D. Akbaba, D. Lange, M. Correll, A. Lex, M. Meyer. Troubling Collaboration: Matters of Care for Visualization Design Study, In Proceedings of the 2023 CHI Conference on Human Factors in Computing Systems (CHI ’23),, pp. 23--28. April, 2023.
A common research process in visualization is for visualization researchers to collaborate with domain experts to solve particular applied data problems. While there is existing guidance and expertise around how to structure collaborations to strengthen research contributions, there is comparatively little guidance on how to navigate the implications of, and power produced through the socio-technical entanglements of collaborations. In this paper, we qualitatively analyze refective interviews of past participants of collaborations from multiple perspectives: visualization graduate students, visualization professors, and domain collaborators. We juxtapose the perspectives of these individuals, revealing tensions about the tools that are built and the relationships that are formed — a complex web of competing motivations. Through the lens of matters of care, we interpret this web, concluding with considerations that both trouble and necessitate reformation of current patterns around collaborative work in visualization design studies to promote more equitable, useful, and care-ful outcomes.
J.W. Beiriger, W. Tao, M.K. Bruce, E. Anstadt, C. Christiensen, J. Smetona, R. Whitaker, J. Goldstein. CranioRate TM: An Image-Based, Deep-Phenotyping Analysis Toolset and Online Clinician Interface for Metopic Craniosynostosis, In Plastic and Reconstructive Surgery, 2023.
The diagnosis and management of metopic craniosynostosis involves subjective decision-making at the point of care. The purpose of this work is to describe a quantitative severity metric and point-of-care user interface to aid clinicians in the management of metopic craniosynostosis and to provide a platform for future research through deep phenotyping.
Two machine-learning algorithms were developed that quantify the severity of craniosynostosis – a supervised model specific to metopic craniosynostosis (Metopic Severity Score) and an unsupervised model used for cranial morphology in general (Cranial Morphology Deviation). CT imaging from multiple institutions were compiled to establish the spectrum of severity and a point-of-care tool was developed and validated.
Over the study period (2019-2021), 254 patients with metopic craniosynostosis and 92 control patients who underwent CT scan between the ages of 6 and 18 months were included. Scans were processed using an unsupervised machine-learning based dysmorphology quantification tool, CranioRate TM. The average Metopic severity score (MSS) for normal controls was 0.0±1.0 and for metopic synostosis was 4.9±2.3 (p<0.001). The average Cranial Morphology Deviation (CMD) for normal controls was 85.2±19.2 and for metopic synostosis was 189.9±43.4 (p<0.001). A point-of-care user interface (craniorate.org) has processed 46 CT images from 10 institutions.
T.C. Bidone, D.J. Odde. Multiscale models of integrins and cellular adhesions, In Current Opinion in Structural Biology, Vol. 80, Elsevier, 2023.
Computational models of integrin-based adhesion complexes have revealed important insights into the mechanisms by which cells establish connections with their external environment. However, how changes in conformation and function of individual adhesion proteins regulate the dynamics of whole adhesion complexes remains largely elusive. This is because of the large separation in time and length scales between the dynamics of individual adhesion proteins (nanoseconds and nanometers) and the emergent dynamics of the whole adhesion complex (seconds and micrometers), and the limitations of molecular simulation approaches in extracting accurate free energies, conformational transitions, reaction mechanisms, and kinetic rates, that can inform mechanisms at the larger scales. In this review, we discuss models of integrin-based adhesion complexes and highlight their main findings regarding: (i) the conformational transitions of integrins at the molecular and macromolecular scales and (ii) the molecular clutch mechanism at the mesoscale. Lastly, we present unanswered questions in the field of modeling adhesions and propose new ideas for future exciting modeling opportunities.
R. Han, A. Narayan, Y. Xu. An approximate control variates approach to multifidelity distribution estimation, Subtitled arXiv:2303.06422v1, 2023.
