Interatomic potentials learned using machine learning methods have been successfully applied to atomistic simulations. However, deep learning pipelines are notoriously data-hungry, while generating reference calculations is computationally demanding. To overcome this difficulty, we propose a transfer learning algorithm that leverages the ability of graph neural networks (GNNs) in describing chemical environments, together with kernel mean embeddings. We extract a feature map from GNNs pre-trained on the OC20 dataset and use it to learn the potential energy surface from system-specific datasets of catalytic processes. Our method is further enhanced by a flexible kernel function that incorporates chemical species information, resulting in improved performance and interpretability. We test our approach on a series of realistic datasets of increasing complexity, showing excellent generalization and transferability performance, and improving on methods that rely on GNNs or ridge regression alone, as well as similar fine-tuning approaches. We make the code available to the community at //github.com/IsakFalk/atomistic_transfer_mekrr.
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This research aims to develop kernel GNG, a kernelized version of the growing neural gas (GNG) algorithm, and to investigate the features of the networks generated by the kernel GNG. The GNG is an unsupervised artificial neural network that can transform a dataset into an undirected graph, thereby extracting the features of the dataset as a graph. The GNG is widely used in vector quantization, clustering, and 3D graphics. Kernel methods are often used to map a dataset to feature space, with support vector machines being the most prominent application. This paper introduces the kernel GNG approach and explores the characteristics of the networks generated by kernel GNG. Five kernels, including Gaussian, Laplacian, Cauchy, inverse multiquadric, and log kernels, are used in this study. The results of this study show that the average degree and the average clustering coefficient decrease as the kernel parameter increases for Gaussian, Laplacian, Cauchy, and IMQ kernels. If we avoid more edges and a higher clustering coefficient (or more triangles), the kernel GNG with a larger value of the parameter will be more appropriate.
Partial differential equations (PDEs) have become an essential tool for modeling complex physical systems. Such equations are typically solved numerically via mesh-based methods, such as finite element methods, the outputs of which consist of the solutions on a set of mesh nodes over the spatial domain. However, these simulations are often prohibitively costly to survey the input space. In this paper, we propose an efficient emulator that simultaneously predicts the outputs on a set of mesh nodes, with theoretical justification of its uncertainty quantification. The novelty of the proposed method lies in the incorporation of the mesh node coordinates into the statistical model. In particular, the proposed method segments the mesh nodes into multiple clusters via a Dirichlet process prior and fits a Gaussian process model in each. Most importantly, by revealing the underlying clustering structures, the proposed method can extract valuable flow physics present in the systems that can be used to guide further investigations. Real examples are demonstrated to show that our proposed method has smaller prediction errors than its main competitors, with competitive computation time, and identifies interesting clusters of mesh nodes that exhibit coherent input-output relationships and possess physical significance, such as satisfying boundary conditions. An R package for the proposed methodology is provided in an open repository.
Accurately estimating parameters in complex nonlinear systems is crucial across scientific and engineering fields. We present a novel approach for parameter estimation using a neural network with the Huber loss function. This method taps into deep learning's abilities to uncover parameters governing intricate behaviors in nonlinear equations. We validate our approach using synthetic data and predefined functions that model system dynamics. By training the neural network with noisy time series data, it fine-tunes the Huber loss function to converge to accurate parameters. We apply our method to damped oscillators, Van der Pol oscillators, Lotka-Volterra systems, and Lorenz systems under multiplicative noise. The trained neural network accurately estimates parameters, evident from closely matching latent dynamics. Comparing true and estimated trajectories visually reinforces our method's precision and robustness. Our study underscores the Huber loss-guided neural network as a versatile tool for parameter estimation, effectively uncovering complex relationships in nonlinear systems. The method navigates noise and uncertainty adeptly, showcasing its adaptability to real-world challenges.
In this work, we are interested in solving large linear systems stemming from the Extra-Membrane-Intra (EMI) model, which is employed for simulating excitable tissues at a cellular scale. After setting the related systems of partial differential equations (PDEs) equipped with proper boundary conditions, we provide numerical approximation schemes for the EMI PDEs and focus on the resulting large linear systems. We first give a relatively complete spectral analysis using tools from the theory of Generalized Locally Toeplitz matrix sequences. The obtained spectral information is used for designing appropriate (preconditioned) Krylov solvers. We show, through numerical experiments, that the presented solution strategy is robust w.r.t. problem and discretization parameters, efficient and scalable.
