Data-driven discovery of coordinates and governing equations

Abstract

The discovery of governing equations from scientific data has the potential to transform data-rich fields that lack well-characterized quantitative descriptions. Advances in sparse regression are currently enabling the tractable identification of both the structure and parameters of a nonlinear dynamical system from data. The resulting models have the fewest terms necessary to describe the dynamics, balancing model complexity with descriptive ability, and thus promoting interpretability and generalizability. This provides an algorithmic approach to Occam's razor for model discovery. However, this approach fundamentally relies on an effective coordinate system in which the dynamics have a simple representation. In this work, we design a custom autoencoder to discover a coordinate transformation into a reduced space where the dynamics may be sparsely represented. Thus, we simultaneously learn the governing equations and the associated coordinate system. We demonstrate this approach on several example high-dimensional dynamical systems with low-dimensional behavior. The resulting modeling framework combines the strengths of deep neural networks for flexible representation and sparse identification of nonlinear dynamics (SINDy) for parsimonious models. It is the first method of its kind to place the discovery of coordinates and models on an equal footing.

Aggregation kinetics of irreversible patches coupled with reversible isotropic interaction leading to chains, bundles and globules

“…To mimic some of these biological structures, colloidal particles which are asymmetric, patterned or patchy [11] have received considerable attention in recent years as their isotropic [12][13][14] counterpart was not able to explain some of the experimental observations like bundling. Particles which are anisotropic in shape or interactions are synthesized as they have very promising applications in electronics [15] , drug delivery [16,17], in fabricating photonic and plasmonic materials [18][19][20][21][22] . Several patchy particles have been developed where colloidal particles undergo DNA-mediated interactions and are known as DNA-functionalised colloids [23][24][25] .…”

Time-series learning of latent-space dynamics for reduced-order model closure

“…In recent literature, there has been a considerable interest in the utility of machine learning techniques for effective ROM construction. Data-driven techniques have been used in various phases of ROM techniques such as in reduced basis construction [18], augmented dynamics evolution [19,20,14,5,21,15,22,6,23,24] and system identification [25,26,27,28].…”

Discovering Unmodeled Components in Astrodynamics with Symbolic Regression

“…symbolically representing the position and the velocity of the spacecraft in some reference system, together with a symbolic representation of a set of constant parameters: the introduction of the latter is justified in Section III-B. Of course, the choice of the reference system influences the complexity of the formulation of the equations of motion: it is however possible, as shown in [17], to develop methods discovering suitable coordinates and models at the same time. The primitives considered are:…”