Forward simulation-based uncertainty quantification that studies the output distribution of quantities of interest (QoI) is a crucial component for computationally robust statistics and engineering. There is a large body of literature devoted to accurately assessing statistics of QoI, and in particular, multilevel or multifidelity approaches are known to be effective, leveraging cost-accuracy tradeoffs between a given ensemble of models. However, effective algorithms that can estimate the full distribution of outputs are still under active development. In this paper, we introduce a general multifidelity framework for estimating the cumulative distribution functions (CDFs) of vector-valued QoI associated with a high-fidelity model under a budget constraint. Given a family of appropriate control variates obtained from lower fidelity surrogates, our framework involves identifying the most cost-effective model subset and then using it to build an approximate control variates estimator for the target CDF. We instantiate the framework by constructing a family of control variates using intermediate linear approximators and rigorously analyze the corresponding algorithm. Our analysis reveals that the resulting CDF estimator is uniformly consistent and budget-asymptotically optimal, with only mild moment and regularity assumptions. The approach provides a robust multifidelity CDF estimator that is adaptive to the available budget, does not require a priori knowledge of cross-model statistics or model hierarchy, and is applicable to general output dimensions. We demonstrate the efficiency and robustness of the approach using several test examples.
K. G. Hicks, A. A. Cluntun, H. L. Schubert, S. R. Hackett, J. A. Berg, P. G. Leonard, M. A. Ajalla Aleixo, Y. Zhou, A. J. Bott, S. R. Salvatore, F. Chang, A. Blevins, P. Barta, S. Tilley, A. Leifer, A. Guzman, A. Arok, S. Fogarty, J. M. Winter, H. Ahn, K. N. Allen, S. Block, I. A. Cardoso, J. Ding, I. Dreveny, C. Gasper, Q. Ho, A. Matsuura, M. J. Palladino, S. Prajapati, P. Sun, K. Tittmann, D. R. Tolan, J. Unterlass, A. P. VanDemark, M. G. Vander Heiden, B. A. Webb, C. Yun, P. Zhap, B. Wang, F. J. Schopfer, C. P. Hill, M. C. Nonato, F. L. Muller, J. E. Cox, J. Rutter. Protein-metabolite interactomics of carbohydrate metabolism reveal regulation of lactate dehydrogenase, In Science, Vol. 379, No. 6636, pp. 996-1003. 2023.
Metabolic networks are interconnected and influence diverse cellular processes. The protein-metabolite interactions that mediate these networks are frequently low affinity and challenging to systematically discover. We developed mass spectrometry integrated with equilibrium dialysis for the discovery of allostery systematically (MIDAS) to identify such interactions. Analysis of 33 enzymes from human carbohydrate metabolism identified 830 protein-metabolite interactions, including known regulators, substrates, and products as well as previously unreported interactions. We functionally validated a subset of interactions, including the isoform-specific inhibition of lactate dehydrogenase by long-chain acyl–coenzyme A. Cell treatment with fatty acids caused a loss of pyruvate-lactate interconversion dependent on lactate dehydrogenase isoform expression. These protein-metabolite interactions may contribute to the dynamic, tissue-specific metabolic flexibility that enables growth and survival in an ever-changing nutrient environment. Understanding how metabolic state influences cellular processes requires systematic analysis of low-affinity interactions of metabolites with proteins. Hicks et al. describe a method called MIDAS (mass spectrometry integrated with equilibrium dialysis for the discovery of allostery systematically), which allowed them to probe such interactions for 33 enzymes of human carbohydrate metabolism and more than 400 metabolites. The authors detected many known and many new interactions, including regulation of lactate dehydrogenase by ATP and long-chain acyl coenzyme A, which may help to explain known physiological relations between fat and carbohydrate metabolism in different tissues. —LBR A mass spectrometry and dialysis method detects metabolite-protein interactions that help to control physiology.
M. Hu, K. Zhang, Q. Nguyen, T. Tasdizen. The effects of passive design on indoor thermal comfort and energy savings for residential buildings in hot climates: A systematic review, In Urban Climate, Vol. 49, pp. 101466. 2023.
