Nature Communications· 2018Q1
Deep learning for universal linear embeddings of nonlinear dynamics
- 1,378citations
- Q1SCImago
- 2018year
Short summary
A deep learning framework discovers interpretable, low-dimensional coordinates that globally linearize complex nonlinear dynamics, generalizing the Koopman operator to systems with continuous spectra.
AI-generated from the title and abstract; the full text is not read.
Key points
- Deep learning discovers nonlinear coordinates that globally linearize complex dynamics.
- The framework uses a modified auto-encoder for interpretable, low-dimensional embeddings.
- It generalizes Koopman operator representations to systems with continuous spectra.
- Auxiliary networks parametrize continuous frequencies, connecting to asymptotic theory.
AI-generated from the title and abstract; the full text is not read.
Abstract
Identifying coordinate transformations that make strongly nonlinear dynamics approximately linear has the potential to enable nonlinear prediction, estimation, and control using linear theory. The Koopman operator is a leading data-driven embedding, and its eigenfunctions provide intrinsic coordinates that globally linearize the dynamics. However, identifying and representing these eigenfunctions has proven challenging. This work leverages deep learning to discover representations of Koopman eigenfunctions from data. Our network is parsimonious and interpretable by construction, embedding the dynamics on a low-dimensional manifold. We identify nonlinear coordinates on which the dynamics are globally linear using a modified auto-encoder. We also generalize Koopman representations to include a ubiquitous class of systems with continuous spectra. Our framework parametrizes the continuous frequency using an auxiliary network, enabling a compact and efficient embedding, while connecting our models to decades of asymptotics. Thus, we benefit from the power of deep learning, while retaining the physical interpretability of Koopman embeddings.
The authors' abstract, as published at the source. Nature Communications, 2018 · DOI ↗
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Field: Statistical and Nonlinear Physics
Statistical and Nonlinear PhysicsPhysics and Astronomy