Physics-inspired deep learning to characterize the signal manifold of quasi-circular, spinning, non-precessing binary black hole mergers
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Khan, Asad (ORCID:000000032055795X)↗
SEARCH · Engineering Papers
Results for “manifold learning”
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Tracking the topology of neural manifolds across populations
Neural manifolds summarize the intrinsic structure of the information encoded by a population of neurons. Advances in experimental techniques have made simultaneous recordings from multiple brain regions increasingly commonplace, raising the possibility of studying how these manifolds relate across populations. However, when the manifolds are nonlinear and possibly code for multiple unknown variables, it is challenging to extract robust and falsifiable information about their relationships. We introduce a framework, called the method of analogous cycles, for matching topological features of neural manifolds using only observed dissimilarity matrices within and between neural populations. We demonstrate via analysis of simulations and in vivo experimental data that this method can be used to correctly identify multiple shared circular coordinate systems across both stimuli and inferred neural manifolds. Conversely, the method rejects matching features that are not intrinsic to one of the systems. Further, as this method is deterministic and does not rely on dimensionality reduction or optimization methods, it is amenable to direct mathematical investigation and interpretation in terms of the underlying neural activity. We thus propose the method of analogous cycles as a suitable foundation for a theory of cross-population analysis via neural manifolds.
97 MATHEMATICS AND COMPUTING↗
Latent map Gaussian processes for mixed variable metamodeling
Gaussian processes (GPs) are ubiquitously used in sciences and engineering as metamodels. Standard GPs, however, can only handle numerical or quantitative variables. Here we introduce latent map Gaussian processes (LMGPs) that inherit the attractive properties of GPs and are also applicable to mixed data which have both quantitative and qualitative inputs. The core idea behind LMGPs is to learn a continuous, low-dimensional latent space or manifold which encodes all qualitative inputs. To learn this manifold, we first assign a unique prior vector representation to each combination of qualitative inputs. We then use a low-rank linear map to project these priors on a manifold that characterizes the posterior representations. As the posteriors are quantitative, they can be directly used in any standard correlation function such as the Gaussian or Matern. Hence, the optimal map and the corresponding manifold, along with other hyperparameters of the correlation function, can be systematically learned via maximum likelihood estimation. Through a wide range of analytic and real-world examples, we demonstrate the advantages of LMGPs over state-of-the-art methods in terms of accuracy and versatility. In particular, we show that LMGPs can handle variable-length inputs, have an explainable neural network interpretation, and provide insights into how qualitative inputs affect the response or interact with each other. We also employ LMGPs in Bayesian optimization and illustrate that they can discover optimal compound compositions more efficiently than conventional methods that convert compositions to qualitative variables via manual featurization.
42 ENGINEERING↗
Graph interpolating activation improves both natural and robust accuracies in data-efficient deep learning
Improving the accuracy and robustness of deep neural nets (DNNs) and adapting them to small training data are primary tasks in deep learning (DL) research. In this paper, we replace the output activation function of DNNs, typically the data-agnostic softmax function, with a graph Laplacian-based high-dimensional interpolating function which, in the continuum limit, converges to the solution of a Laplace–Beltrami equation on a high-dimensional manifold. Furthermore, we propose end-to-end training and testing algorithms for this new architecture. The proposed DNN with graph interpolating activation integrates the advantages of both deep learning and manifold learning. Compared to the conventional DNNs with the softmax function as output activation, the new framework demonstrates the following major advantages: First, it is better applicable to data-efficient learning in which we train high capacity DNNs without using a large number of training data. Second, it remarkably improves both natural accuracy on the clean images and robust accuracy on the adversarial images crafted by both white-box and black-box adversarial attacks. Third, it is a natural choice for semi-supervised learning. This paper is a significant extension of our earlier work published in NeurIPS, 2018. For reproducibility, the code is available at https://github.com/BaoWangMath/DNN-DataDependentActivation .
