Lift & Learn: Physics-informed machine learning for large-scale nonlinear dynamical systems

Abstract

We present Lift & Learn, a physics-informed method for learning low-dimensional models for large-scale dynamical systems. The method exploits knowledge of a system's governing equations to identify a coordinate transformation in which the system dynamics have quadratic structure. This transformation is called a lifting map because it often adds auxiliary variables to the system state. The lifting map is applied to data obtained by evaluating a model for the original nonlinear system. This lifted data is projected onto its leading principal components, and low-dimensional linear and quadratic matrix operators are fit to the lifted reduced data using a least-squares operator inference procedure. Analysis of our method shows that the Lift & Learn models are able to capture the system physics in the lifted coordinates at least as accurately as traditional intrusive model reduction approaches. This preservation of system physics makes the Lift & Learn models robust to changes in inputs. Numerical experiments on the FitzHugh-Nagumo neuron activation model and the compressible Euler equations demonstrate the generalizability of our model.

Data-driven reduced-order models via regularised Operator Inference for a single-injector combustion process

“…Here T :Rñ R n is the map representing a reversible transformation (e.g. from density to specific volume) or a lifting transformation (Qian et al 2020). (3) Compute the proper orthogonal decomposition (POD) basis V of the transformed snapshots.…”

A global eigenvalue reassignment method for the stabilization of nonlinear reduced‐order models

“…The specific volume form of the Euler equations is a simple example of a lifted nonlinear model. A systematic framework has been developed in the literature for efficient model reduction of general nonlinear systems via lifting transformations 18,70 . This is particularly important as directly handling the nonlinear ROMs in nonpolynomial form multiplies the cost of stabilization.…”

“…The physically and mathematically rigorous nature of projection‐based ROMs becomes critical in sensitive applications, while the nonintrusive framework of the black‐box methods prevails in the absence of comprehensive knowledge about the equations driving the dynamical system. The achievements of physics‐informed machine learning models 17‐21 further magnify the value of projection‐based ROMs that contain strong physical and mathematical connections with the full‐order model (FOM), in contrast to purely data‐driven models with little knowledge about the conservation laws that govern the fluid dynamics.…”

Physically sound, self-learning digital twins for sloshing fluids

“…In order to overcome these limitations, the machine learning community has recently taken a step forward physically-informed learning [9], whose application field ranges from neural networks [10] [11] [3] [12] to linear regression learning [13] [14]. This new perspective of machine learning aims to guide the learning process with criteria that fulfills the laws of physics.…”