLIPs) provide an effective approach for accurately and efficiently modeling atomic interactions, expanding the capabilities of atomistic simulations to complex systems. However, a priori feature selection leads to high complexity, which can be detrimental to both computational cost and generalization, resulting in a need for hyperparameter tuning. We demonstrate the benefits of active set algorithms for automated data-driven feature selection. The proposed methods are implemented within the Atomic Cluster Expansion (ACE) framework. Computational tests conducted on a variety of benchmark datasets indicate that sparse ACE models consistently enhance computational efficiency, generalization accuracy and interpretability over dense ACE models. An added benefit of the proposed algorithms is that they produce entire paths of models with varying cost/accuracy ratio.

physics.comp-ph↗

arXiv2025

Transferable Machine Learning Potential X-MACE for Excited States using Integrated DeepSets

Conical intersections serve as critical gateways in photochemical reactions, enabling rapid nonradiative transitions between potential energy surfaces that underpin fundamental processes such as photosynthesis or vision. Their calculation with quantum chemistry is, however, extremely computationally intensive and their modeling with machine learning poses a significant challenge due to their inherently non-smooth and complex nature. To address this challenge, we introduce a deep learning architecture designed to precisely model excited states and improve their accuracy around these critical, non-smooth regions. Our model integrates Deep Sets into the Message Passing Atomic Cluster Expansion (MACE) framework resulting in a smooth representation of the non-smooth excited-state potential energy surfaces. We validate our method using numerous molecules, showcasing a significant improvement in accurately modeling the energy landscape around conical intersections compared to conventional excited-state models. Additionally, we apply ground-state foundational machine learning models as a basis for excited states. By doing so, we showcase that the developed model is capable of transferring not only from the ground state to excited states, but also within chemical space to molecular systems beyond those included in the training dataset. This advancement not only enhances the fidelity of excited-state modeling, but also lays the foundations for the investigation of more complex molecular systems.

physics.chem-ph↗

arXiv2025

Many-Body Coarse-Grained Molecular Dynamics with the Atomic Cluster Expansion

Molecular dynamics (MD) simulations provide detailed insight into atomic-scale mechanisms but are inherently restricted to small spatio-temporal scales. Coarse-grained molecular dynamics (CGMD) techniques allow simulations of much larger systems over extended timescales. In theory, these techniques can be quantitatively accurate, but common practice is to only target qualitatively correct behaviour of coarse-grained models. Recent advances in applying machine learning methodology in this setting are now being applied to create also quantitatively accurate CGMD models. We demonstrate how the Atomic Cluster Expansion parameterization (Drautz, 2019) can be used in this task to construct highly efficient, interpretable and accurate CGMD models. We focus in particular on exploring the role of many-body effects.

physics.comp-ph↗

arXiv2025

Nearsightedness in Materials with Indirect Band Gaps

We investigate the nearsightedness property in the linear tight binding model at zero Fermi-temperature. We focus on the decay property of the density matrix for materials with indirect band gaps. By representing the density matrix in reciprocal space, we establish a qualitatively sharp estimate for the exponential decay rate in homogeneous systems, possibly with localized perturbations. This work refines the estimates presented in (Ortner, Thomas & Chen, 2020) for systems with small band gaps.

math-ph↗

arXiv2024

Fast automatically differentiable matrix functions and applications in molecular simulations

We describe efficient differentiation methods for computing Jacobians and gradients of a large class of matrix functions including the matrix logarithm $\log(A)$ and $p$-th roots $A^{\frac{1}{p}}$. We exploit contour integrals and conformal maps as described by (Hale et al., SIAM J. Numer. Anal. 2008) for evaluation and differentiation and analyze the computational complexity as well as numerical accuracy compared to high accuracy finite difference methods. As a demonstrator application we compute properties of structural defects in silicon crystals at positive temperatures, requiring efficient and accurate gradients of matrix trace-logarithms.

physics.comp-ph↗

arXiv2024

Analysis of local structure of mechanical and thermal rearrangements in glasses with the atomic cluster expansion

We explore the structural signatures of excitations in amorphous materials with the atomic cluster expansion (ACE), a universal and complete linear basis of descriptors of the atomic environment. Body-orderd linear classifiers are constructed that distinguish between active and inactive particles in three different model glass formers, in which structural relaxation occurs either through spontaneous thermal activation or by simple shear. We find that in binary mixtures, maximum prediction accuracy is already achieved with very few two-body correlations, while a polymer glass requires both two- and three-body correlations. Trends are robust across both activation mechanisms.

cond-mat.dis-nn↗

arXiv2024

Equivariant Representation of Configuration-Dependent Friction Tensors in Langevin Heatbaths

Dynamics of coarse-grained particle systems derived via the Mori-Zwanzig projection formalism commonly take the form of a (generalized) Langevin equation with configuration-dependent friction and diffusion tensors. In this article, we introduce a class of equivariant representations of tensor-valued functions based on the Atomic Cluster Expansion (ACE) framework that allows for efficient learning of such configuration-dependent friction and diffusion tensors from data. Besides satisfying the correct equivariance properties with respect to the Euclidean group E(3), the resulting heat bath models satisfy a fluctuation-dissipation relation. Moreover, our models can be extended to include additional symmetries, such as momentum conservation, to preserve the hydrodynamic properties of the particle system. We demonstrate the capabilities of the model by constructing a model of configuration-dependent tensorial electronic friction calculated from first principles that arises during reactive molecular dynamics at metal surfaces.

cond-mat.mtrl-sci↗

arXiv2024

Parameterizing Intersecting Surfaces via Invariants

We introduce and analyze numerical companion matrix methods for the reconstruction of hypersurfaces with crossings from smooth interpolants given unordered or, without loss of generality, value-sorted data. The problem is motivated by the desire to machine learn potential energy surfaces arising in molecular excited state computational chemistry applications. We present simplified models which reproduce the analytically predicted convergence and stability behaviors as well as two application-oriented numerical experiments: the electronic excited states of Graphene featuring Dirac conical cusps and energy surfaces corresponding to a sulfur dioxide ($SO_2$) molecule in different configurations.

math.NA↗

arXiv2024

Self-consistent Coulomb interactions for machine learning interatomic potentials

A ubiquitous approach to obtain transferable machine learning-based models of potential energy surfaces for atomistic systems is to decompose the total energy into a sum of local atom-centred contributions. However, in many systems non-negligible long-range electrostatic effects must be taken into account as well. We introduce a general mathematical framework to study how such long-range effects can be included in a way that (i) allows charge equilibration and (ii) retains the locality of the learnable atom-centred contributions to ensure transferability. Our results give partial explanations for the success of existing machine learned potentials that include equilibriation and provide perspectives how to design such schemes in a systematic way. To complement the rigorous theoretical results, we describe a practical scheme for fitting the energy and electron density of water clusters.

physics.comp-ph↗