Mathematics↗
Reduced order modeling for flow and transport problems with Barlow Twins self-supervised learning
Abstract We propose a unified data-driven reduced order model (ROM) that bridges the performance gap between linear and nonlinear manifold approaches. Deep learning ROM (DL-ROM) using deep-convolutional autoencoders (DC–AE) has been shown to capture nonlinear solution manifolds but fails to perform adequately when linear subspace approaches such as proper orthogonal decomposition (POD) would be optimal. Besides, most DL-ROM models rely on convolutional layers, which might limit its application to only a structured mesh. The proposed framework in this study relies on the combination of an autoencoder (AE) and Barlow Twins (BT) self-supervised learning, where BT maximizes the information content of the embedding with the latent space through a joint embedding architecture. Through a series of benchmark problems of natural convection in porous media, BT–AE performs better than the previous DL-ROM framework by providing comparable results to POD-based approaches for problems where the solution lies within a linear subspace as well as DL-ROM autoencoder-based techniques where the solution lies on a nonlinear manifold; consequently, bridges the gap between linear and nonlinear reduced manifolds. We illustrate that a proficient construction of the latent space is key to achieving these results, enabling us to map these latent spaces using regression models. The proposed framework achieves a relative error of 2% on average and 12% in the worst-case scenario (i.e., the training data is small, but the parameter space is large.). We also show that our framework provides a speed-up of $$7 \times 10^{6}$$ 7 × 10 6 times, in the best case, and $$7 \times 10^{3}$$ 7 × 10 3 times on average compared to a finite element solver. Furthermore, this BT–AE framework can operate on unstructured meshes, which provides flexibility in its application to standard numerical solvers, on-site measurements, experimental data, or a combination of these sources.
97 MATHEMATICS AND COMPUTING↗
Operator inference for non-intrusive model reduction with quadratic manifolds
This paper proposes a novel approach for learning a data-driven quadratic manifold from high-dimensional data, then employing this quadratic manifold to derive efficient physics-based reduced-order models. The key ingredient of the approach is a polynomial mapping between high-dimensional states and a low-dimensional embedding. This mapping consists of two parts: a representation in a linear subspace (computed in this work using the proper orthogonal decomposition) and a quadratic component. The approach can be viewed as a form of data-driven closure modeling, since the quadratic component introduces directions into the approximation that lie in the orthogonal complement of the linear subspace, but without introducing any additional degrees of freedom to the low-dimensional representation. Combining the quadratic manifold approximation with the operator inference method for projection-based model reduction leads to a scalable non-intrusive approach for learning reduced-order models of dynamical systems. Further, applying the new approach to transport-dominated systems of partial differential equations illustrates the gains in efficiency that can be achieved over approximation in a linear subspace.
42 ENGINEERING↗
A Geometry-Driven Longitudal Topic Model
A simple and scalable framework for longitudinal analysis of Twitter data is developed that combines latent topic models with computational geometric methods. Dimensionality reduction tools from computational geometry are applied to learn the intrinsic manifold on which the latent, temporal topics reside. Then shortest path distances on the manifold are used to link together these topics. The proposed framework permits visualization of the low-dimensional embedding which provides clear interpretation of the complex, high-dimensional trajectories that may exist among latent topics. Practical application of the proposed framework is demonstrated through its ability to capture and effectively visualize natural progression of latent COVID-19 related topics learned from Twitter data. Interpretability of the trajectories is achieved by comparing to real-world events. In addition, the framework permits study of spatial variation in Twitter behavior for learned topics. The analysis demonstrates that the proposed framework is able to capture granular-level impact of COVID-19 on public discussions. We end by arguing that Twitter data, when analyzed within the proposed framework, can serve as a valuable supplementary data stream for COVID-related studies.
97 MATHEMATICS AND COMPUTING↗
Identifying Vehicle Signals in Continuous Seismic Data Using Unsupervised Machine-Learning Techniques
Seismic sensors deployed near roadways effectively capture ground vibrations generated by passing vehicles. Although both traditional and machine‐learning algorithms have been utilized for analyzing such signals, independent validation of detected vehicle events remains limited. We applied two unsupervised machine‐learning algorithms, uniform manifold approximation and projection for dimension reduction, and hierarchical density‐based spatial clustering of applications with noise, to continuous seismic data collected along a road on the main campus of Oak Ridge National Laboratory. The algorithms identified seven distinct cluster labels across the entire dataset. By comparing these cluster labels with precipitation records from a nearby weather station and image‐derived labels from a local camera system, we identified one cluster associated with rainfall and another with vehicle activity. Our algorithms identified a greater number of vehicle‐related labels compared to the camera‐derived labels because seismic data are unaffected by poor lighting conditions. The arrival times of the newly detected vehicle signals corresponded well with the road’s speed limit, supporting our findings. Our algorithm outperformed the short‐term average/long‐term average method and k‐means clustering. Our results suggest that seismic data, when analyzed with machine‐learning algorithms, can complement existing vehicle monitoring systems, particularly under challenging environmental conditions.