In this study, a systematic review and meta-analysis were conducted to identify, categorize, and investigate the effectiveness of passive cooling strategies (PCSs) for residential buildings. Forty-two studies published between 2000 and 2021 were reviewed; they examined the effects of PCSs on indoor temperature decrease, cooling load reduction, energy savings, and thermal comfort hour extension. In total, 30 passive strategies were identified and classified into three categories: design approach, building envelope, and passive cooling system. The review found that using various passive strategies can achieve, on average, (i) an indoor temperature decrease of 2.2 °C, (ii) a cooling load reduction of 31%, (iii) energy savings of 29%, and (v) a thermal comfort hour extension of 23%. Moreover, the five most effective passive strategies were identified as well as the differences between hot and dry climates and hot and humid climates.
R. Kamali, E. Kwan, M. Regouski, T.J. Bunch, D.J. Dosdall, E. Hsu, R. S. Macleod, I. Polejaeva, R. Ranjan. Contribution of atrial myofiber architecture to atrial fibrillation, In PLOS ONE, Vol. 18, No. 1, Public Library of Science, pp. 1--16. Jan, 2023.
The role of fiber orientation on a global chamber level in sustaining atrial fibrillation (AF) is unknown. The goal of this study was to correlate the fiber direction derived from Diffusion Tensor Imaging (DTI) with AF inducibility.
T. Kataria, S. Rajamani, A.B. Ayubi, M. Bronner, J. Jedrzkiewicz, B. Knudsen, S. Elhabian. Automating Ground Truth Annotations For Gland Segmentation Through Immunohistochemistry, 2023.
The microscopic evaluation of glands in the colon is of utmost importance in the diagnosis of inflammatory bowel disease (IBD) and cancer. When properly trained, deep learning pipelines can provide a systematic, reproducible, and quantitative assessment of disease-related changes in glandular tissue architecture. The training and testing of deep learning models require large amounts of manual annotations, which are difficult, time-consuming, and expensive to obtain. Here, we propose a method for the automated generation of ground truth in digital H&E slides using immunohistochemistry (IHC) labels. The image processing pipeline generates annotations of glands in H&E histopathology images from colon biopsies by transfer of gland masks from CK8/18, CDX2, or EpCAM IHC. The IHC gland outlines are transferred to co-registered H&E images for the training of deep learning models. We compare the performance of the deep learning models to manual annotations using an internal held-out set of biopsies as well as two public data sets. Our results show that EpCAM IHC provides gland outlines that closely match manual gland annotations (DICE = 0.89) and are robust to damage by inflammation. In addition, we propose a simple data sampling technique that allows models trained on data from several sources to be adapted to a new data source using just a few newly annotated samples. The best-performing models achieved average DICE scores of 0.902 and 0.89, respectively, on GLAS and CRAG colon cancer public datasets when trained with only 10% of annotated cases from either public cohort. Altogether, the performances of our models indicate that automated annotations using cell type-specific IHC markers can safely replace manual annotations. The automated IHC labels from single institution cohorts can be combined with small numbers of hand-annotated cases from multi-institutional cohorts to train models that generalize well to diverse data sources.
S. Leventhal, A. Gyulassy, M. Heimann, V. Pascucci. Exploring Classification of Topological Priors with Machine Learning for Feature Extraction, In IEEE Transactions on Visualization and Computer Graphics, pp. 1--12. 2023.
