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  1. Book ; Online: Stochastic Optimal Control via Local Occupation Measures

    Holtorf, Flemming / Rackauckas, Christopher

    2022  

    Abstract: Viewing stochastic processes through the lens of occupation measures has proven to be a powerful angle of attack for the theoretical and computational analysis for a wide range of stochastic optimal control problems. We present a simple modification of ... ...

    Abstract Viewing stochastic processes through the lens of occupation measures has proven to be a powerful angle of attack for the theoretical and computational analysis for a wide range of stochastic optimal control problems. We present a simple modification of the traditional occupation measure framework derived from resolving the occupation measures locally on a partition of the state space and control horizon. This modification bridges the gap between discretization based approximations to the solution of the Hamilton-Jacobi-Bellmann equations and convex optimization based approaches relying on the moment-sum-of-squares hierarchy. When combined with the moment-sum-of-squares hierarchy, the notion of local occupation measures provides fine-grained control over the construction of highly structured semidefinite programming relaxations for a rich class of stochastic optimal control problems with embedded diffusion and jump processes. We show how these relaxations are constructed, analyze their favorable properties, and demonstrate with examples that they hold the potential to allow for the computation of tighter bounds orders of magnitude faster than is possible via naive combination of the moment-sum-of-squares hierarchy with the traditional occupation measure framework.

    Comment: 22 pages, 10 figures, associated implementation: https://github.com/FHoltorf/MarkovBounds.jl
    Keywords Mathematics - Optimization and Control ; Electrical Engineering and Systems Science - Systems and Control
    Subject code 510
    Publishing date 2022-11-28
    Publishing country us
    Document type Book ; Online
    Database BASE - Bielefeld Academic Search Engine (life sciences selection)

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  2. Article ; Online: Forecasting virus outbreaks with social media data via neural ordinary differential equations.

    Núñez, Matías / Barreiro, Nadia L / Barrio, Rafael A / Rackauckas, Christopher

    Scientific reports

    2023  Volume 13, Issue 1, Page(s) 10870

    Abstract: During the Covid-19 pandemic, real-time social media data could in principle be used as an early predictor of a new epidemic wave. This possibility is examined here by employing a neural ordinary differential equation (neural ODE) trained to forecast ... ...

    Abstract During the Covid-19 pandemic, real-time social media data could in principle be used as an early predictor of a new epidemic wave. This possibility is examined here by employing a neural ordinary differential equation (neural ODE) trained to forecast viral outbreaks in a specific geographic region. It learns from multivariate time series of signals derived from a novel set of large online polls regarding COVID-19 symptoms. Once trained, the neural ODE can capture the dynamics of interconnected local signals and effectively estimate the number of new infections up to two months in advance. In addition, it may predict the future consequences of changes in the number of infected at a certain period, which might be related with the flow of individuals entering or exiting a region. This study provides persuasive evidence for the predictive ability of widely disseminated social media surveys for public health applications.
    MeSH term(s) Humans ; COVID-19/epidemiology ; Pandemics ; Social Media ; SARS-CoV-2 ; Disease Outbreaks ; Forecasting
    Language English
    Publishing date 2023-07-05
    Publishing country England
    Document type Journal Article ; Research Support, Non-U.S. Gov't
    ZDB-ID 2615211-3
    ISSN 2045-2322 ; 2045-2322
    ISSN (online) 2045-2322
    ISSN 2045-2322
    DOI 10.1038/s41598-023-37118-9
    Database MEDical Literature Analysis and Retrieval System OnLINE

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  3. Thesis ; Online: Simulation and Control of Biological Stochasticity

    Rackauckas, Christopher Vincent

    2018  

    Abstract: Stochastic models of biochemical interactions elucidate essential properties of the network which are not accessible to deterministic modeling. In this thesis it is described how a network motif, the proportional-reversibility interaction with active ... ...