Chai, Chengping [Oak Ridge National Laboratory (OR↗
GeN-ROM—An OpenFOAM®-based multiphysics reduced-order modeling framework for the analysis of Molten Salt Reactors
This work presents a projection-based multiphysics Model Order Reduction (MOR) framework for the analysis of nuclear systems and its application to parametric simulations of Molten Salt Reactors (MSR). The framework, named GeN-ROM, is developed using OpenFOAM® and employs a Proper Orthogonal Decomposition aided Reduced-Basis technique (POD-RB). It can be used to reduce steady-state and transient multiphysics problems involving parametric fluid dynamics, heat exchange, and neutronics phenomena. For the treatment of structural elements in the hydraulic systems, a porous medium approach has been adopted. The reduction process is data-driven and snapshot information is extracted via POD to learn the solution manifold and to build global spatial basis functions. At the data collection phase, GeN-ROM makes use of the solvers available in GeN-Foam, a similarly OpenFOAM®-based multiphysics framework developed for the analysis of nuclear reactors. The global bases are used both to approximate the solution fields and to project the full-order equations onto lower-dimensional subspaces, thus considerably reducing the number of unknowns in a numerical system. This reduction leads to significant computational speedups, which is ideal for multi-query applications such as uncertainty quantification or design optimization. The developed tool has been tested using a 2D multiphysics model of the Molten Salt Fast Reactor (MSFR) with steady-state and transient scenarios, with speedups on the order of 10 – 10 5 .
22 GENERAL STUDIES OF NUCLEAR REACTORS↗
Generative Patch Priors for Practical Compressive Image Recovery
In this work, we propose the generative patch prior (GPP) that defines a generative prior for compressive image recovery, based on patch-manifold models. Unlike learned, image-level priors that are restricted to the range space of a pre-trained generator, GPP can recover a wide variety of natural images using a pre-trained patch generator. Additionally, GPP retains the benefits of generative priors like high reconstruction quality at extremely low sensing rates, while also being much more generally applicable. We show that GPP outperforms several unsupervised and supervised techniques on three different sensing models – linear compressive sensing with known, and unknown calibration settings, and the non-linear phase retrieval problem. Finally, we propose an alternating optimization strategy using GPP for joint calibration-and-reconstruction which performs favorably against several baselines on a real world, uncalibrated compressive sensing dataset. The code and models for GPP are available on github.
compressed sensing↗
Neural Active Manifolds: Nonlinear Dimensionality Reduction for Uncertainty Quantification
We present a new approach for nonlinear dimensionality reduction, specifically designed for computationally expensive mathematical models. We leverage autoencoders to discover a one-dimensional neural active manifold (NeurAM) capturing the model output variability, through the aid of a simultaneously learnt surrogate model with inputs on this manifold. Our method only relies on model evaluations and does not require the knowledge of gradients. The proposed dimensionality reduction framework can then be applied to assist outer loop many-query tasks in scientific computing, like sensitivity analysis and multifidelity uncertainty propagation. In particular, we prove, both theoretically under idealized conditions, and numerically in challenging test cases, how NeurAM can be used to obtain multifidelity sampling estimators with reduced variance by sampling the models on the discovered low-dimensional and shared manifold among models. Several numerical examples illustrate the main features of the proposed dimensionality reduction strategy and highlight its advantages with respect to existing approaches in the literature.
Autoencoders↗
Reduced order modeling with Barlow Twins self-supervised learning: Navigating the space between linear and nonlinear solution manifolds.
Abstract not provided.
Kadeethum, Teeratorn↗
Learning the Short- and Medium-Range Enthalpic Structural Manifolds of Disordered Materials.
Abstract not provided.
Hardin, Thomas↗
An adaptive and stability-promoting layerwise training approach for sparse deep neural network architecture
This work presents a two-stage adaptive framework for progressively developing deep neural network (DNN) architectures that generalize well for a given training data set. In the first stage, a layerwise training approach is adopted where a new layer is added each time and trained independently by freezing parameters in the previous layers. We impose desirable structures on the DNN by employing manifold regularization, sparsity regularization, and physics-informed terms. We introduce a ε – δ – stability-promoting concept as a desirable property for a learning algorithm and show that employing manifold regularization yields a ε – δ stability-promoting algorithm. Further, we also derive the necessary conditions for the trainability of a newly added layer and investigate the training saturation problem. In the second stage of the algorithm (post-processing), a sequence of shallow networks is employed to extract information from the residual produced in the first stage, thereby improving the prediction accuracy. Numerical investigations on prototype regression and classification problems demonstrate that the proposed approach can outperform fully connected DNNs of the same size. Moreover, by equipping the physics-informed neural network (PINN) with the proposed adaptive architecture strategy to solve partial differential equations, we numerically show that adaptive PINNs not only are superior to standard PINNs but also produce interpretable hidden layers with provable stability. As a result, we also apply our architecture design strategy to solve inverse problems governed by elliptic partial differential equations.
