Invertible Koopman neural operator for data-driven modeling of partial differential equations

INN is introduced to eliminate dependency on reconstruction loss. Koopman operator is parameterized in frequency space to ensure resolution-invariance. By preprocessing, such as interpolation, IKNO is available for non-Cartesian domains. In various numerical and real-world examples, IKNO performs over FNO and KNO. R. ChinaA R T I C L E I N F OKeywords: Deep learning Invertible neural network Koopman operator Data-driven modeling Neural operator Partial differential equations A B S T R A C T Koopman operator theory is a popular candidate for data-driven modeling because it provides a global linearization representation for nonlinear dynamical systems. However, existing Koopman operator-based methods suffer from shortcomings in constructing the well-behaved observable function and its inverse and are inefficient enough when dealing with partial differential equations (PDEs). To address these issues, this paper proposes the Invertible Koopman Neural Operator (IKNO), a novel data-driven modeling approach inspired by the Koopman operator theory and neural operator. IKNO leverages an Invertible Neural Network to parameterize observable function and its inverse simultaneously under the same learnable parameters, explicitly guaranteeing the reconstruction relation, thus eliminating the dependency on the reconstruction loss, which is an essential improvement over the original Koopman Neural Operator (KNO). The structured linear matrix inspired by the Koopman operator theory is parameterized to learn the evolution of observables' low-frequency modes in the frequency space rather than directly in the observable space, sustaining IKNO is resolution-invariant like other neural operators. Moreover, with preprocessing such as interpolation and dimension expansion, IKNO can be extended to operator learning tasks defined on non-Cartesian domains. We fully support the above claims based on rich numerical and real-world examples and demonstrate the effectiveness of IKNO and superiority over other neural operators.1. Introduction Complex nonlinear dynamical systems are ubiquitous in many engineering fields, such as aerospace and vibration control, and modeling these systems is an important research topic [1-3]. Traditional knowledge-driven modeling approaches usually use a priori expertise to build a set of differential or algebraic equations to describe or explain phenomena of interest, having achieved relative maturity. However, in many scenarios, some key parameters, even expressions of systems of concern, may be difficult to measure or give accurately, making establishing a physical model that can accurately characterize systems' evolution challenging. In recent years, as big data technology and computer performance have improved, data-driven modeling approaches have gained extensive attention from researchers, providing a feasible route to solve the aforementioned problems [4-6].