In many scientific endeavors, increasingly abstract representations of data allow for new interpretive methodologies and conceptualization of phenomena. For example, moving from raw imaged pixels to segmented and reconstructed objects allows researchers new insights and means to direct their studies toward relevant areas. Thus, the development of new and improved methods for segmentation remains an active area of research. With advances in machine learning and neural networks, scientists have been focused on employing deep neural networks such as U-Net to obtain pixel-level segmentations, namely, defining associations between pixels and corresponding/referent objects and gathering those objects afterward. Topological analysis, such as the use of the Morse-Smale complex to encode regions of uniform gradient flow behavior, offers an alternative approach: first, create geometric priors, and then apply machine learning to classify. This approach is empirically motivated since phenomena of interest often appear as subsets of topological priors in many applications. Using topological elements not only reduces the learning space but also introduces the ability to use learnable geometries and connectivity to aid the classification of the segmentation target. In this paper, we describe an approach to creating learnable topological elements, explore the application of ML techniques to classification tasks in a number of areas, and demonstrate this approach as a viable alternative to pixel-level classification, with similar accuracy, improved execution time, and requiring marginal training data.
H. Oh, R. Amici, G. Bomarito, S. Zhe, R. Kirby, J. Hochhalter. Genetic Programming Based Symbolic Regression for Analytical Solutions to Differential Equations, Subtitled arXiv:2302.03175v1, 2023.
In this paper, we present a machine learning method for the discovery of analytic solutions to differential equations. The method utilizes an inherently interpretable algorithm, genetic programming based symbolic regression. Unlike conventional accuracy measures in machine learning we demonstrate the ability to recover true analytic solutions, as opposed to a numerical approximation. The method is verified by assessing its ability to recover known analytic solutions for two separate differential equations. The developed method is compared to a conventional, purely data-driven genetic programming based symbolic regression algorithm. The reliability of successful evolution of the true solution, or an algebraic equivalent, is demonstrated.
M. Penwarden, S. Zhe, A. Narayan, R.M. Kirby. A Metalearning Approach for Physics-Informed Neural Networks (PINNs): Application to Parameterized PDEs, In Journal of Computational Physics, Elsevier, 2023.
Physics-informed neural networks (PINNs) as a means of discretizing partial differential equations (PDEs) are garnering much attention in the Computational Science and Engineering (CS&E) world. At least two challenges exist for PINNs at present: an understanding of accuracy and convergence characteristics with respect to tunable parameters and identification of optimization strategies that make PINNs as efficient as other computational science tools. The cost of PINNs training remains a major challenge of Physics-informed Machine Learning (PiML) – and, in fact, machine learning (ML) in general. This paper is meant to move towards addressing the latter through the study of PINNs on new tasks, for which parameterized PDEs provides a good testbed application as tasks can be easily defined in this context. Following the ML world, we introduce metalearning of PINNs with application to parameterized PDEs. By introducing metalearning and transfer learning concepts, we can greatly accelerate the PINNs optimization process. We present a survey of model-agnostic metalearning, and then discuss our model-aware metalearning applied to PINNs as well as implementation considerations and algorithmic complexity. We then test our approach on various canonical forward parameterized PDEs that have been presented in the emerging PINNs literature.
M. Penwarden, A.D. Jagtap, S. Zhe, G.E. Karniadakis, R.M. Kirby. A unified scalable framework for causal sweeping strategies for Physics-Informed Neural Networks (PINNs) and their temporal decompositions, Subtitled arXiv:2302.14227v1, 2023.
Physics-informed neural networks (PINNs) as a means of solving partial differential equations (PDE) have garnered much attention in the Computational Science and Engineering (CS&E) world. However, a recent topic of interest is exploring various training (i.e., optimization) challenges – in particular, arriving at poor local minima in the optimization landscape results in a PINN approximation giving an inferior, and sometimes trivial, solution when solving forward time-dependent PDEs with no data. This problem is also found in, and in some sense more difficult, with domain decomposition strategies such as temporal decomposition using XPINNs. To address this problem, we first enable a general categorization for previous causality methods, from which we identify a gap (e.g., opportunity) in the previous approaches. We then furnish examples and explanations for different training challenges, their cause, and how they relate to information propagation and temporal decomposition. We propose a solution to fill this gap by reframing these causality concepts into a generalized information propagation framework in which any prior method or combination of methods can be described. This framework is easily modifiable via user parameters in the open-source code accompanying this paper. Our unified framework moves toward reducing the number of PINN methods to consider and the reimplementation and retuning cost for thorough comparisons rather than increasing it. Using the idea of information propagation, we propose a new stacked-decomposition method that bridges the gap between time-marching PINNs and XPINNs. We also introduce significant computational speed-ups by using transfer learning concepts to initialize subnetworks in the domain and loss tolerance-based propagation for the subdomains. Finally, we formulate a new time-sweeping collocation point algorithm inspired by the previous PINNs causality literature, which our framework can still describe, and provides a significant computational speed-up via reduced-cost collocation point segmentation. The proposed methods overcome training challenges in PINNs and XPINNs for time-dependent PDEs by respecting the causality in multiple forms and improving scalability by limiting the computation required per optimization iteration. Finally, we provide numerical results for these methods on baseline PDE problems for which unmodified PINNs and XPINNs struggle to train.
