Reduced operator inference for nonlinear PDEs

Elizabeth Qian (Department of Computing and Mathematical Sciences (Caltech))

The development of computationally efficient reduced models for nonlinear PDEs is an enabling technology for many computational algorithms used in engineering settings, including for optimization and uncertainty quantification. In this talk, I present a recent formulation for scientific machine learning that learns from data the operators of such a reduced model nonlinear PDEs. Our formulation generalizes to the PDE setting an Operator Inference method previously developed for systems of ODEs. The method brings together two main elements. First, ideas from projection-based model reduction are used to explicitly parametrize the learned model by low-dimensional polynomial operators which reflect the known form of the governing PDE. Second, supervised machine learning tools are used to infer from data the reduced operators of this physics-informed parametrization. For systems whose governing PDEs contain more general (non-polynomial) nonlinearities, the learned model performance can be improved through the use of lifting variable transformations, which expose polynomial structure in the PDE. The proposed method is demonstrated on a three-dimensional combustion simulation with over 18 million degrees of freedom, for which the learned reduced models achieve accurate predictions with a dimension reduction of six orders of magnitude and model runtime reduction of 5-6 orders of magnitude.