arXiv subjects

Paul J. Atzberger

Publications and source records attributed to Paul J. Atzberger.

At least 19 recordsLinked to original sources

arXiv2026

Extending Neural Operators: Robust Handling of Functions Beyond the Training Set

We develop a rigorous framework for extending neural operators to handle out-of-distribution input functions. We leverage kernel approximation techniques and provide theory for characterizing the input-output function spaces in terms of Reproducing Kernel Hilbert Spaces (RKHSs). We provide theorems on the requirements for reliable extensions and their predicted approximation accuracy. We also establish formal relationships between specific kernel choices and their corresponding Sobolev Native Spaces. This connection further allows the extended neural operators to reliably capture not only function values but also their derivatives. Our methods are empirically validated through the solution of elliptic partial differential equations (PDEs) involving operators on manifolds having point-cloud representations and handling geometric contributions. We report results on key factors impacting the accuracy and computational performance of the extension approaches.

cs.LG↗

arXiv2025

Transferable Foundation Models for Geometric Tasks on Point Cloud Representations: Geometric Neural Operators

We introduce methods for obtaining pretrained Geometric Neural Operators (GNPs) that can serve as basal foundation models for use in obtaining geometric features. These can be used within data processing pipelines for machine learning tasks and numerical methods. We show how our GNPs can be trained to learn robust latent representations for the differential geometry of point-clouds to provide estimates of metric, curvature, and other shape-related features. We demonstrate how our pre-trained GNPs can be used (i) to estimate the geometric properties of surfaces of arbitrary shape and topologies with robustness in the presence of noise, (ii) to approximate solutions of geometric partial differential equations (PDEs) on manifolds, and (iii) to solve equations for shape deformations such as curvature driven flows. We release codes and weights for using GNPs in the package geo_neural_op. This allows for incorporating our pre-trained GNPs as components for reuse within existing and new data processing pipelines. The GNPs also can be used as part of numerical solvers involving geometry or as part of methods for performing inference and other geometric tasks.

cs.LG↗

arXiv2025

GD-VAEs: Geometric Dynamic Variational Autoencoders for Learning Nonlinear Dynamics and Dimension Reductions

We develop data-driven methods incorporating geometric and topological information to learn parsimonious representations of nonlinear dynamics from observations. The approaches learn nonlinear state-space models of the dynamics for general manifold latent spaces using training strategies related to Variational Autoencoders (VAEs). Our methods are referred to as Geometric Dynamic (GD) Variational Autoencoders (GD-VAEs). We learn encoders and decoders for the system states and evolution based on deep neural network architectures that include general Multilayer Perceptrons (MLPs), Convolutional Neural Networks (CNNs), and other architectures. Motivated by problems arising in parameterized PDEs and physics, we investigate the performance of our methods on tasks for learning reduced dimensional representations of the nonlinear Burgers Equations, Constrained Mechanical Systems, and spatial fields of Reaction-Diffusion Systems. GD-VAEs provide methods that can be used to obtain representations in manifold latent spaces for diverse learning tasks involving dynamics.

cs.LG↗

arXiv2024

Simulation of Stochastic Non-Equilibrium Thermal Effects of Particle Inclusions within Fluid Interfaces and Membranes

We formulate theoretical modeling approaches and develop practical computational simulation methods for investigating the non-equilibrium statistical mechanics of fluid interfaces with passive and active immersed particles. Our approaches capture phenomena taking into account thermal and dissipative energy exchanges, hydrodynamic coupling, and correlated spontaneous fluctuations. Our methods allow for modeling non-uniform time-varying temperature fields, fluid momentum fields, and their impacts on particle drift-diffusion dynamics. We show how practical stochastic numerical methods can be developed for these systems by performing analysis to factor operators analytically to obtain efficient algorithms for generating the fluctuating fields. We demonstrate our methods in a few simulation studies showing how they can be used to capture particle heating of the interface and the roles of thermal gradients on hydrodynamic fluctuations. The methods developed provide modeling and simulation approaches that can be used for further investigations of non-equilibrium phenomena in active soft materials, complex fluids, and biophysical systems.

cond-mat.soft↗

arXiv2024

Geometric Neural Operators (GNPs) for Data-Driven Deep Learning of Non-Euclidean Operators