K. Shukla, V. Oommen, A. Peyvan, M. Penwarden, L. Bravo, A. Ghoshal, R.M. Kirby, G. Karniadakis. Deep neural operators can serve as accurate surrogates for shape optimization: A case study for airfoils, Subtitled arXiv:2302.00807v1, 2023.
Deep neural operators, such as DeepONets, have changed the paradigm in high-dimensional nonlinear regression from function regression to (differential) operator regression, paving the way for significant changes in computational engineering applications. Here, we investigate the use of DeepONets to infer flow fields around unseen airfoils with the aim of shape optimization, an important design problem in aerodynamics that typically taxes computational resources heavily. We present results which display little to no degradation in prediction accuracy, while reducing the online optimization cost by orders of magnitude. We consider NACA airfoils as a test case for our proposed approach, as their shape can be easily defined by the four-digit parametrization. We successfully optimize the constrained NACA four-digit problem with respect to maximizing the lift-to-drag ratio and validate all results by comparing them to a high-order CFD solver. We find that DeepONets have low generalization error, making them ideal for generating solutions of unseen shapes. Specifically, pressure, density, and velocity fields are accurately inferred at a fraction of a second, hence enabling the use of general objective functions beyond the maximization of the lift-to-drag ratio considered in the current work.
J. Adams, N. Khan, A. Morris, S. Elhabian. Spatiotemporal Cardiac Statistical Shape Modeling: A Data-Driven Approach, Subtitled arXiv preprint arXiv:2209.02736, 2022.
Clinical investigations of anatomy’s structural changes over time could greatly benefit from population-level quantification of shape, or spatiotemporal statistic shape modeling (SSM). Such a tool enables characterizing patient organ cycles or disease progression in relation to a cohort of interest. Constructing shape models requires establishing a quantitative shape representation (e.g., corresponding landmarks). Particle-based shape modeling (PSM) is a data-driven SSM approach that captures population-level shape variations by optimizing landmark placement. However, it assumes cross-sectional study designs and hence has limited statistical power in representing shape changes over time. Existing methods for modeling spatiotemporal or longitudinal shape changes require predefined shape atlases and pre-built shape models that are typically constructed cross-sectionally. This paper proposes a data-driven approach inspired by the PSM method to learn population-level spatiotemporal shape changes directly from shape data. We introduce a novel SSM optimization scheme that produces landmarks that are in correspondence both across the population (inter-subject) and across time-series (intra-subject). We apply the proposed method to 4D cardiac data from atrial-fibrillation patients and demonstrate its efficacy in representing the dynamic change of the left atrium. Furthermore, we show that our method outperforms an image-based approach for spatiotemporal SSM with respect to a generative time-series model, the Linear Dynamical System (LDS). LDS fit using a spatiotemporal shape model optimized via our approach provides better generalization and specificity, indicating it accurately captures the underlying time-dependency.
M. Alirezaei, T. Tasdizen. Adversarially Robust Classification by Conditional Generative Model Inversion, Subtitled arXiv preprint arXiv:2201.04733, 2022.
