arXiv subjects

Peter Zaspel

Publications and source records attributed to Peter Zaspel.

At least 19 recordsLinked to original sources

arXiv2026

Bayesian Inference: Kernel-Based Model for Surface Temperature Reconstruction in Ice Borehole Thermometry

Reconstructing past surface temperature from shallow ice borehole temperature profiles requires solving an ill-posed inverse problem while quantifying uncertainties arising from measurements and prior assumptions. Bayesian formulations enable probabilistic reconstruction of surface temperature histories and uncertainty quantification. Existing reversible jump-Markov chain Monte Carlo approach based on adaptive piecewise-linear surface temperature models can, however, be computationally demanding. Here, we introduce a kernel-based surface temperature model that enables the use of a parallel ensemble Markov chain Monte Carlo sampler for efficient exploration of the solution space and quantification of the posterior. Using synthetic experiments, we investigate the effects of kernel configuration, measurement uncertainty, measurement density, and temporal smearing on reconstruction performance. We find that reconstruction quality is largely insensitive to the number of kernels once the kernel basis is sufficiently dense. Reducing measurement uncertainty substantially improves reconstructions, whereas increasing the number of borehole temperature measurements provides only marginal benefit. Finally, we evaluate the method using realistic surrogate climate histories that combine long-term temperature changes with stochastic climate variability. The kernel-based surface temperature model cannot represent short-term variability and therefore cannot fully explain the realistic measurements, highlighting the need to account for this approximation uncertainty. The likelihood is adapted to include the approximation uncertainty of the surface temperature model, yielding robust reconstructions with reliable posterior uncertainties. Overall, our results demonstrate that kernel-based Bayesian inversion provides an efficient framework for shallow ice borehole based climate reconstructions.

physics.comp-ph↗

arXiv2026

Variational Free Energy Pivot Selection for Pivoted Cholesky

Pivoted Cholesky factorizations construct low-rank approximations of symmetric positive definite matrices by sequentially selecting pivots from the residual diagonal. Classical greedy and randomized rules, such as randomly pivoted Cholesky, target the algebraic trace-norm error of the residual. In many applications, however, the matrix enters a nonlinear matrix functional whose value, not the trace-norm error, determines solution quality, and residual-based rules ignore this structure. We derive a pivot rule that maximizes the exact one-step change of such a functional under Cholesky-consistent rank-1 updates, for a functional combining log-determinant, quadratic, and trace terms. This functional arises as the variational free energy in Gaussian process regression, where the matrix is a kernel matrix. The resulting per-step gain admits a closed-form additive decomposition into complexity, data-fit, and trace contributions, and is used directly as a pivot-selection criterion. We refer to the resulting method as $\Delta$-VFE pivoted Cholesky. At each iteration, the criterion is evaluated on a batch of $s$ candidate pivots sampled proportionally to the residual diagonal via incremental Woodbury updates, at a total cost of $\mathcal{O}(snr^2)$ for an $n\times n$ matrix and target rank $r$. This matches the asymptotic complexity of randomly pivoted Cholesky up to the batch factor $s$. Cholesky-consistent rank-1 updates yield monotonically non-decreasing functional values, and the proposed rule maximizes the per-step gain among them. Numerical experiments show improved objective values and predictive accuracy at low to moderate ranks compared to classical and randomly pivoted Cholesky, while preserving trace-norm approximation quality.

math.NA↗

arXiv2026

Improvise, Adapt, Overcome: An On-The-Fly Multifidelity Algorithm for Efficient Machine Learning

Machine learning has accelerated quantum chemistry but is hindered by the prohibitive cost of generating high fidelity training data. Multifidelity machine learning (MFML) mitigates this overhead by systematically combining abundant low fidelity data with sparse high fidelity data. In spite of its success, standard MFML schemes rely on pre-defined scaling factors to determine sparse data ratio across fidelities, often generating redundant multifidelity data resulting in a loss of efficiency. Here, we introduce an adaptive on-the-fly multifidelity framework for machine learning that autonomously determines training dataset composition. By dynamically querying training samples at each fidelity, the algorithm saturates model accuracy at lower fidelities before moving up to more expensive reference calculations. We benchmark the novel adaptive-MFML across diverse chemical properties including the computational chemistry gold standard coupled cluster energies, and the more chemically challenging excitation energies. In our numerical experiments we show that our adaptive algorithm reduces data generation costs by up to a factor of 30 compared to single fidelity methods and improves upon standard MFML by up to a factor of 5. The mitigation of data redundancy establishes a high-accuracy low-cost pathway for sustainable cost-aware machine learning in quantum chemistry.