We introduce Geometric Neural Operators (GNPs) for accounting for geometric contributions in data-driven deep learning of operators. We show how GNPs can be used (i) to estimate geometric properties, such as the metric and curvatures, (ii) to approximate Partial Differential Equations (PDEs) on manifolds, (iii) learn solution maps for Laplace-Beltrami (LB) operators, and (iv) to solve Bayesian inverse problems for identifying manifold shapes. The methods allow for handling geometries of general shape including point-cloud representations. The developed GNPs provide approaches for incorporating the roles of geometry in data-driven learning of operators.

cs.LG↗

arXiv2023

Surface Fluctuating Hydrodynamics Methods for the Drift-Diffusion Dynamics of Particles and Microstructures within Curved Fluid Interfaces

We introduce fluctuating hydrodynamics approaches on surfaces for capturing the drift-diffusion dynamics of particles and microstructures immersed within curved fluid interfaces of spherical shape. We take into account the interfacial hydrodynamic coupling, traction coupling with the surrounding bulk fluid, and thermal fluctuations. For fluid-structure interactions, we introduce Immersed Boundary Methods (IBM) and related Stochastic Eulerian-Lagrangian Methods (SELM) for curved surfaces. We use these approaches to investigate the statistics of surface fluctuating hydrodynamics and microstructures. For velocity autocorrelations, we find characteristic power-law scalings $τ^{-1}$, $τ^{-2}$, and plateaus can emerge. This depends on the physical regime associated with the geometry, surface viscosity, and bulk viscosity. This differs from the characteristic $τ^{-3/2}$ scaling for bulk three dimensional fluids. We develop theory explaining these observed power-laws associated with time-scales for dissipation within the fluid interface and coupling to the surrounding fluid. We then use our introduced methods to investigate a few example systems and roles of hydrodynamic coupling and thermal fluctuations including for the kinetics of passive particles and active microswimmers in curved fluid interfaces.

cond-mat.soft↗

arXiv2023

Systematic Stochastic Reduction of Inertial Fluid-Structure Interactions subject to Thermal Fluctuations

We investigate the dynamics of elastic microstructures within a fluid that are subjected to thermal fluctuations. We perform analysis to obtain systematically simplified descriptions of the mechanics in the limiting regimes when (i) the coupling forces that transfer momentum between the fluid and microstructures is strong, (ii) the mass of the microstructures is small relative to the displaced mass of the fluid, and (iii) the response to stresses results in hydrodynamics that relax rapidly to a quasi-steady-state relative to the motions of the microstructure. We derive effective equations using a singular perturbation analysis of the Backward Kolmogorov equations of the stochastic process. Our continuum mechanics description is based on the Stochastic Eulerian Lagrangian Method (SELM) which provides a framework for approximation of the fluid-structure interactions when subject to thermal fluctuations.

cond-mat.soft↗

arXiv2023

MLMOD: Machine Learning Methods for Data-Driven Modeling in LAMMPS

MLMOD is a software package for incorporating machine learning approaches and models into simulations of microscale mechanics and molecular dynamics in LAMMPS. Recent machine learning approaches provide promising data-driven approaches for learning representations for system behaviors from experimental data and high fidelity simulations. The package faciliates learning and using data-driven models for (i) dynamics of the system at larger spatial-temporal scales (ii) interactions between system components, (iii) features yielding coarser degrees of freedom, and (iv) features for new quantities of interest characterizing system behaviors. MLMOD provides hooks in LAMMPS for (i) modeling dynamics and time-step integration, (ii) modeling interactions, and (iii) computing quantities of interest characterizing system states. The package allows for use of machine learning methods with general model classes including Neural Networks, Gaussian Process Regression, Kernel Models, and other approaches. Here we discuss our prototype C++/Python package, aims, and example usage. The package is integrated currently with the mesocale and molecular dynamics simulation package LAMMPS and PyTorch. For related papers, examples, updates, and additional information see https://github.com/atzberg/mlmod and http://atzberger.org/.

cs.LG↗

arXiv2023

Coarse-Grained Methods for Heterogeneous Vesicles with Phase-Separated Domains: Elastic Mechanics of Shape Fluctuations, Plate Compression, and Channel Insertion

We develop coarse-grained particle approaches for studying the elastic mechanics of vesicles with heterogeneous membranes having phase-separated domains. We perform simulations both of passive shape fluctuations and of active systems where vesicles are subjected to compression between two plates or subjected to insertion into narrow channels. Analysis methods are developed for mapping particle configurations to continuum fields with spherical harmonics representations. Heterogeneous vesicles are found to exhibit rich behaviors where the heterogeneity can amplify surface two-point correlations, reduce resistance during compression, and augment vesicle transport times in channels. The developed methods provide general approaches for characterizing the mechanics of coarse-grained heterogeneous systems taking into account the roles of thermal fluctuations, geometry, and phase separation.