Most adversarial attack defense methods rely on obfuscating gradients. These methods are successful in defending against gradient-based attacks; however, they are easily circumvented by attacks which either do not use the gradient or by attacks which approximate and use the corrected gradient. Defenses that do not obfuscate gradients such as adversarial training exist, but these approaches generally make assumptions about the attack such as its magnitude. We propose a classification model that does not obfuscate gradients and is robust by construction without assuming prior knowledge about the attack. Our method casts classification as an optimization problem where we "invert" a conditional generator trained on unperturbed, natural images to find the class that generates the closest sample to the query image. We hypothesize that a potential source of brittleness against adversarial attacks is the high-to-low-dimensional nature of feed-forward classifiers which allows an adversary to find small perturbations in the input space that lead to large changes in the output space. On the other hand, a generative model is typically a low-to-high-dimensional mapping. While the method is related to Defense-GAN, the use of a conditional generative model and inversion in our model instead of the feed-forward classifier is a critical difference. Unlike Defense-GAN, which was shown to generate obfuscated gradients that are easily circumvented, we show that our method does not obfuscate gradients. We demonstrate that our model is extremely robust against black-box attacks and has improved robustness against white-box attacks compared to naturally trained, feed-forward classifiers.
E.E. Anstadt, W. Tao, E. Guo, L. Dvoracek, M.K. Bruce, P.J. Grosse, L. Wang, L. Kavan, R. Whitaker, J.A. Goldstein. Quantifying the Severity of Metopic Craniosynostosis Using Unsupervised Machine Learning, In Plastic and Reconstructive Surgery, November, 2022.
Quantifying the severity of head shape deformity and establishing a threshold for operative intervention remains challenging in patients with Metopic Craniosynostosis (MCS). This study combines 3D skull shape analysis with an unsupervised machine-learning algorithm to generate a quantitative shape severity score (CMD) and provide an operative threshold score.
Head computed tomography (CT) scans from subjects with MCS and normal controls (age 5-15 months) were used for objective 3D shape analysis using ShapeWorks software and in a survey for craniofacial surgeons to rate head-shape deformity and report whether they would offer surgical correction based on head shape alone. An unsupervised machine-learning algorithm was developed to quantify the degree of shape abnormality of MCS skulls compared to controls.
124 CTs were used to develop the model; 50 (24% MCS, 76% controls) were rated by 36 craniofacial surgeons, with an average of 20.8 ratings per skull. The interrater reliability was high (ICC=0.988). The algorithm performed accurately and correlates closely with the surgeons assigned severity ratings (Spearman’s Correlation coefficient r=0.817). The median CMD for affected skulls was 155.0 (IQR 136.4-194.6, maximum 231.3). Skulls with ratings ≥150.2 were highly likely to be offered surgery by the experts in this study.
This study describes a novel metric to quantify the head shape deformity associated with metopic craniosynostosis and contextualizes the results using clinical assessments of head shapes by craniofacial experts. This metric may be useful in supporting clinical decision making around operative intervention as well as in describing outcomes and comparing patient population across centers.
A. Arzani, K.W. Cassel, R.M. D'Souza. Theory-guided physics-informed neural networks for boundary layer problems with singular perturbation, In Journal of Computational Physics, 2022.
Physics-informed neural networks (PINNs) are a recent trend in scientific machine learning research and modeling of differential equations. Despite progress in PINN research, large gradients and highly nonlinear patterns remain challenging to model. Thin boundary layer problems are prominent examples of large gradients that commonly arise in transport problems. In this study, boundary-layer PINN (BL-PINN) is proposed to enable a solution to thin boundary layers by considering them as a singular perturbation problem. Inspired by the classical perturbation theory and asymptotic expansions, BL-PINN is designed to replicate the procedure in singular perturbation theory. Namely, different parallel PINN networks are defined to represent different orders of approximation to the boundary layer problem in the inner and outer regions. In different benchmark problems (forward and inverse), BL-PINN shows superior performance compared to the traditional PINN approach and is able to produce accurate results, whereas the classical PINN approach could not provide meaningful solutions. BL-PINN also demonstrates significantly better results compared to other extensions of PINN such as the extended PINN (XPINN) approach. The natural incorporation of the perturbation parameter in BL-PINN provides the opportunity to evaluate parametric solutions without the need for retraining. BL-PINN demonstrates an example of how classical mathematical theory could be used to guide the design of deep neural networks for solving challenging problems.