    Abstract Stochastic models of biochemical interactions elucidate essential properties of the network which are not accessible to deterministic modeling. In this thesis it is described how a network motif, the proportional-reversibility interaction with active intermediate states, gives rise to the ability for the variance of biochemical signals to be controlled without changing the mean, a property designated as mean-independent noise control (MINC). This noise control is demonstrated to be essential for macro-scale biological processes via spatial models of the zebrafish hindbrain boundary sharpening. Additionally, the ability to deduce noise origin from the aggregate noise properties is shown. However, these large-scale stochastic models of developmental processes required significant advances in the methodology and tooling for solving stochastic differential equations. Two improvements to stochastic integration methods, an efficient method for time stepping adaptivity on high order stochastic Runge-Kutta methods termed Rejection Sampling with Memory (RSwM) and optimal-stability stochastic Runge-Kutta methods, are combined to give over 1000 times speedups on biological models over previously used methodologies. In addition, a new software for solving differential equations in the Julia programming language is detailed. Its unique features for handling complex biological models, along with its high performance (routinely benchmarking as faster than classic C++ and Fortran integrators of similar implementations) and new methods, give rise to an accessible tool for simulation of large-scale stochastic biological models.
    Keywords Applied Mathematics|Systematic biology|Developmental biology
    Subject code 612 ; 510
    Language ENG
    Publishing date 2018-01-01 00:00:01.0
    Publisher University of California, Irvine
    Publishing country us
    Document type Thesis ; Online
    Database BASE - Bielefeld Academic Search Engine (life sciences selection)

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  4. Article ; Online: Collocation based training of neural ordinary differential equations.

    Roesch, Elisabeth / Rackauckas, Christopher / Stumpf, Michael P H

    Statistical applications in genetics and molecular biology

    2021  Volume 20, Issue 2, Page(s) 37–49

    Abstract: The predictive power of machine learning models often exceeds that of mechanistic modeling approaches. However, the interpretability of purely data-driven models, without any mechanistic basis is often complicated, and predictive power by itself can be a ...

    Abstract The predictive power of machine learning models often exceeds that of mechanistic modeling approaches. However, the interpretability of purely data-driven models, without any mechanistic basis is often complicated, and predictive power by itself can be a poor metric by which we might want to judge different methods. In this work, we focus on the relatively new modeling techniques of neural ordinary differential equations. We discuss how they relate to machine learning and mechanistic models, with the potential to narrow the gulf between these two frameworks: they constitute a class of hybrid model that integrates ideas from data-driven and dynamical systems approaches. Training neural ODEs as representations of dynamical systems data has its own specific demands, and we here propose a collocation scheme as a fast and efficient training strategy. This alleviates the need for costly ODE solvers. We illustrate the advantages that collocation approaches offer, as well as their robustness to qualitative features of a dynamical system, and the quantity and quality of observational data. We focus on systems that exemplify some of the hallmarks of complex dynamical systems encountered in systems biology, and we map out how these methods can be used in the analysis of mathematical models of cellular and physiological processes.
    MeSH term(s) Machine Learning ; Models, Theoretical ; Systems Biology/methods
    Language English
    Publishing date 2021-07-09
    Publishing country Germany
    Document type Journal Article ; Research Support, Non-U.S. Gov't ; Research Support, U.S. Gov't, Non-P.H.S.
    ISSN 1544-6115
    ISSN (online) 1544-6115
    DOI 10.1515/sagmb-2020-0025
    Database MEDical Literature Analysis and Retrieval System OnLINE

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  5. Book ; Online: Performance Bounds for Quantum Control

    Holtorf, Flemming / Schäfer, Frank / Arnold, Julian / Rackauckas, Christopher / Edelman, Alan

    2023  

    Abstract: Quantum feedback controllers often lack performance targets and optimality certificates. We combine quantum filtering theory and moment-sum-of-squares techniques to construct a hierarchy of convex optimization problems that furnish monotonically ... ...

    Abstract Quantum feedback controllers often lack performance targets and optimality certificates. We combine quantum filtering theory and moment-sum-of-squares techniques to construct a hierarchy of convex optimization problems that furnish monotonically improving, computable bounds on the best attainable performance for a large class of quantum feedback control problems. We prove convergence of the bounds under technical assumptions and demonstrate the practical utility of our approach by designing certifiably near-optimal controllers for a qubit in a cavity subjected to continuous photon counting and homodyne detection measurements.