cs.LG↗

arXiv2025

LFaB: Low fidelity as Bias for Active Learning in the chemical configuration space

Active learning promises to provide an optimal training sample selection procedure in the construction of machine learning models. It often relies on minimizing the model's variance, which is assumed to decrease the prediction error. Still, it is frequently even less efficient than pure random sampling. Motivated by the bias-variance decomposition, we propose to minimize the model's bias instead of its variance. By doing so, we are able to almost exactly match the best-case error over all possible greedy sample selection procedures for a relevant application. Our bias approximation is based on using cheap to calculate low fidelity data as known from $\Delta$-ML or multifidelity machine learning. We exemplify our approach for a wider class of applications in quantum chemistry including predicting excitation energies and ab initio potential energy surfaces. Here, the proposed method reduces training data consumption by up to an order of magnitude compared to standard active learning.

physics.chem-ph↗

arXiv2024

Evaluation of uncertainty estimations for Gaussian process regression based machine learning interatomic potentials

Uncertainty estimations for machine learning interatomic potentials (MLIPs) are crucial for quantifying model error and identifying informative training samples in active learning strategies. In this study, we evaluate uncertainty estimations of Gaussian process regression (GPR)-based MLIPs, including the predictive GPR standard deviation and ensemble-based uncertainties. We do this in terms of calibration and in terms of impact on model performance in an active learning scheme. We consider GPR models with Coulomb and Smooth Overlap of Atomic Positions (SOAP) representations as inputs to predict potential energy surfaces and excitation energies of molecules. Regarding calibration, we find that ensemble-based uncertainty estimations show already poor global calibration (e.g., averaged over the whole test set). In contrast, the GPR standard deviation shows good global calibration, but when grouping predictions by their uncertainty, we observe a systematical bias for predictions with high uncertainty. Although an increasing uncertainty correlates with an increasing bias, the bias is not captured quantitatively by the uncertainty. Therefore, the GPR standard deviation can be useful to identify predictions with a high bias and error but, without further knowledge, should not be interpreted as a quantitative measure for a potential error range. Selecting the samples with the highest GPR standard deviation from a fixed configuration space leads to a model that overemphasizes the borders of the configuration space represented in the fixed dataset. This may result in worse performance in more densely sampled areas but better generalization for extrapolation tasks.

cs.LG↗

arXiv2024

Excitation Energy Transfer between Porphyrin Dyes on a Clay Surface: A study employing Multifidelity Machine Learning

Natural light-harvesting antenna complexes efficiently capture solar energy using chlorophyll, i.e., magnesium porphyrin pigments, embedded in a protein matrix. Inspired by this natural configuration, artificial clay-porphyrin antenna structures have been experimentally synthesized and have demonstrated remarkable excitation energy transfer properties. The study presents the computational design and simulation of a synthetic light-harvesting system that emulates natural mechanisms by arranging cationic free-base porphyrin molecules on an anionic clay surface. We investigated the transfer of excitation energy among the porphyrin dyes using a multiscale quantum mechanics/molecular mechanics (QM/MM) approach based on the semi-empirical density functional-based tight-binding (DFTB) theory for the ground state dynamics. To improve the accuracy of our results, we incorporated an innovative multifidelity machine learning (MFML) approach, which allows the prediction of excitation energies at the numerically demanding time-dependent density functional theory level with the Def2-SVP basis set. This approach was applied to an extensive dataset of 640K geometries for the 90-atom porphyrin structures, facilitating a thorough analysis of the excitation energy diffusion among the porphyrin molecules adsorbed to the clay surface. The insights gained from this study, inspired by natural light-harvesting complexes, demonstrate the potential of porphyrin-clay systems as effective energy transfer systems.

physics.chem-ph↗

arXiv2024

Benchmarking Data Efficiency in $\Delta$-ML and Multifidelity Models for Quantum Chemistry