cond-mat.soft↗

arXiv2023

SDYN-GANs: Adversarial Learning Methods for Multistep Generative Models for General Order Stochastic Dynamics

We introduce adversarial learning methods for data-driven generative modeling of the dynamics of $n^{th}$-order stochastic systems. Our approach builds on Generative Adversarial Networks (GANs) with generative model classes based on stable $m$-step stochastic numerical integrators. We introduce different formulations and training methods for learning models of stochastic dynamics based on observation of trajectory samples. We develop approaches using discriminators based on Maximum Mean Discrepancy (MMD), training protocols using conditional and marginal distributions, and methods for learning dynamic responses over different time-scales. We show how our approaches can be used for modeling physical systems to learn force-laws, damping coefficients, and noise-related parameters. The adversarial learning approaches provide methods for obtaining stable generative models for dynamic tasks including long-time prediction and developing simulations for stochastic systems.

cs.LG↗

arXiv2022

Drift-Diffusion Dynamics and Phase Separation in Curved Cell Membranes and Dendritic Spines: Hybrid Discrete-Continuum Methods

We develop methods for investigating protein drift-diffusion dynamics in heterogeneous cell membranes and the roles played by geometry, diffusion, chemical kinetics, and phase separation. Our hybrid stochastic numerical methods combine discrete particle descriptions with continuum-level models for tracking the individual protein drift-diffusion dynamics when coupled to continuum fields. We show how our approaches can be used to investigate phenomena motivated by protein kinetics within dendritic spines. The spine geometry is hypothesized to play an important biological role regulating synaptic strength, protein kinetics, and self-assembly of clusters. We perform simulation studies for model spine geometries varying the neck size to investigate how phase-separation and protein organization is influenced by different shapes. We also show how our methods can be used to study the roles of geometry in reaction-diffusion systems including Turing instabilities. Our methods provide general approaches for investigating protein kinetics and drift-diffusion dynamics within curved membrane structures.

physics.bio-ph↗

arXiv2021

Variational Autoencoders for Learning Nonlinear Dynamics of Physical Systems

We develop data-driven methods for incorporating physical information for priors to learn parsimonious representations of nonlinear systems arising from parameterized PDEs and mechanics. Our approach is based on Variational Autoencoders (VAEs) for learning from observations nonlinear state space models. We develop ways to incorporate geometric and topological priors through general manifold latent space representations. We investigate the performance of our methods for learning low dimensional representations for the nonlinear Burgers equation and constrained mechanical systems.

cs.LG↗

arXiv2019

GMLS-Nets: A framework for learning from unstructured data

Data fields sampled on irregularly spaced points arise in many applications in the sciences and engineering. For regular grids, Convolutional Neural Networks (CNNs) have been successfully used to gaining benefits from weight sharing and invariances. We generalize CNNs by introducing methods for data on unstructured point clouds based on Generalized Moving Least Squares (GMLS). GMLS is a non-parametric technique for estimating linear bounded functionals from scattered data, and has recently been used in the literature for solving partial differential equations. By parameterizing the GMLS estimator, we obtain learning methods for operators with unstructured stencils. In GMLS-Nets the necessary calculations are local, readily parallelizable, and the estimator is supported by a rigorous approximation theory. We show how the framework may be used for unstructured physical data sets to perform functional regression to identify associated differential operators and to regress quantities of interest. The results suggest the architectures to be an attractive foundation for data-driven model development in scientific machine learning applications.

cs.LG↗

arXiv2019

Topological Methods for Polymeric Materials: Characterizing the Relationship Between Polymer Entanglement and Viscoelasticity

We develop topological methods for characterizing the relationship between polymer chain entanglement and bulk viscoelastic responses. We introduce generalized Linking Number and Writhe characteristics that are applicable to open linear chains. We investigate the rheology of polymeric chains entangled into weaves with varying topologies and levels of chain density. To investigate viscoelastic responses, we perform non-equilibrium molecular simulations over a range of frequencies using sheared Lees-Edwards boundary conditions. We show how our topological characteristics can be used to capture key features of the polymer entanglements related to the viscoelastic responses. We find there is a linear relation over a significant range of frequencies between the mean absolute Writhe $Wr$ and the Loss Tangent $\tan(δ)$. We also find an approximate inverse linear relationship between the mean absolute Periodic Linking Number $LK_P$ and the Loss Tangent $\tan(δ)$. Our results show some of the ways topological methods can be used to characterize chain entanglements to better understand the origins of mechanical responses in polymeric materials.