    Comment: 7 pages, 2 figures
    Keywords Quantum Physics ; Electrical Engineering and Systems Science - Systems and Control ; Mathematics - Optimization and Control
    Publishing date 2023-04-06
    Publishing country us
    Document type Book ; Online
    Database BASE - Bielefeld Academic Search Engine (life sciences selection)

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  6. Book ; Online: Continuous Deep Equilibrium Models

    Pal, Avik / Edelman, Alan / Rackauckas, Christopher

    Training Neural ODEs faster by integrating them to Infinity

    2022  

    Abstract: Implicit models separate the definition of a layer from the description of its solution process. While implicit layers allow features such as depth to adapt to new scenarios and inputs automatically, this adaptivity makes its computational expense ... ...

    Abstract Implicit models separate the definition of a layer from the description of its solution process. While implicit layers allow features such as depth to adapt to new scenarios and inputs automatically, this adaptivity makes its computational expense challenging to predict. In this manuscript, we increase the "implicitness" of the DEQ by redefining the method in terms of an infinite time neural ODE, which paradoxically decreases the training cost over a standard neural ODE by 2-4x. Additionally, we address the question: is there a way to simultaneously achieve the robustness of implicit layers while allowing the reduced computational expense of an explicit layer? To solve this, we develop Skip and Skip Reg. DEQ, an implicit-explicit (IMEX) layer that simultaneously trains an explicit prediction followed by an implicit correction. We show that training this explicit predictor is free and even decreases the training time by 1.11-3.19x. Together, this manuscript shows how bridging the dichotomy of implicit and explicit deep learning can combine the advantages of both techniques.
    Keywords Computer Science - Machine Learning ; Computer Science - Computer Vision and Pattern Recognition ; Mathematics - Dynamical Systems
    Subject code 000
    Publishing date 2022-01-28
    Publishing country us
    Document type Book ; Online
    Database BASE - Bielefeld Academic Search Engine (life sciences selection)

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  7. Book ; Online: Parallelizing Explicit and Implicit Extrapolation Methods for Ordinary Differential Equations

    Utkarsh / Elrod, Chris / Ma, Yingbo / Rackauckas, Christopher

    2022  

    Abstract: Numerically solving ordinary differential equations (ODEs) is a naturally serial process and as a result the vast majority of ODE solver software are serial. In this manuscript we developed a set of parallelized ODE solvers using extrapolation methods ... ...

    Abstract Numerically solving ordinary differential equations (ODEs) is a naturally serial process and as a result the vast majority of ODE solver software are serial. In this manuscript we developed a set of parallelized ODE solvers using extrapolation methods which exploit "parallelism within the method" so that arbitrary user ODEs can be parallelized. We describe the specific choices made in the implementation of the explicit and implicit extrapolation methods which allow for generating low overhead static schedules to then exploit with optimized multi-threaded implementations. We demonstrate that while the multi-threading gives a noticeable acceleration on both explicit and implicit problems, the explicit parallel extrapolation methods gave no significant improvement over state-of-the-art even with a multi-threading advantage against current optimized high order Runge-Kutta tableaus. However, we demonstrate that the implicit parallel extrapolation methods are able to achieve state-of-the-art performance (2x-4x) on standard multicore x86 CPUs for systems of $<200$ stiff ODEs solved at low tolerance, a typical setup for a vast majority of users of high level language equation solver suites. The resulting method is distributed as the first widely available open source software for within-method parallel acceleration targeting typical modest compute architectures.

    Comment: 6 figures
    Keywords Mathematics - Numerical Analysis ; Computer Science - Mathematical Software
    Subject code 000
    Publishing date 2022-07-17
    Publishing country us
    Document type Book ; Online
    Database BASE - Bielefeld Academic Search Engine (life sciences selection)

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  8. Article: ADAPTIVE METHODS FOR STOCHASTIC DIFFERENTIAL EQUATIONS VIA NATURAL EMBEDDINGS AND REJECTION SAMPLING WITH MEMORY.

