Learning nonparametric ordinary differential equations from noisy data

Learning nonparametric systems of Ordinary Differential Equations (ODEs) x˙=f(t,x) from noisy data is an emerging machine learning topic. We use the well-developed theory of Reproducing Kernel Hilbert Spaces (RKHS) to define candidates for f for which the solution of the ODE exists and is unique. Le...

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Bibliographic Details
Published inJournal of computational physics Vol. 507; p. 112971
Main Authors Lahouel, Kamel, Wells, Michael, Rielly, Victor, Lew, Ethan, Lovitz, David, Jedynak, Bruno M.
Format Journal Article
LanguageEnglish
Published United States Elsevier Inc 15.06.2024
Subjects
Amyloid accumulation
Kernel methods
Nonlinear dynamical systems
Penalty method
System identification
Amyloid accumulation
Kernel methods
Nonlinear dynamical systems
System identification
Penalty method
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Summary:Learning nonparametric systems of Ordinary Differential Equations (ODEs) x˙=f(t,x) from noisy data is an emerging machine learning topic. We use the well-developed theory of Reproducing Kernel Hilbert Spaces (RKHS) to define candidates for f for which the solution of the ODE exists and is unique. Learning f consists of solving a constrained optimization problem in an RKHS. We propose a penalty method that iteratively uses the Representer theorem and Euler approximations to provide a numerical solution. We prove a generalization bound for the L2 distance between x and its estimator. Experiments are provided for the FitzHugh–Nagumo oscillator, the Lorenz system, and for predicting the Amyloid level in the cortex of aging subjects. In all cases, we show competitive results compared with the state-of-the-art.
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JCOMP author statement TermDefinitionParticipants (in alphabetic order)ConceptualizationIdeas; formulation or evolution of overarching research goals and aimsJedynak, LahouelMethodologyDevelopment or design of methodology; creation of modelsJedynak, Lahouel, Lew, Lovitz Rielly, WellsSoftwareProgramming, software development; designing computer programs; implementation of the computer code and supporting algorithms; testing of existing code componentsJedynak, Lahouel, Lew, Lovitz, Rielly, WellsValidationVerification, whether as a part of the activity or separate, of the overall replication/reproducibility of results/experiments and other research outputsJedynak, Lahouel, Lew, Lovitz Rielly, WellsFormal analysisApplication of statistical, mathematical, computational, or other formal techniques to analyze or synthesize study dataJedynak, Lahouel, Lew, Lovitz Rielly, WellsInvestigationConducting a research and investigation process, specifically performing the experiments, or data/evidence collectionJedynak, Lahouel, Lew, Lovitz Rielly, WellsResourcesProvision of study materials, reagents, materials, patients, laboratory samples, animals, instrumentation, computing resources, or other analysis toolsJedynakData CurationManagement activities to annotate (produce metadata), scrub data and maintain research data (including software code, where it is necessary for interpreting the data itself) for initial use and later reuseJedynak, Lew, Lovitz, Rielly, WellsWriting - Original DraftPreparation, creation and/or presentation of the published work, specifically writing the initial draft (including substantive translation)Jedynak, Lahouel, Lew, Lovitz Rielly, WellsWriting - Review & EditingPreparation, creation and/or presentation of the published work by those from the original research group, specifically critical review, commentary or revision - including pre-or postpublication stagesJedynak, Lahouel, Lew, Lovitz Rielly, WellsVisualizationPreparation, creation and/or presentation of the published work, specifically visualization/data presentationJedynak, Lahouel, Lew, Lovitz Rielly, WellsSupervisionOversight and leadership responsibility for the research activity planning and execution, including mentorship external to the core teamJedynak, LahouelProject administrationManagement and coordination responsibility for the research activity planning and executionJedynak, LahouelFunding acquisitionAcquisition of the financial support for the project leading to this publicationJedynak, Lahouel
ISSN:0021-9991
1090-2716
DOI:10.1016/j.jcp.2024.112971