T. M. Athawale, D. Maljovec. L. Yan, C. R. Johnson, V. Pascucci, B. Wang. Uncertainty Visualization of 2D Morse Complex Ensembles Using Statistical Summary Maps, In IEEE Transactions on Visualization and Computer Graphics, Vol. 28, No. 4, pp. 1955-1966. April, 2022.
Morse complexes are gradient-based topological descriptors with close connections to Morse theory. They are widely applicable in scientific visualization as they serve as important abstractions for gaining insights into the topology of scalar fields. Data uncertainty inherent to scalar fields due to randomness in their acquisition and processing, however, limits our understanding of Morse complexes as structural abstractions. We, therefore, explore uncertainty visualization of an ensemble of 2D Morse complexes that arises from scalar fields coupled with data uncertainty. We propose several statistical summary maps as new entities for quantifying structural variations and visualizing positional uncertainties of Morse complexes in ensembles. Specifically, we introduce three types of statistical summary maps – the probabilistic map , the significance map , and the survival map – to characterize the uncertain behaviors of gradient flows. We demonstrate the utility of our proposed approach using wind, flow, and ocean eddy simulation datasets.
J. Baker, E. Cherkaev, A. Narayan, B. Wang. Learning POD of Complex Dynamics Using Heavy-ball Neural ODEs, Subtitled arXiv:2202.12373, 2022.
Proper orthogonal decomposition (POD) allows reduced-order modeling of complex dynamical systems at a substantial level, while maintaining a high degree of accuracy in modeling the underlying dynamical systems. Advances in machine learning algorithms enable learning POD-based dynamics from data and making accurate and fast predictions of dynamical systems. In this paper, we leverage the recently proposed heavy-ball neural ODEs (HBNODEs) [Xia et al. NeurIPS, 2021] for learning data-driven reduced-order models (ROMs) in the POD context, in particular, for learning dynamics of time-varying coefficients generated by the POD analysis on training snapshots generated from solving full order models. HBNODE enjoys several practical advantages for learning POD-based ROMs with theoretical guarantees, including 1) HBNODE can learn long-term dependencies effectively from sequential observations and 2) HBNODE is computationally efficient in both training and testing. We compare HBNODE with other popular ROMs on several complex dynamical systems, including the von Kármán Street flow, the Kurganov-Petrova-Popov equation, and the one-dimensional Euler equations for fluids modeling.
J. Baker, H. Xia, Y. Wang, E. Cherkaev, A. Narayan, L. Chen, J. Xin, A. L. Bertozzi, S. J. Osher, B. Wang. Proximal Implicit ODE Solvers for Accelerating Learning Neural ODEs, Subtitled arXiv preprint arXiv:2204.08621, 2022.
Learning neural ODEs often requires solving very stiff ODE systems, primarily using explicit adaptive step size ODE solvers. These solvers are computationally expensive, requiring the use of tiny step sizes for numerical stability and accuracy guarantees. This paper considers learning neural ODEs using implicit ODE solvers of different orders leveraging proximal operators. The proximal implicit solver consists of inner-outer iterations: the inner iterations approximate each implicit update step using a fast optimization algorithm, and the outer iterations solve the ODE system over time. The proximal implicit ODE solver guarantees superiority over explicit solvers in numerical stability and computational efficiency. We validate the advantages of proximal implicit solvers over existing popular neural ODE solvers on various challenging benchmark tasks, including learning continuous-depth graph neural networks and continuous normalizing flows.