The development of machine learning (ML) methods has made quantum chemistry (QC) calculations more accessible by reducing the compute cost incurred in conventional QC methods. This has since been translated into the overhead cost of generating training data. Increased work in reducing the cost of generating training data resulted in the development of $\Delta$-ML and multifidelity machine learning methods which use data at more than one QC level of accuracy, or fidelity. This work compares the data costs associated with $\Delta$-ML, multifidelity machine learning (MFML), and optimized MFML (o-MFML) in contrast with a newly introduced Multifidelity$\Delta$-Machine Learning (MF$\Delta$ML) method for the prediction of ground state energies, vertical excitation energies, and the magnitude of electronic contribution of molecular dipole moments from the multifidelity benchmark dataset QeMFi. This assessment is made on the basis of training data generation cost associated with each model and is compared with the single fidelity kernel ridge regression (KRR) case. The results indicate that the use of multifidelity methods surpasses the standard $\Delta$-ML approaches in cases of a large number of predictions. For applications which require only a few evaluations to be made using ML models, while the $\Delta$-ML method might be favored, the MF$\Delta$ML method is shown to be more efficient.

physics.chem-ph↗

arXiv2024

Investigating Data Hierarchies in Multifidelity Machine Learning for Excitation Energies

Recent progress in machine learning (ML) has made high-accuracy quantum chemistry (QC) calculations more accessible. Of particular interest are multifidelity machine learning (MFML) methods where training data from differing accuracies or fidelities are used. These methods usually employ a fixed scaling factor, $\gamma$, to relate the number of training samples across different fidelities, which reflects the cost and assumed sparsity of the data. This study investigates the impact of modifying $\gamma$ on model efficiency and accuracy for the prediction of vertical excitation energies using the QeMFi benchmark dataset. Further, this work introduces QC compute time informed scaling factors, denoted as $\theta$, that vary based on QC compute times at different fidelities. A novel error metric, error contours of MFML, is proposed to provide a comprehensive view of model error contributions from each fidelity. The results indicate that high model accuracy can be achieved with just 2 training samples at the target fidelity when a larger number of samples from lower fidelities are used. This is further illustrated through a novel concept, the $\Gamma$-curve, which compares model error against the time-cost of generating training samples, demonstrating that multifidelity models can achieve high accuracy while minimizing training data costs.

physics.chem-ph↗

arXiv2024

Assessing Non-Nested Configurations of Multifidelity Machine Learning for Quantum-Chemical Properties

Multifidelity machine learning (MFML) for quantum chemical (QC) properties has seen strong development in the recent years. The method has been shown to reduce the cost of generating training data for high-accuracy low-cost ML models. In such a set-up, the ML models are trained on molecular geometries and some property of interest computed at various computational chemistry accuracies, or fidelities. These are then combined in training the MFML models. In some multifidelity models, the training data is required to be nested, that is the same molecular geometries are included to calculate the property across all the fidelities. In these multifidelity models, the requirement of a nested configuration restricts the kind of sampling that can be performed while selection training samples at different fidelities. This work assesses the use of non-nested training data for two of these multifidelity methods, namely MFML and optimized MFML (o-MFML). The assessment is carried out for the prediction of ground state energies and first vertical excitation energies of a diverse collection of molecules of the CheMFi dataset. Results indicate that the MFML method still requires a nested structure of training data across the fidelities. However, the o-MFML method shows promising results for non-nested multifidelity training data with model errors comparable to the nested configurations.

physics.chem-ph↗

arXiv2024

Data-driven identification of port-Hamiltonian DAE systems by Gaussian processes

Port-Hamiltonian systems (pHS) allow for a structure-preserving modeling of dynamical systems. Coupling pHS via linear relations between input and output defines an overall pHS, which is structure preserving. However, in multiphysics applications, some subsystems do not allow for a physical pHS description, as (a) this is not available or (b) too expensive. Here, data-driven approaches can be used to deliver a pHS for such subsystems, which can then be coupled to the other subsystems in a structure-preserving way. In this work, we derive a data-driven identification approach for port-Hamiltonian differential algebraic equation (DAE) systems. The approach uses input and state space data to estimate nonlinear effort functions of pH-DAEs. As underlying technique, we us (multi-task) Gaussian processes. This work thereby extends over the current state of the art, in which only port-Hamiltonian ordinary differential equation systems could be identified via Gaussian processes. We apply this approach successfully to two applications from network design and constrained multibody system dynamics, based on pH-DAE system of index one and three, respectively.

