The development of data-informed predictive models for dynamical systems is of widespread interest in many disciplines. We present a unifying framework for blending mechanistic and machine-learning approaches to identify dynamical systems from noisily and partially observed data. We compare pure data-driven learning with hybrid models which incorporate imperfect domain knowledge, referring to the discrepancy between an assumed truth model and the imperfect mechanistic model as model error. Our formulation is agnostic to the chosen machine learning model, is presented in both continuous- and discrete-time settings, and is compatible both with model errors that exhibit substantial memory and errors that are memoryless. First, we study memoryless linear (w.r.t. parametric-dependence) model error from a learning theory perspective, defining excess risk and generalization error. For ergodic continuous-time systems, we prove that both excess risk and generalization error are bounded above by terms that diminish with the square-root of T T , the time-interval over which training data is specified. Secondly, we study scenarios that benefit from modeling with memory, proving universal approximation theorems for two classes of continuous-time recurrent neural networks (RNNs): both can learn memory-dependent model error, assuming that it is governed by a finite-dimensional hidden variable and that, together, the observed and hidden variables form a continuous-time Markovian system. In addition, we connect one class of RNNs to reservoir computing, thereby relating learning of memory-dependent error to recent work on supervised learning between Banach spaces using random features. Numerical results are presented (Lorenz ’63, Lorenz ’96 Multiscale systems) to compare purely data-driven and hybrid approaches, finding hybrid methods less datahungry and more parametrically efficient. We also find that, while a continuous-time framing allows for robustness to irregular sampling and desirable domain- interpretability, a discrete-time framing can provide similar or better predictive performance, especially when data are undersampled and the vector field defining the true dynamics cannot be identified. Finally, we demonstrate numerically how data assimilation can be leveraged to learn hidden dynamics from noisy, partially-observed data, and illustrate challenges in representing memory by this approach, and in the training of such models.
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A Standardized Framework for Communicating and Modelling Parametrically Defined Mesostructure Patterns
Intricate mesostructures in additive manufacturing (AM) designs can offer enhanced strength-to-weight performance. However, complex mesostructures can also hinder designers, often resulting in unpalatably large digital files that are difficult to modify. Similarly, existing methods for defining and representing complex mesostructures are highly variable, which further increases the challenge in realizing such structures for AM. To address these gaps, we propose a standardized framework for designing and representing mesostructured components tailored to AM. Our method uses a parametric language to describe complex patterns, defined by a combination of macrostructural, mesostructural, and vector field information. We show how various mesostructures, ranging from simple rectilinear patterns to complex, vector field-driven cellular cutouts can be represented using few parameters (unit cell dimensions, orientation, and spacing). Our proposed framework has the potential to significantly reduce file size, while its extensible nature enables it to be expanded in the future.
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- Award ID(s):
- 1825535
- NSF-PAR ID:
- 10171176
- Date Published:
- Journal Name:
- Solid Freeform Fabrication 2019: Proceedings of the 30th Annual International Solid Freeform Fabrication Symposium – An Additive Manufacturing Conference
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract Deterministic transformations of 2D patterns of materials into well‐controlled 3D mesostructures serve as the basis for manufacturing methods that can bypass limitations of conventional 3D micro/nanofabrication. Here, guided mechanical buckling processes provide access to a rich range of complex 3D mesostructures in high‐performance materials, from inorganic and organic semiconductors, metals and dielectrics, to ceramics and even 2D materials (e.g., graphene, MoS2). Previous studies demonstrate that iterative computational procedures can define design parameters for certain targeted 3D configurations, but without the ability to address complex shapes. A technical need is in efficient, generalized inverse design algorithms that directly yield sets of optimized parameters. Here, such schemes are introduced, where the distributions of thicknesses across arrays of separated or interconnected ribbons provide scalable routes to 3D surfaces with a broad range of targeted shapes. Specifically, discretizing desired shapes into 2D ribbon components allows for analytic solutions to the inverse design of centrally symmetric and even general surfaces, in an approximate manner. Combined theoretical, numerical, and experimental studies of ≈20 different 3D structures with characteristic sizes (e.g., ribbon width) ranging from ≈200 µm to ≈2 cm and with geometries that resemble hemispheres, fire balloons, flowers, concave lenses, saddle surfaces, waterdrops, and rodents, illustrate the essential ideas.
