Parameterized neural ordinary differential equations: applications to computational physics problems

Abstract

This work proposes an extension of neural ordinary differential equations (NODEs) by introducing an additional set of ODE input parameters to NODEs. This extension allows NODEs to learn multiple dynamics specified by the input parameter instances. Our extension is inspired by the concept of parameterized ODEs, which are widely investigated in computational science and engineering contexts, where characteristics of the governing equations vary over the input parameters. We apply the proposed parameterized NODEs (PNODEs) for learning latent dynamics of complex dynamical processes that arise in computational physics, which is an essential component for enabling rapid numerical simulations for time-critical physics applications. For this, we propose an encoder–decoder-type framework, which models latent dynamics as PNODEs. We demonstrate the effectiveness of PNODEs on benchmark problems from computational physics.

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“…It is a particularly promising approach for learning latent dynamics of dynamical systems. NODE naturally fits well as a latent-dynamics model in reduced-order modeling of physical processes because it learns the latent dynamics in the form of ODEs [40]. Furthermore, NODE is flexible in learning from irregularly sampled time-series data [40][41][42][43].…”

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“…Neural ODEs offer a promising approach for hybrid modeling and system identification [12][13][14][15]. Furthermore, the neural network's architecture can be optimized to represent experimental data better.…”

State Key Laboratory for Geomechanics and Deep Underground Engineering, School of Mechanics & Civil Engineering, China University of Mining & Technology, Xuzhou 221116, China

“…How do you model the operator Ψ in Eq (7)? Many neural dynamical models for complex networks (e.g., [8]) follow a common encoder-decoder architecture in graph neural networks [2] and also in general neural ODEs [15]. In NDCN [8] (see Fig 1 .(a)), the authors defined an encoder f E : R → R d and a decoder f D : R d → R, where d ∈ Z + is the embedding dimension, so that each initial value x i (0) is encoded jointly as d state variables in the vector h i (0) = f E (x(0)).…”