eess.SY↗

arXiv2024

QeMFi: A Multifidelity Dataset of Quantum Chemical Properties of Diverse Molecules

Progress in both Machine Learning (ML) and Quantum Chemistry (QC) methods have resulted in high accuracy ML models for QC properties. Datasets such as MD17 and WS22 have been used to benchmark these models at some level of QC method, or fidelity, which refers to the accuracy of the chosen QC method. Multifidelity ML (MFML) methods, where models are trained on data from more than one fidelity, have shown to be effective over single fidelity methods. Much research is progressing in this direction for diverse applications ranging from energy band gaps to excitation energies. One hurdle for effective research here is the lack of a diverse multifidelity dataset for benchmarking. We provide the Quantum chemistry MultiFidelity (QeMFi) dataset consisting of five fidelities calculated with the TD-DFT formalism. The fidelities differ in their basis set choice: STO-3G, 3-21G, 6-31G, def2-SVP, and def2-TZVP. QeMFi offers to the community a variety of QC properties such as vertical excitation properties and molecular dipole moments, further including QC computation times allowing for a time benefit benchmark of multifidelity models for ML-QC.

physics.chem-ph↗

arXiv2023

Optimized Multifidelity Machine Learning for Quantum Chemistry

Machine learning (ML) provides access to fast and accurate quantum chemistry (QC) calculations for various properties of interest such as excitation energies. It is often the case that high accuracy in prediction using an ML model, demands a large and costly training set. Various solutions and procedures have been presented to reduce this cost. These include methods such as $\Delta$-ML, hierarchical-ML, and multifidelity machine learning (MFML). MFML combines various $\Delta$-ML like sub-models for various fidelities according to a fixed scheme derived from the sparse grid combination technique. In this work we implement an optimization procedure to combine multifidelity models in a flexible scheme resulting in optimized MFML (o-MFML) that provides superior prediction capabilities. This hyper-parameter optimization is carried out on a holdout validation set of the property of interest. This work benchmarks the o-MFML method in predicting the atomization energies on the QM7b dataset, and again in the prediction of excitation energies for three molecules of growing size. The results indicate that o-MFML is a strong methodological improvement over MFML and provides lower error of prediction. Even in cases of poor data distributions and lack of clear hierarchies among the fidelities, which were previously identified as issues for multifidelity methods, the o-MFML provides advantage to the prediction of quantum chemical properties.

physics.chem-ph↗

arXiv2023

Multi-Fidelity Machine Learning for Excited State Energies of Molecules

The accurate but fast calculation of molecular excited states is still a very challenging topic. For many applications, detailed knowledge of the energy funnel in larger molecular aggregates is of key importance requiring highly accurate excited state energies. To this end, machine learning techniques can be an extremely useful tool though the cost of generating highly accurate training datasets still remains a severe challenge. To overcome this hurdle, this work proposes the use of multi-fidelity machine learning where very little training data from high accuracies is combined with cheaper and less accurate data to achieve the accuracy of the costlier level. In the present study, the approach is employed to predict the first excited state energies for three molecules of increasing size, namely, benzene, naphthalene, and anthracene. The energies are trained and tested for conformations stemming from classical molecular dynamics simulations and from real-time density functional tight-binding calculations. It can be shown that the multi-fidelity machine learning model can achieve the same accuracy as a machine learning model built only on high cost training data while having a much lower computational effort to generate the data. The numerical gain observed in these benchmark test calculations was over a factor of 30 but certainly can be much higher for high accuracy data.

physics.chem-ph↗

arXiv2021

Towards data-driven filters in Paraview

Recent progress in scientific visualization has expanded the scope of visualization from being merely a way of presentation to an analysis and discovery tool. A given visualization result is usually generated by applying a series of transformations or filters to the underlying data. Nowadays, such filters use deterministic algorithms to process the data. In this work, we aim at extending this methodology towards data-driven filters, thus filters that expose the abilities of pre-trained machine learning models to the visualization system. The use of such data-driven filters is of particular interest in fields like segmentation, classification, etc., where machine learning models regularly outperform existing algorithmic approaches. To showcase this idea, we couple Paraview, the well-known flow visualization tool, with PyTorch, a deep learning framework. Paraview is extended by plugins that allow users to load pre-trained models of their choice in the form of newly developed filters. The filters transform the input data by feeding it into the model and then provide the model's output as input to the remaining visualization pipeline. A series of simplistic use cases for segmentation and classification on image and fluid data is presented to showcase the technical applicability of such data-driven transformations in Paraview for future complex analysis tasks.

