TY - JOUR
T1 - Data-driven learning of differential equations
T2 - combining data and model uncertainty
AU - Glasner, Karl
N1 - Funding Information:
The author was supported through NSF award DMS-1908968.
Publisher Copyright:
© 2023, The Author(s) under exclusive licence to Sociedade Brasileira de Matemática Aplicada e Computacional.
PY - 2023/2
Y1 - 2023/2
N2 - Data-driven discovery of differential equations relies on estimating model parameters using information about a solution that is often incomplete and corrupted by noise. Moreover, the sizes of the uncertainties in the model and data are usually unknown as well. This paper develops a likelihood-type cost function which incorporates both sources of uncertainty and provides a theoretically justified way of optimizing the balance between them. This approach accommodates missing information about model solutions, allows for considerable noise in the data, and is demonstrated to provide estimates which are often superior to regression methods currently used for model discovery and calibration. Practical implementation and optimization strategies are discussed both for systems of ordinary differential and partial differential equations. Numerical experiments using synthetic data are performed for a variety of test problems, including those exhibiting chaotic or complex spatiotemporal behavior.
AB - Data-driven discovery of differential equations relies on estimating model parameters using information about a solution that is often incomplete and corrupted by noise. Moreover, the sizes of the uncertainties in the model and data are usually unknown as well. This paper develops a likelihood-type cost function which incorporates both sources of uncertainty and provides a theoretically justified way of optimizing the balance between them. This approach accommodates missing information about model solutions, allows for considerable noise in the data, and is demonstrated to provide estimates which are often superior to regression methods currently used for model discovery and calibration. Practical implementation and optimization strategies are discussed both for systems of ordinary differential and partial differential equations. Numerical experiments using synthetic data are performed for a variety of test problems, including those exhibiting chaotic or complex spatiotemporal behavior.
KW - Learning differential equations
KW - Model error
KW - Parameter estimation
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U2 - 10.1007/s40314-022-02180-y
DO - 10.1007/s40314-022-02180-y
M3 - Article
AN - SCOPUS:85145775790
SN - 0101-8205
VL - 42
JO - Computational and Applied Mathematics
JF - Computational and Applied Mathematics
IS - 1
M1 - 36
ER -