cond-mat.soft↗

arXiv2018

Importance of the Mathematical Foundations of Machine Learning Methods for Scientific and Engineering Applications

There has been a lot of recent interest in adopting machine learning methods for scientific and engineering applications. This has in large part been inspired by recent successes and advances in the domains of Natural Language Processing (NLP) and Image Classification (IC). However, scientific and engineering problems have their own unique characteristics and requirements raising new challenges for effective design and deployment of machine learning approaches. There is a strong need for further mathematical developments on the foundations of machine learning methods to increase the level of rigor of employed methods and to ensure more reliable and interpretable results. Also as reported in the recent literature on state-of-the-art results and indicated by the No Free Lunch Theorems of statistical learning theory incorporating some form of inductive bias and domain knowledge is essential to success. Consequently, even for existing and widely used methods there is a strong need for further mathematical work to facilitate ways to incorporate prior scientific knowledge and related inductive biases into learning frameworks and algorithms. We briefly discuss these topics and discuss some ideas proceeding in this direction.

stat.ML↗

arXiv2018

Stochastic Discontinuous Galerkin Methods (SDGM) Based on Fluctuation-Dissipation Balance

We introduce a general framework for approximating parabolic Stochastic Partial Differential Equations (SPDEs) based on fluctuation-dissipation balance. Using this approach we formulate Stochastic Discontinuous Galerkin Methods (SDGM). We show how methods with linear-time computational complexity can be developed for handling domains with general geometry and generating stochastic terms handling both Dirichlet and Neumann boundary conditions. We demonstrate our approach on example systems and contrast with alternative approaches using direct stochastic discretizations based on random fluxes. We show how our Fluctuation-Dissipation Discretizations (FDD) framework allows for compensating for differences in dissipative properties of discrete numerical operators relative to their continuum counter-parts. This allows us to handle general heterogeneous discretizations capturing accurately statistical relations. Our FDD framework provides a general approach for formulating SDGM discretizations and other numerical methods for robust approximation of stochastic differential equations.

math.NA↗

arXiv2018

Hydrodynamic Flows on Curved Surfaces: Spectral Numerical Methods for Radial Manifold Shapes

We formulate hydrodynamic equations and spectrally accurate numerical methods for investigating the role of geometry in flows within two-dimensional fluid interfaces. To achieve numerical approximations having high precision and level of symmetry for radial manifold shapes, we develop spectral Galerkin methods based on hyperinterpolation with Lebedev quadratures for $L^2$-projection to spherical harmonics. We demonstrate our methods by investigating hydrodynamic responses as the surface geometry is varied. Relative to the case of a sphere, we find significant changes can occur in the observed hydrodynamic flow responses as exhibited by quantitative and topological transitions in the structure of the flow. We present numerical results based on the Rayleigh-Dissipation principle to gain further insights into these flow responses. We investigate the roles played by the geometry especially concerning the positive and negative Gaussian curvature of the interface. We provide general approaches for taking geometric effects into account for investigations of hydrodynamic phenomena within curved fluid interfaces.

cond-mat.soft↗

arXiv2018

Electrostatics of Colloidal Particles Confined in Nanochannels: Role of Double-Layer Interactions and Ion-Ion Correlations

We perform computational investigations of electrolyte-mediated interactions of charged colloidal particles confined within nanochannels. We investigate the role of discrete ion effects, valence, and electrolyte strength on colloid-wall interactions. We find for some of the multivalent charge regimes that the like-charged colloids and walls can have attractive interactions. We study in detail these interactions and the free energy profile for the colloid-wall separation. We find there are energy barriers and energy minima giving preferred colloid locations in the channel near the center and at a distance near to but separated from the channel walls. We characterize contributions from surface overcharging, condensed layers, and overlap of ion double-layers. We perform our investigations using Coarse-Grained Brownian Dynamics simulations (BD), classical Density Functional Theory (cDFT), and mean-field Poisson-Boltzmann Theory (PB). We discuss the implications of our results for phenomena in nanoscale devices.

cond-mat.soft↗