cs.LG↗

arXiv2018

Kernel-based stochastic collocation for the random two-phase Navier-Stokes equations

In this work, we apply stochastic collocation methods with radial kernel basis functions for an uncertainty quantification of the random incompressible two-phase Navier-Stokes equations. Our approach is non-intrusive and we use the existing fluid dynamics solver NaSt3DGPF to solve the incompressible two-phase Navier-Stokes equation for each given realization. We are able to empirically show that the resulting kernel-based stochastic collocation is highly competitive in this setting and even outperforms some other standard methods.

math.NA↗

arXiv2018

Ensemble Kalman Filters for reliability estimation in perfusion inference

We consider the solution of inverse problems in dynamic contrast-enhanced imaging by means of Ensemble Kalman Filters. Our quantity of interest is blood perfusion, i.e. blood flow rates in tissue. While existing approaches to compute blood perfusion parameters for given time series of radiological measurements mainly rely on deterministic, deconvolution-based methods, we aim at recovering probabilistic solution information for given noisy measurements. To this end, we model radiological image capturing as sequential data assimilation process and solve it by an Ensemble Kalman Filter. Thereby, we recover deterministic results as ensemble-based mean and are able to compute reliability information such as probabilities for the perfusion to be in a given range. Our target application is the inference of blood perfusion parameters in the human brain. A numerical study shows promising results for artificial measurements generated by a Digital Perfusion Phantom.

math.NA↗

arXiv2018

Boosting quantum machine learning models with multi-level combination technique: Pople diagrams revisited

Inspired by Pople diagrams popular in quantum chemistry, we introduce a hierarchical scheme, based on the multi-level combination (C) technique, to combine various levels of approximations made when calculating molecular energies within quantum chemistry. When combined with quantum machine learning (QML) models, the resulting CQML model is a generalized unified recursive kernel ridge regression which exploits correlations implicitly encoded in training data comprised of multiple levels in multiple dimensions. Here, we have investigated up to three dimensions: Chemical space, basis set, and electron correlation treatment. Numerical results have been obtained for atomization energies of a set of $\sim$7'000 organic molecules with up to 7 atoms (not counting hydrogens) containing CHONFClS, as well as for $\sim$6'000 constitutional isomers of C$_7$H$_{10}$O$_2$. CQML learning curves for atomization energies suggest a dramatic reduction in necessary training samples calculated with the most accurate and costly method. In order to generate milli-second estimates of CCSD(T)/cc-pvdz atomization energies with prediction errors reaching chemical accuracy ($\sim$1 kcal/mol), the CQML model requires only $\sim$100 training instances at CCSD(T)/cc-pvdz level, rather than thousands within conventional QML, while more training molecules are required at lower levels. Our results suggest a possibly favourable trade-off between various hierarchical approximations whose computational cost scales differently with electron number.

physics.chem-ph↗

arXiv2018

A scalable H-matrix approach for the solution of boundary integral equations on multi-GPU clusters

In this work, we consider the solution of boundary integral equations by means of a scalable hierarchical matrix approach on clusters equipped with graphics hardware, i.e. graphics processing units (GPUs). To this end, we extend our existing single-GPU hierarchical matrix library hmglib such that it is able to scale on many GPUs and such that it can be coupled to arbitrary application codes. Using a model GPU implementation of a boundary element method (BEM) solver, we are able to achieve more than 67 percent relative parallel speed-up going from 128 to 1024 GPUs for a model geometry test case with 1.5 million unknowns and a real-world geometry test case with almost 1.2 million unknowns. On 1024 GPUs of the cluster Titan, it takes less than 6 minutes to solve the 1.5 million unknowns problem, with 5.7 minutes for the setup phase and 20 seconds for the iterative solver. To the best of the authors' knowledge, we here discuss the first fully GPU-based distributed-memory parallel hierarchical matrix Open Source library using the traditional H-matrix format and adaptive cross approximation with an application to BEM problems.

cs.MS↗