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null (Ed.)Abstract Arbuscular mycorrhizal (AM) and ectomycorrhizal (EcM) associations are critical for host-tree performance. However, how mycorrhizal associations correlate with the latitudinal tree beta-diversity remains untested. Using a global dataset of 45 forest plots representing 2,804,270 trees across 3840 species, we test how AM and EcM trees contribute to total beta-diversity and its components (turnover and nestedness) of all trees. We find AM rather than EcM trees predominantly contribute to decreasing total beta-diversity and turnover and increasing nestedness with increasing latitude, probably because wide distributions of EcM trees do not generate strong compositional differences among localities. Environmental variables, especially temperature and precipitation, are strongly correlated with beta-diversity patterns for both AM trees and all trees rather than EcM trees. Results support our hypotheses that latitudinal beta-diversity patterns and environmental effects on these patterns are highly dependent on mycorrhizal types. Our findings highlight the importance of AM-dominated forests for conserving global forest biodiversity.more » « less
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Urban dispersal events occur when an unexpectedly large number of people leave an area in a relatively short period of time. It is beneficial for the city authorities, such as law enforcement and city management, to have an advance knowledge of such events, as it can help them mitigate the safety risks and handle important challenges such as managing traffic, and so forth. Predicting dispersal events is also beneficial to Taxi drivers and/or ride-sharing services, as it will help them respond to an unexpected demand and gain competitive advantage. Large urban datasets such as detailed trip records and point of interest ( POI ) data make such predictions achievable. The related literature mainly focused on taxi demand prediction. The pattern of the demand was assumed to be repetitive and proposed methods aimed at capturing those patterns. However, dispersal events are, by definition, violations of those patterns and are, understandably, missed by the methods in the literature. We proposed a different approach in our prior work [32]. We showed that dispersal events can be predicted by learning the complex patterns of arrival and other features that precede them in time. We proposed a survival analysis formulation of this problem and proposed a two-stage framework (DILSA), where a deep learning model predicted the survival function at each point in time in the future. We used that prediction to determine the time of the dispersal event in the future, or its non-occurrence. However, DILSA is subject to a few limitations. First, based on evidence from the data, mobility patterns can vary through time at a given location. DILSA does not distinguish between different mobility patterns through time. Second, mobility patterns are also different for different locations. DILSA does not have the capability to directly distinguish between different locations based on their mobility patterns. In this article, we address these limitations by proposing a method to capture the interaction between POIs and mobility patterns and we create vector representations of locations based on their mobility patterns. We call our new method DILSA+. We conduct extensive case studies and experiments on the NYC Yellow taxi dataset from 2014 to 2016. Results show that DILSA+ can predict events in the next 5 hours with an F1-score of 0.66. It is significantly better than DILSA and the state-of-the-art deep learning approaches for taxi demand prediction.more » « less
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We propose STSRNet, a joint space-time super-resolution deep learning based model for time-varying vector field data. Our method is designed to reconstruct high temporal resolution (HTR) and high spatial resolution (HSR) vector fields sequence from the corresponding low-resolution key frames. For large scale simulations, only data from a subset of time steps with reduced spatial resolution can be stored for post-hoc analysis. In this paper, we leverage a deep learning model to capture the non-linear complex changes of vector field data with a two-stage architecture: the first stage deforms a pair of low spatial resolution (LSR) key frames forward and backward to generate the intermediate LSR frames, and the second stage performs spatial super-resolution to output the high-resolution sequence. Our method is scalable and can handle different data sets. We demonstrate the effectiveness of our framework with several data sets through quantitative and qualitative evaluations.more » « less
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Accepted Manuscript1.0
- Conference Paper:
- The DOI is not currently available.