In this paper, we study min-max optimization problems on Riemannian manifolds. We introduce a Riemannian Hamiltonian function, minimization of which serves as a proxy for solving the original min-max problems. Under the Riemannian Polyak--{\L}ojasiewicz condition on the Hamiltonian function, its minimizer corresponds to the desired min-max saddle point. We also provide cases where this condition is satisfied. For geodesic-bilinear optimization in particular, solving the proxy problem leads to the correct search direction towards global optimality, which becomes challenging with the min-max formulation. To minimize the Hamiltonian function, we propose Riemannian Hamiltonian methods (RHM) and present their convergence analyses. We extend RHM to include consensus regularization and to the stochastic setting. We illustrate the efficacy of the proposed RHM in applications such as subspace robust Wasserstein distance, robust training of neural networks, and generative adversarial networks.
Numerical methods such as the Finite Element Method (FEM) have been successfully adapted to utilize the computational power of GPU accelerators. However, much of the effort around applying FEM to GPU's has been focused on high-order FEM due to higher arithmetic intensity and order of accuracy. For applications such as the simulation of subsurface processes, high levels of heterogeneity results in high-resolution grids characterized by highly discontinuous (cell-wise) material property fields. Moreover, due to the significant uncertainties in the characterization of the domain of interest, e.g. geologic reservoirs, the benefits of high order accuracy are reduced, and low-order methods are typically employed. In this study, we present a strategy for implementing highly performant low-order matrix-free FEM operator kernels in the context of the conjugate gradient (CG) method. Performance results of matrix-free Laplace and isotropic elasticity operator kernels are presented and are shown to compare favorably to matrix-based SpMV operators on V100, A100, and MI250X GPUs.
This work presents a numerical analysis of a master equation modeling the interaction of a system with a noisy environment in the particular context of open quantum systems. It is shown that our transformed master equation has a reduced computational cost in comparison to a Wigner-Fokker-Planck model of the same system for the general case of any potential. Specifics of a NIPG-DG numerical scheme adequate for the convection-diffusion system obtained are then presented. This will let us solve computationally the transformed system of interest modeling our open quantum system. A benchmark problem, the case of a harmonic potential, is then presented, for which the numerical results are compared against the analytical steady-state solution of this problem.
In this study, Synthetic Aperture Radar (SAR) and optical data are both considered for Earth surface classification. Specifically, the integration of Sentinel-1 (S-1) and Sentinel-2 (S-2) data is carried out through supervised Machine Learning (ML) algorithms implemented on the Google Earth Engine (GEE) platform for the classification of a particular region of interest. Achieved results demonstrate how in this case radar and optical remote detection provide complementary information, benefiting surface cover classification and generally leading to increased mapping accuracy. In addition, this paper works in the direction of proving the emerging role of GEE as an effective cloud-based tool for handling large amounts of satellite data.
Current research on bias in machine learning often focuses on fairness, while overlooking the roots or causes of bias. However, bias was originally defined as a "systematic error," often caused by humans at different stages of the research process. This article aims to bridge the gap between past literature on bias in research by providing taxonomy for potential sources of bias and errors in data and models. The paper focus on bias in machine learning pipelines. Survey analyses over forty potential sources of bias in the machine learning (ML) pipeline, providing clear examples for each. By understanding the sources and consequences of bias in machine learning, better methods can be developed for its detecting and mitigating, leading to fairer, more transparent, and more accurate ML models.
Most algorithms for representation learning and link prediction in relational data have been designed for static data. However, the data they are applied to usually evolves with time, such as friend graphs in social networks or user interactions with items in recommender systems. This is also the case for knowledge bases, which contain facts such as (US, has president, B. Obama, [2009-2017]) that are valid only at certain points in time. For the problem of link prediction under temporal constraints, i.e., answering queries such as (US, has president, ?, 2012), we propose a solution inspired by the canonical decomposition of tensors of order 4. We introduce new regularization schemes and present an extension of ComplEx (Trouillon et al., 2016) that achieves state-of-the-art performance. Additionally, we propose a new dataset for knowledge base completion constructed from Wikidata, larger than previous benchmarks by an order of magnitude, as a new reference for evaluating temporal and non-temporal link prediction methods.
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