42 ENGINEERING↗
Efficient bifurcation and tabulation of multi-dimensional combustion manifolds using deep mixture of experts: An a priori study
This work describes and validates an approach for autonomously bifurcating turbulent combustion man-ifolds to divide regression tasks amongst specialized artificial neural networks (ANNs). This approach relies on the mixture of experts (MoE) framework, where each neural network is trained to be specialized in a given portion of the input space. The assignment of different input regions to the experts is determined by a gating network, which is a neural network classifier. In some previous studies, it has been demonstrated that bifurcation of a complex combustion manifold and fitting different ANNs for each part leads to better fits or faster inference speeds. However, the manner of bifurcation in these studies was based on heuristic approaches or clustering techniques. In contrast, the proposed technique enables automatic bifurcation using non-linear planes in high-dimensional turbulent combustion manifolds that are often associated with complex behavior due to different dominating physics in various zones. Furthermore, the proposed concept is validated using 4-dimensional (4D) and 5D flamelet tables, showing that the errors obtained with a given network size, or conversely the network size required to achieve a given accuracy, is considerably reduced. The effect of the number of experts on inference speed is also investigated, showing that by increasing the number of experts from 1 to 8, the inference time can be approximately reduced by a factor of two. Moreover, it is shown that the MoE approach divides the input manifold in a physically intuitive manner, suggesting that the MoE framework can elucidate high-dimensional datasets in a physically meaningful way.
42 ENGINEERING↗
Fast Characterization of Inducible Regions of Atrial Fibrillation Models With Multi-Fidelity Gaussian Process Classification
Computational models of atrial fibrillation have successfully been used to predict optimal ablation sites. A critical step to assess the effect of an ablation pattern is to pace the model from different, potentially random, locations to determine whether arrhythmias can be induced in the atria. In this work, we propose to use multi-fidelity Gaussian process classification on Riemannian manifolds to efficiently determine the regions in the atria where arrhythmias are inducible. We build a probabilistic classifier that operates directly on the atrial surface. We take advantage of lower resolution models to explore the atrial surface and combine seamlessly with high-resolution models to identify regions of inducibility. We test our methodology in 9 different cases, with different levels of fibrosis and ablation treatments, totalling 1,800 high resolution and 900 low resolution simulations of atrial fibrillation. When trained with 40 samples, our multi-fidelity classifier that combines low and high resolution models, shows a balanced accuracy that is, on average, 5.7% higher than a nearest neighbor classifier. We hope that this new technique will allow faster and more precise clinical applications of computational models for atrial fibrillation. All data and code accompanying this manuscript will be made publicly available at: https://github.com/fsahli/AtrialMFclass.
59 BASIC BIOLOGICAL SCIENCES↗
Hunting for Polluted White Dwarfs and Other Treasures with Gaia XP Spectra and Unsupervised Machine Learning
White dwarfs (WDs) polluted by exoplanetary material provide the unprecedented opportunity to directly observe the interiors of exoplanets. However, spectroscopic surveys are often limited by brightness constraints, and WDs tend to be very faint, making detections of large populations of polluted WDs difficult. In this paper, we aim to increase considerably the number of WDs with multiple metals in their atmospheres. Using 96,134 WDs with Gaia DR3 BP/RP (XP) spectra, we constructed a 2D map using an unsupervised machine-learning technique called Uniform Manifold Approximation and Projection (UMAP) to organize the WDs into identifiable spectral regions. The polluted WDs are among the distinct spectral groups identified in our map. We have shown that this selection method could potentially increase the number of known WDs with five or more metal species in their atmospheres by an order of magnitude. Such systems are essential for characterizing exoplanet diversity and geology.
79 ASTRONOMY AND ASTROPHYSICS↗
State predictive information bottleneck
We report the ability to make sense of the massive amounts of high-dimensional data generated from molecular dynamics simulations is heavily dependent on the knowledge of a low-dimensional manifold (parameterized by a reaction coordinate or RC) that typically distinguishes between relevant metastable states, and which captures the relevant slow dynamics of interest. Methods based on machine learning and artificial intelligence have been proposed over the years to deal with learning such low-dimensional manifolds, but they are often criticized for a disconnect from more traditional and physically interpretable approaches. To deal with such concerns, in this work we propose a deep learning based state predictive information bottleneck approach to learn the RC from high-dimensional molecular simulation trajectories. We demonstrate analytically and numerically how the RC learnt in this approach is connected to the committor in chemical physics and can be used to accurately identify transition states. A crucial hyperparameter in this approach is the time delay or how far into the future the algorithm should make predictions about. Through careful comparisons for benchmark systems, we demonstrate that this hyperparameter choice gives useful control over how coarse-grained we want the metastable state classification of the system to be. We thus believe that this work represents a step forward in systematic application of deep learning based ideas to molecular simulations.
37 INORGANIC, ORGANIC, PHYSICAL, AND ANALYTICAL CH↗