    Rackauckas, Christopher / Nie, Qing

    Discrete and continuous dynamical systems. Series B

    2018  Volume 22, Issue 7, Page(s) 2731–2761

    Abstract: Adaptive time-stepping with high-order embedded Runge-Kutta pairs and rejection sampling provides efficient approaches for solving differential equations. While many such methods exist for solving deterministic systems, little progress has been made for ... ...

    Abstract Adaptive time-stepping with high-order embedded Runge-Kutta pairs and rejection sampling provides efficient approaches for solving differential equations. While many such methods exist for solving deterministic systems, little progress has been made for stochastic variants. One challenge in developing adaptive methods for stochastic differential equations (SDEs) is the construction of embedded schemes with direct error estimates. We present a new class of embedded stochastic Runge-Kutta (SRK) methods with strong order 1.5 which have a natural embedding of strong order 1.0 methods. This allows for the derivation of an error estimate which requires no additional function evaluations. Next we derive a general method to reject the time steps without losing information about the future Brownian path termed Rejection Sampling with Memory (RSwM). This method utilizes a stack data structure to do rejection sampling, costing only a few floating point calculations. We show numerically that the methods generate statistically-correct and tolerance-controlled solutions. Lastly, we show that this form of adaptivity can be applied to systems of equations, and demonstrate that it solves a stiff biological model 12.28x faster than common fixed timestep algorithms. Our approach only requires the solution to a bridging problem and thus lends itself to natural generalizations beyond SDEs.
    Language English
    Publishing date 2018-01-30
    Publishing country United States
    Document type Journal Article
    ISSN 1531-3492
    ISSN 1531-3492
    DOI 10.3934/dcdsb.2017133
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  9. Article ; Online: Stiff neural ordinary differential equations.

    Kim, Suyong / Ji, Weiqi / Deng, Sili / Ma, Yingbo / Rackauckas, Christopher

    Chaos (Woodbury, N.Y.)

    2021  Volume 31, Issue 9, Page(s) 93122

    Abstract: Neural Ordinary Differential Equations (ODEs) are a promising approach to learn dynamical models from time-series data in science and engineering applications. This work aims at learning neural ODEs for stiff systems, which are usually raised from ... ...

    Abstract Neural Ordinary Differential Equations (ODEs) are a promising approach to learn dynamical models from time-series data in science and engineering applications. This work aims at learning neural ODEs for stiff systems, which are usually raised from chemical kinetic modeling in chemical and biological systems. We first show the challenges of learning neural ODEs in the classical stiff ODE systems of Robertson's problem and propose techniques to mitigate the challenges associated with scale separations in stiff systems. We then present successful demonstrations in stiff systems of Robertson's problem and an air pollution problem. The demonstrations show that the usage of deep networks with rectified activations, proper scaling of the network outputs as well as loss functions, and stabilized gradient calculations are the key techniques enabling the learning of stiff neural ODEs. The success of learning stiff neural ODEs opens up possibilities of using neural ODEs in applications with widely varying time-scales, such as chemical dynamics in energy conversion, environmental engineering, and life sciences.
    Language English
    Publishing date 2021-09-23
    Publishing country United States
    Document type Journal Article
    ZDB-ID 1472677-4
    ISSN 1089-7682 ; 1054-1500
    ISSN (online) 1089-7682
    ISSN 1054-1500
    DOI 10.1063/5.0060697
    Database MEDical Literature Analysis and Retrieval System OnLINE

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  10. Article ; Online: Author Correction: Julia for biologists.

    Roesch, Elisabeth / Greener, Joe G / MacLean, Adam L / Nassar, Huda / Rackauckas, Christopher / Holy, Timothy E / Stumpf, Michael P H

    Nature methods

    2023  Volume 20, Issue 5, Page(s) 771

    Language English
    Publishing date 2023-04-29
    Publishing country United States
    Document type Published Erratum
    ZDB-ID 2169522-2
    ISSN 1548-7105 ; 1548-7091
    ISSN (online) 1548-7105
    ISSN 1548-7091
    DOI 10.1038/s41592-023-01887-y
    Database MEDical Literature Analysis and Retrieval System OnLINE

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